LMFDB - Embedded newform 850.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(1,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.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:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +2.41421 q^{6} +2.41421 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +2.41421 q^{6} +2.41421 q^{7} +1.00000 q^{8} +2.82843 q^{9} +2.00000 q^{11} +2.41421 q^{12} -4.65685 q^{13} +2.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -4.82843 q^{19} +5.82843 q^{21} +2.00000 q^{22} -3.65685 q^{23} +2.41421 q^{24} -4.65685 q^{26} -0.414214 q^{27} +2.41421 q^{28} -4.00000 q^{29} +4.41421 q^{31} +1.00000 q^{32} +4.82843 q^{33} +1.00000 q^{34} +2.82843 q^{36} -5.65685 q^{37} -4.82843 q^{38} -11.2426 q^{39} -3.65685 q^{41} +5.82843 q^{42} +11.3137 q^{43} +2.00000 q^{44} -3.65685 q^{46} +8.82843 q^{47} +2.41421 q^{48} -1.17157 q^{49} +2.41421 q^{51} -4.65685 q^{52} +0.171573 q^{53} -0.414214 q^{54} +2.41421 q^{56} -11.6569 q^{57} -4.00000 q^{58} -8.00000 q^{59} +7.65685 q^{61} +4.41421 q^{62} +6.82843 q^{63} +1.00000 q^{64} +4.82843 q^{66} +15.3137 q^{67} +1.00000 q^{68} -8.82843 q^{69} -4.75736 q^{71} +2.82843 q^{72} -9.65685 q^{73} -5.65685 q^{74} -4.82843 q^{76} +4.82843 q^{77} -11.2426 q^{78} -7.24264 q^{79} -9.48528 q^{81} -3.65685 q^{82} +13.6569 q^{83} +5.82843 q^{84} +11.3137 q^{86} -9.65685 q^{87} +2.00000 q^{88} -2.00000 q^{89} -11.2426 q^{91} -3.65685 q^{92} +10.6569 q^{93} +8.82843 q^{94} +2.41421 q^{96} -1.17157 q^{98} +5.65685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 6 q^{21} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{26} + 2 q^{27} + 2 q^{28} - 8 q^{29} + 6 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} - 4 q^{38} - 14 q^{39} + 4 q^{41} + 6 q^{42} + 4 q^{44} + 4 q^{46} + 12 q^{47} + 2 q^{48} - 8 q^{49} + 2 q^{51} + 2 q^{52} + 6 q^{53} + 2 q^{54} + 2 q^{56} - 12 q^{57} - 8 q^{58} - 16 q^{59} + 4 q^{61} + 6 q^{62} + 8 q^{63} + 2 q^{64} + 4 q^{66} + 8 q^{67} + 2 q^{68} - 12 q^{69} - 18 q^{71} - 8 q^{73} - 4 q^{76} + 4 q^{77} - 14 q^{78} - 6 q^{79} - 2 q^{81} + 4 q^{82} + 16 q^{83} + 6 q^{84} - 8 q^{87} + 4 q^{88} - 4 q^{89} - 14 q^{91} + 4 q^{92} + 10 q^{93} + 12 q^{94} + 2 q^{96} - 8 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 6 * q^21 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^26 + 2 * q^27 + 2 * q^28 - 8 * q^29 + 6 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^34 - 4 * q^38 - 14 * q^39 + 4 * q^41 + 6 * q^42 + 4 * q^44 + 4 * q^46 + 12 * q^47 + 2 * q^48 - 8 * q^49 + 2 * q^51 + 2 * q^52 + 6 * q^53 + 2 * q^54 + 2 * q^56 - 12 * q^57 - 8 * q^58 - 16 * q^59 + 4 * q^61 + 6 * q^62 + 8 * q^63 + 2 * q^64 + 4 * q^66 + 8 * q^67 + 2 * q^68 - 12 * q^69 - 18 * q^71 - 8 * q^73 - 4 * q^76 + 4 * q^77 - 14 * q^78 - 6 * q^79 - 2 * q^81 + 4 * q^82 + 16 * q^83 + 6 * q^84 - 8 * q^87 + 4 * q^88 - 4 * q^89 - 14 * q^91 + 4 * q^92 + 10 * q^93 + 12 * q^94 + 2 * q^96 - 8 * 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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.41421	0.985599
$7$	2.41421	0.912487	0.456243	−	0.889855i	$-0.349195\pi$
$7$	2.41421	0.912487	0.456243	+	0.889855i	$0.349195\pi$
$8$	1.00000	0.353553
$9$	2.82843	0.942809
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.41421	0.696923
$13$	−4.65685	−1.29158	−0.645789	−	0.763516i	$-0.723472\pi$
$13$	−4.65685	−1.29158	−0.645789	+	0.763516i	$0.723472\pi$
$14$	2.41421	0.645226
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.82843	0.666667
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	0	0
$21$	5.82843	1.27187
$22$	2.00000	0.426401
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	2.41421	0.492799
$25$	0	0
$26$	−4.65685	−0.913284
$27$	−0.414214	−0.0797154
$28$	2.41421	0.456243
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	4.41421	0.792816	0.396408	−	0.918074i	$-0.370257\pi$
$31$	4.41421	0.792816	0.396408	+	0.918074i	$0.370257\pi$
$32$	1.00000	0.176777
$33$	4.82843	0.840521
$34$	1.00000	0.171499
$35$	0	0
$36$	2.82843	0.471405
$37$	−5.65685	−0.929981	−0.464991	−	0.885316i	$-0.653942\pi$
$37$	−5.65685	−0.929981	−0.464991	+	0.885316i	$0.653942\pi$
$38$	−4.82843	−0.783274
$39$	−11.2426	−1.80026
$40$	0	0
$41$	−3.65685	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$41$	−3.65685	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$42$	5.82843	0.899346
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−3.65685	−0.539174
$47$	8.82843	1.28776	0.643879	−	0.765127i	$-0.277324\pi$
$47$	8.82843	1.28776	0.643879	+	0.765127i	$0.277324\pi$
$48$	2.41421	0.348462
$49$	−1.17157	−0.167368
$50$	0	0
$51$	2.41421	0.338058
$52$	−4.65685	−0.645789
$53$	0.171573	0.0235673	0.0117837	−	0.999931i	$-0.496249\pi$
$53$	0.171573	0.0235673	0.0117837	+	0.999931i	$0.496249\pi$
$54$	−0.414214	−0.0563673
$55$	0	0
$56$	2.41421	0.322613
$57$	−11.6569	−1.54399
$58$	−4.00000	−0.525226
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	4.41421	0.560606
$63$	6.82843	0.860301
$64$	1.00000	0.125000
$65$	0	0
$66$	4.82843	0.594338
$67$	15.3137	1.87087	0.935434	−	0.353502i	$-0.115009\pi$
$67$	15.3137	1.87087	0.935434	+	0.353502i	$0.115009\pi$
$68$	1.00000	0.121268
$69$	−8.82843	−1.06282
$70$	0	0
$71$	−4.75736	−0.564595	−0.282297	−	0.959327i	$-0.591096\pi$
$71$	−4.75736	−0.564595	−0.282297	+	0.959327i	$0.591096\pi$
$72$	2.82843	0.333333
$73$	−9.65685	−1.13025	−0.565125	−	0.825006i	$-0.691172\pi$
$73$	−9.65685	−1.13025	−0.565125	+	0.825006i	$0.691172\pi$
$74$	−5.65685	−0.657596
$75$	0	0
$76$	−4.82843	−0.553859
$77$	4.82843	0.550250
$78$	−11.2426	−1.27298
$79$	−7.24264	−0.814861	−0.407430	−	0.913236i	$-0.633575\pi$
$79$	−7.24264	−0.814861	−0.407430	+	0.913236i	$0.633575\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	−3.65685	−0.403832
$83$	13.6569	1.49903	0.749517	−	0.661985i	$-0.230286\pi$
$83$	13.6569	1.49903	0.749517	+	0.661985i	$0.230286\pi$
$84$	5.82843	0.635934
$85$	0	0
$86$	11.3137	1.21999
$87$	−9.65685	−1.03532
$88$	2.00000	0.213201
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−11.2426	−1.17855
$92$	−3.65685	−0.381253
$93$	10.6569	1.10506
$94$	8.82843	0.910583
$95$	0	0
$96$	2.41421	0.246400
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−1.17157	−0.118347
$99$	5.65685	0.568535
$100$	0	0
$101$	−12.6569	−1.25940	−0.629702	−	0.776837i	$-0.716823\pi$
$101$	−12.6569	−1.25940	−0.629702	+	0.776837i	$0.716823\pi$
$102$	2.41421	0.239043
$103$	−18.4853	−1.82141	−0.910704	−	0.413059i	$-0.864460\pi$
$103$	−18.4853	−1.82141	−0.910704	+	0.413059i	$0.864460\pi$
$104$	−4.65685	−0.456642
$105$	0	0
$106$	0.171573	0.0166646
$107$	1.92893	0.186477	0.0932385	−	0.995644i	$-0.470278\pi$
$107$	1.92893	0.186477	0.0932385	+	0.995644i	$0.470278\pi$
$108$	−0.414214	−0.0398577
$109$	−7.31371	−0.700526	−0.350263	−	0.936651i	$-0.613908\pi$
$109$	−7.31371	−0.700526	−0.350263	+	0.936651i	$0.613908\pi$
$110$	0	0
$111$	−13.6569	−1.29625
$112$	2.41421	0.228122
$113$	17.6569	1.66102	0.830509	−	0.557006i	$-0.188050\pi$
$113$	17.6569	1.66102	0.830509	+	0.557006i	$0.188050\pi$
$114$	−11.6569	−1.09176
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−13.1716	−1.21771
$118$	−8.00000	−0.736460
$119$	2.41421	0.221311
$120$	0	0
$121$	−7.00000	−0.636364
$122$	7.65685	0.693219
$123$	−8.82843	−0.796032
$124$	4.41421	0.396408
$125$	0	0
$126$	6.82843	0.608325
$127$	9.65685	0.856907	0.428454	−	0.903564i	$-0.359059\pi$
$127$	9.65685	0.856907	0.428454	+	0.903564i	$0.359059\pi$
$128$	1.00000	0.0883883
$129$	27.3137	2.40484
$130$	0	0
$131$	6.89949	0.602812	0.301406	−	0.953496i	$-0.402544\pi$
$131$	6.89949	0.602812	0.301406	+	0.953496i	$0.402544\pi$
$132$	4.82843	0.420261
$133$	−11.6569	−1.01078
$134$	15.3137	1.32290
$135$	0	0
$136$	1.00000	0.0857493
$137$	3.48528	0.297768	0.148884	−	0.988855i	$-0.452432\pi$
$137$	3.48528	0.297768	0.148884	+	0.988855i	$0.452432\pi$
$138$	−8.82843	−0.751526
$139$	15.3848	1.30492	0.652460	−	0.757823i	$-0.273737\pi$
$139$	15.3848	1.30492	0.652460	+	0.757823i	$0.273737\pi$
$140$	0	0
$141$	21.3137	1.79494
$142$	−4.75736	−0.399229
$143$	−9.31371	−0.778851
$144$	2.82843	0.235702
$145$	0	0
$146$	−9.65685	−0.799207
$147$	−2.82843	−0.233285
$148$	−5.65685	−0.464991
$149$	19.1421	1.56818	0.784092	−	0.620644i	$-0.213129\pi$
$149$	19.1421	1.56818	0.784092	+	0.620644i	$0.213129\pi$
$150$	0	0
$151$	−14.4853	−1.17880	−0.589398	−	0.807843i	$-0.700635\pi$
$151$	−14.4853	−1.17880	−0.589398	+	0.807843i	$0.700635\pi$
$152$	−4.82843	−0.391637
$153$	2.82843	0.228665
$154$	4.82843	0.389086
$155$	0	0
$156$	−11.2426	−0.900132
$157$	−21.8284	−1.74210	−0.871049	−	0.491196i	$-0.836560\pi$
$157$	−21.8284	−1.74210	−0.871049	+	0.491196i	$0.836560\pi$
$158$	−7.24264	−0.576194
$159$	0.414214	0.0328493
$160$	0	0
$161$	−8.82843	−0.695778
$162$	−9.48528	−0.745234
$163$	−22.4142	−1.75562	−0.877808	−	0.479012i	$-0.840995\pi$
$163$	−22.4142	−1.75562	−0.877808	+	0.479012i	$0.840995\pi$
$164$	−3.65685	−0.285552
$165$	0	0
$166$	13.6569	1.05998
$167$	3.65685	0.282976	0.141488	−	0.989940i	$-0.454811\pi$
$167$	3.65685	0.282976	0.141488	+	0.989940i	$0.454811\pi$
$168$	5.82843	0.449673
$169$	8.68629	0.668176
$170$	0	0
$171$	−13.6569	−1.04437
$172$	11.3137	0.862662
$173$	22.9706	1.74642	0.873210	−	0.487345i	$-0.162034\pi$
$173$	22.9706	1.74642	0.873210	+	0.487345i	$0.162034\pi$
$174$	−9.65685	−0.732084
$175$	0	0
$176$	2.00000	0.150756
$177$	−19.3137	−1.45171
$178$	−2.00000	−0.149906
$179$	19.1716	1.43295	0.716475	−	0.697612i	$-0.245754\pi$
$179$	19.1716	1.43295	0.716475	+	0.697612i	$0.245754\pi$
$180$	0	0
$181$	8.34315	0.620141	0.310071	−	0.950714i	$-0.399647\pi$
$181$	8.34315	0.620141	0.310071	+	0.950714i	$0.399647\pi$
$182$	−11.2426	−0.833360
$183$	18.4853	1.36647
$184$	−3.65685	−0.269587
$185$	0	0
$186$	10.6569	0.781398
$187$	2.00000	0.146254
$188$	8.82843	0.643879
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	2.48528	0.179829	0.0899143	−	0.995950i	$-0.471341\pi$
$191$	2.48528	0.179829	0.0899143	+	0.995950i	$0.471341\pi$
$192$	2.41421	0.174231
$193$	1.65685	0.119263	0.0596315	−	0.998220i	$-0.481007\pi$
$193$	1.65685	0.119263	0.0596315	+	0.998220i	$0.481007\pi$
$194$	0	0
$195$	0	0
$196$	−1.17157	−0.0836838
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	5.65685	0.402015
$199$	−15.6569	−1.10988	−0.554942	−	0.831889i	$-0.687260\pi$
$199$	−15.6569	−1.10988	−0.554942	+	0.831889i	$0.687260\pi$
$200$	0	0
$201$	36.9706	2.60770
$202$	−12.6569	−0.890533
$203$	−9.65685	−0.677778
$204$	2.41421	0.169029
$205$	0	0
$206$	−18.4853	−1.28793
$207$	−10.3431	−0.718898
$208$	−4.65685	−0.322895
$209$	−9.65685	−0.667979
$210$	0	0
$211$	15.7279	1.08275	0.541377	−	0.840780i	$-0.317903\pi$
$211$	15.7279	1.08275	0.541377	+	0.840780i	$0.317903\pi$
$212$	0.171573	0.0117837
$213$	−11.4853	−0.786959
$214$	1.92893	0.131859
$215$	0	0
$216$	−0.414214	−0.0281837
$217$	10.6569	0.723434
$218$	−7.31371	−0.495347
$219$	−23.3137	−1.57539
$220$	0	0
$221$	−4.65685	−0.313254
$222$	−13.6569	−0.916588
$223$	−1.51472	−0.101433	−0.0507165	−	0.998713i	$-0.516151\pi$
$223$	−1.51472	−0.101433	−0.0507165	+	0.998713i	$0.516151\pi$
$224$	2.41421	0.161306
$225$	0	0
$226$	17.6569	1.17452
$227$	−4.75736	−0.315757	−0.157879	−	0.987459i	$-0.550465\pi$
$227$	−4.75736	−0.315757	−0.157879	+	0.987459i	$0.550465\pi$
$228$	−11.6569	−0.771994
$229$	10.3137	0.681549	0.340775	−	0.940145i	$-0.389311\pi$
$229$	10.3137	0.681549	0.340775	+	0.940145i	$0.389311\pi$
$230$	0	0
$231$	11.6569	0.766965
$232$	−4.00000	−0.262613
$233$	20.0000	1.31024	0.655122	−	0.755523i	$-0.272617\pi$
$233$	20.0000	1.31024	0.655122	+	0.755523i	$0.272617\pi$
$234$	−13.1716	−0.861053
$235$	0	0
$236$	−8.00000	−0.520756
$237$	−17.4853	−1.13579
$238$	2.41421	0.156490
$239$	−24.9706	−1.61521	−0.807606	−	0.589723i	$-0.799237\pi$
$239$	−24.9706	−1.61521	−0.807606	+	0.589723i	$0.799237\pi$
$240$	0	0
$241$	−24.6274	−1.58639	−0.793196	−	0.608967i	$-0.791584\pi$
$241$	−24.6274	−1.58639	−0.793196	+	0.608967i	$0.791584\pi$
$242$	−7.00000	−0.449977
$243$	−21.6569	−1.38929
$244$	7.65685	0.490180
$245$	0	0
$246$	−8.82843	−0.562880
$247$	22.4853	1.43070
$248$	4.41421	0.280303
$249$	32.9706	2.08942
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	6.82843	0.430150
$253$	−7.31371	−0.459809
$254$	9.65685	0.605925
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.4853	−1.71448	−0.857242	−	0.514913i	$-0.827824\pi$
$257$	−27.4853	−1.71448	−0.857242	+	0.514913i	$0.827824\pi$
$258$	27.3137	1.70048
$259$	−13.6569	−0.848596
$260$	0	0
$261$	−11.3137	−0.700301
$262$	6.89949	0.426252
$263$	21.6569	1.33542	0.667709	−	0.744422i	$-0.267275\pi$
$263$	21.6569	1.33542	0.667709	+	0.744422i	$0.267275\pi$
$264$	4.82843	0.297169
$265$	0	0
$266$	−11.6569	−0.714728
$267$	−4.82843	−0.295495
$268$	15.3137	0.935434
$269$	−16.6274	−1.01379	−0.506896	−	0.862007i	$-0.669207\pi$
$269$	−16.6274	−1.01379	−0.506896	+	0.862007i	$0.669207\pi$
$270$	0	0
$271$	−5.51472	−0.334995	−0.167498	−	0.985872i	$-0.553569\pi$
$271$	−5.51472	−0.334995	−0.167498	+	0.985872i	$0.553569\pi$
$272$	1.00000	0.0606339
$273$	−27.1421	−1.64272
$274$	3.48528	0.210554
$275$	0	0
$276$	−8.82843	−0.531409
$277$	12.3431	0.741628	0.370814	−	0.928707i	$-0.379079\pi$
$277$	12.3431	0.741628	0.370814	+	0.928707i	$0.379079\pi$
$278$	15.3848	0.922718
$279$	12.4853	0.747474
$280$	0	0
$281$	−4.65685	−0.277805	−0.138902	−	0.990306i	$-0.544357\pi$
$281$	−4.65685	−0.277805	−0.138902	+	0.990306i	$0.544357\pi$
$282$	21.3137	1.26921
$283$	14.9706	0.889908	0.444954	−	0.895554i	$-0.353220\pi$
$283$	14.9706	0.889908	0.444954	+	0.895554i	$0.353220\pi$
$284$	−4.75736	−0.282297
$285$	0	0
$286$	−9.31371	−0.550731
$287$	−8.82843	−0.521126
$288$	2.82843	0.166667
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	−9.65685	−0.565125
$293$	2.68629	0.156935	0.0784674	−	0.996917i	$-0.474997\pi$
$293$	2.68629	0.156935	0.0784674	+	0.996917i	$0.474997\pi$
$294$	−2.82843	−0.164957
$295$	0	0
$296$	−5.65685	−0.328798
$297$	−0.828427	−0.0480702
$298$	19.1421	1.10887
$299$	17.0294	0.984838
$300$	0	0
$301$	27.3137	1.57434
$302$	−14.4853	−0.833534
$303$	−30.5563	−1.75542
$304$	−4.82843	−0.276929
$305$	0	0
$306$	2.82843	0.161690
$307$	−1.65685	−0.0945617	−0.0472808	−	0.998882i	$-0.515056\pi$
$307$	−1.65685	−0.0945617	−0.0472808	+	0.998882i	$0.515056\pi$
$308$	4.82843	0.275125
$309$	−44.6274	−2.53877
$310$	0	0
$311$	−22.2132	−1.25960	−0.629798	−	0.776759i	$-0.716862\pi$
$311$	−22.2132	−1.25960	−0.629798	+	0.776759i	$0.716862\pi$
$312$	−11.2426	−0.636489
$313$	10.9706	0.620093	0.310046	−	0.950721i	$-0.399655\pi$
$313$	10.9706	0.620093	0.310046	+	0.950721i	$0.399655\pi$
$314$	−21.8284	−1.23185
$315$	0	0
$316$	−7.24264	−0.407430
$317$	25.3137	1.42176	0.710880	−	0.703314i	$-0.248297\pi$
$317$	25.3137	1.42176	0.710880	+	0.703314i	$0.248297\pi$
$318$	0.414214	0.0232279
$319$	−8.00000	−0.447914
$320$	0	0
$321$	4.65685	0.259920
$322$	−8.82843	−0.491989
$323$	−4.82843	−0.268661
$324$	−9.48528	−0.526960
$325$	0	0
$326$	−22.4142	−1.24141
$327$	−17.6569	−0.976426
$328$	−3.65685	−0.201916
$329$	21.3137	1.17506
$330$	0	0
$331$	2.48528	0.136603	0.0683017	−	0.997665i	$-0.478242\pi$
$331$	2.48528	0.136603	0.0683017	+	0.997665i	$0.478242\pi$
$332$	13.6569	0.749517
$333$	−16.0000	−0.876795
$334$	3.65685	0.200094
$335$	0	0
$336$	5.82843	0.317967
$337$	−4.97056	−0.270764	−0.135382	−	0.990793i	$-0.543226\pi$
$337$	−4.97056	−0.270764	−0.135382	+	0.990793i	$0.543226\pi$
$338$	8.68629	0.472472
$339$	42.6274	2.31520
$340$	0	0
$341$	8.82843	0.478086
$342$	−13.6569	−0.738478
$343$	−19.7279	−1.06521
$344$	11.3137	0.609994
$345$	0	0
$346$	22.9706	1.23491
$347$	36.4142	1.95482	0.977409	−	0.211358i	$-0.0677886\pi$
$347$	36.4142	1.95482	0.977409	+	0.211358i	$0.0677886\pi$
$348$	−9.65685	−0.517662
$349$	21.8284	1.16845	0.584224	−	0.811592i	$-0.301399\pi$
$349$	21.8284	1.16845	0.584224	+	0.811592i	$0.301399\pi$
$350$	0	0
$351$	1.92893	0.102959
$352$	2.00000	0.106600
$353$	32.6569	1.73815	0.869074	−	0.494681i	$-0.164715\pi$
$353$	32.6569	1.73815	0.869074	+	0.494681i	$0.164715\pi$
$354$	−19.3137	−1.02651
$355$	0	0
$356$	−2.00000	−0.106000
$357$	5.82843	0.308473
$358$	19.1716	1.01325
$359$	8.14214	0.429725	0.214863	−	0.976644i	$-0.431070\pi$
$359$	8.14214	0.429725	0.214863	+	0.976644i	$0.431070\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	8.34315	0.438506
$363$	−16.8995	−0.886993
$364$	−11.2426	−0.589274
$365$	0	0
$366$	18.4853	0.966241
$367$	−19.0416	−0.993965	−0.496983	−	0.867761i	$-0.665559\pi$
$367$	−19.0416	−0.993965	−0.496983	+	0.867761i	$0.665559\pi$
$368$	−3.65685	−0.190627
$369$	−10.3431	−0.538443
$370$	0	0
$371$	0.414214	0.0215049
$372$	10.6569	0.552532
$373$	−27.0000	−1.39801	−0.699004	−	0.715118i	$-0.746373\pi$
$373$	−27.0000	−1.39801	−0.699004	+	0.715118i	$0.746373\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	8.82843	0.455291
$377$	18.6274	0.959361
$378$	−1.00000	−0.0514344
$379$	−10.0711	−0.517316	−0.258658	−	0.965969i	$-0.583280\pi$
$379$	−10.0711	−0.517316	−0.258658	+	0.965969i	$0.583280\pi$
$380$	0	0
$381$	23.3137	1.19440
$382$	2.48528	0.127158
$383$	17.7990	0.909486	0.454743	−	0.890623i	$-0.349731\pi$
$383$	17.7990	0.909486	0.454743	+	0.890623i	$0.349731\pi$
$384$	2.41421	0.123200
$385$	0	0
$386$	1.65685	0.0843317
$387$	32.0000	1.62665
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	−3.65685	−0.184935
$392$	−1.17157	−0.0591734
$393$	16.6569	0.840227
$394$	−2.00000	−0.100759
$395$	0	0
$396$	5.65685	0.284268
$397$	28.9706	1.45399	0.726995	−	0.686642i	$-0.240916\pi$
$397$	28.9706	1.45399	0.726995	+	0.686642i	$0.240916\pi$
$398$	−15.6569	−0.784807
$399$	−28.1421	−1.40887
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	36.9706	1.84392
$403$	−20.5563	−1.02398
$404$	−12.6569	−0.629702
$405$	0	0
$406$	−9.65685	−0.479262
$407$	−11.3137	−0.560800
$408$	2.41421	0.119521
$409$	3.97056	0.196332	0.0981658	−	0.995170i	$-0.468702\pi$
$409$	3.97056	0.196332	0.0981658	+	0.995170i	$0.468702\pi$
$410$	0	0
$411$	8.41421	0.415043
$412$	−18.4853	−0.910704
$413$	−19.3137	−0.950365
$414$	−10.3431	−0.508338
$415$	0	0
$416$	−4.65685	−0.228321
$417$	37.1421	1.81886
$418$	−9.65685	−0.472332
$419$	29.3137	1.43207	0.716034	−	0.698065i	$-0.245955\pi$
$419$	29.3137	1.43207	0.716034	+	0.698065i	$0.245955\pi$
$420$	0	0
$421$	4.31371	0.210237	0.105119	−	0.994460i	$-0.466478\pi$
$421$	4.31371	0.210237	0.105119	+	0.994460i	$0.466478\pi$
$422$	15.7279	0.765623
$423$	24.9706	1.21411
$424$	0.171573	0.00833232
$425$	0	0
$426$	−11.4853	−0.556464
$427$	18.4853	0.894565
$428$	1.92893	0.0932385
$429$	−22.4853	−1.08560
$430$	0	0
$431$	−3.44365	−0.165875	−0.0829374	−	0.996555i	$-0.526430\pi$
$431$	−3.44365	−0.165875	−0.0829374	+	0.996555i	$0.526430\pi$
$432$	−0.414214	−0.0199289
$433$	37.3137	1.79318	0.896591	−	0.442859i	$-0.146036\pi$
$433$	37.3137	1.79318	0.896591	+	0.442859i	$0.146036\pi$
$434$	10.6569	0.511545
$435$	0	0
$436$	−7.31371	−0.350263
$437$	17.6569	0.844642
$438$	−23.3137	−1.11397
$439$	17.8701	0.852891	0.426446	−	0.904513i	$-0.359766\pi$
$439$	17.8701	0.852891	0.426446	+	0.904513i	$0.359766\pi$
$440$	0	0
$441$	−3.31371	−0.157796
$442$	−4.65685	−0.221504
$443$	−22.4853	−1.06831	−0.534154	−	0.845387i	$-0.679370\pi$
$443$	−22.4853	−1.06831	−0.534154	+	0.845387i	$0.679370\pi$
$444$	−13.6569	−0.648126
$445$	0	0
$446$	−1.51472	−0.0717240
$447$	46.2132	2.18581
$448$	2.41421	0.114061
$449$	29.6569	1.39959	0.699797	−	0.714342i	$-0.253274\pi$
$449$	29.6569	1.39959	0.699797	+	0.714342i	$0.253274\pi$
$450$	0	0
$451$	−7.31371	−0.344389
$452$	17.6569	0.830509
$453$	−34.9706	−1.64306
$454$	−4.75736	−0.223274
$455$	0	0
$456$	−11.6569	−0.545882
$457$	3.14214	0.146983	0.0734915	−	0.997296i	$-0.476586\pi$
$457$	3.14214	0.146983	0.0734915	+	0.997296i	$0.476586\pi$
$458$	10.3137	0.481928
$459$	−0.414214	−0.0193338
$460$	0	0
$461$	32.6274	1.51961	0.759805	−	0.650151i	$-0.225294\pi$
$461$	32.6274	1.51961	0.759805	+	0.650151i	$0.225294\pi$
$462$	11.6569	0.542326
$463$	13.6569	0.634688	0.317344	−	0.948311i	$-0.397209\pi$
$463$	13.6569	0.634688	0.317344	+	0.948311i	$0.397209\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	20.0000	0.926482
$467$	−29.1127	−1.34718	−0.673588	−	0.739107i	$-0.735248\pi$
$467$	−29.1127	−1.34718	−0.673588	+	0.739107i	$0.735248\pi$
$468$	−13.1716	−0.608856
$469$	36.9706	1.70714
$470$	0	0
$471$	−52.6985	−2.42822
$472$	−8.00000	−0.368230
$473$	22.6274	1.04041
$474$	−17.4853	−0.803126
$475$	0	0
$476$	2.41421	0.110655
$477$	0.485281	0.0222195
$478$	−24.9706	−1.14213
$479$	−10.6863	−0.488269	−0.244135	−	0.969741i	$-0.578504\pi$
$479$	−10.6863	−0.488269	−0.244135	+	0.969741i	$0.578504\pi$
$480$	0	0
$481$	26.3431	1.20114
$482$	−24.6274	−1.12175
$483$	−21.3137	−0.969807
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−21.6569	−0.982375
$487$	−8.62742	−0.390946	−0.195473	−	0.980709i	$-0.562624\pi$
$487$	−8.62742	−0.390946	−0.195473	+	0.980709i	$0.562624\pi$
$488$	7.65685	0.346610
$489$	−54.1127	−2.44706
$490$	0	0
$491$	−9.65685	−0.435808	−0.217904	−	0.975970i	$-0.569922\pi$
$491$	−9.65685	−0.435808	−0.217904	+	0.975970i	$0.569922\pi$
$492$	−8.82843	−0.398016
$493$	−4.00000	−0.180151
$494$	22.4853	1.01166
$495$	0	0
$496$	4.41421	0.198204
$497$	−11.4853	−0.515185
$498$	32.9706	1.47745
$499$	−40.3553	−1.80655	−0.903277	−	0.429059i	$-0.858845\pi$
$499$	−40.3553	−1.80655	−0.903277	+	0.429059i	$0.858845\pi$
$500$	0	0
$501$	8.82843	0.394425
$502$	−4.00000	−0.178529
$503$	2.97056	0.132451	0.0662254	−	0.997805i	$-0.478904\pi$
$503$	2.97056	0.132451	0.0662254	+	0.997805i	$0.478904\pi$
$504$	6.82843	0.304162
$505$	0	0
$506$	−7.31371	−0.325134
$507$	20.9706	0.931335
$508$	9.65685	0.428454
$509$	6.85786	0.303969	0.151985	−	0.988383i	$-0.451434\pi$
$509$	6.85786	0.303969	0.151985	+	0.988383i	$0.451434\pi$
$510$	0	0
$511$	−23.3137	−1.03134
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−27.4853	−1.21232
$515$	0	0
$516$	27.3137	1.20242
$517$	17.6569	0.776548
$518$	−13.6569	−0.600048
$519$	55.4558	2.43424
$520$	0	0
$521$	18.2843	0.801048	0.400524	−	0.916286i	$-0.368828\pi$
$521$	18.2843	0.801048	0.400524	+	0.916286i	$0.368828\pi$
$522$	−11.3137	−0.495188
$523$	13.6569	0.597173	0.298586	−	0.954383i	$-0.403485\pi$
$523$	13.6569	0.597173	0.298586	+	0.954383i	$0.403485\pi$
$524$	6.89949	0.301406
$525$	0	0
$526$	21.6569	0.944284
$527$	4.41421	0.192286
$528$	4.82843	0.210130
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−22.6274	−0.981946
$532$	−11.6569	−0.505389
$533$	17.0294	0.737627
$534$	−4.82843	−0.208946
$535$	0	0
$536$	15.3137	0.661451
$537$	46.2843	1.99731
$538$	−16.6274	−0.716859
$539$	−2.34315	−0.100926
$540$	0	0
$541$	7.02944	0.302219	0.151110	−	0.988517i	$-0.451715\pi$
$541$	7.02944	0.302219	0.151110	+	0.988517i	$0.451715\pi$
$542$	−5.51472	−0.236877
$543$	20.1421	0.864382
$544$	1.00000	0.0428746
$545$	0	0
$546$	−27.1421	−1.16158
$547$	−22.0711	−0.943691	−0.471845	−	0.881681i	$-0.656412\pi$
$547$	−22.0711	−0.943691	−0.471845	+	0.881681i	$0.656412\pi$
$548$	3.48528	0.148884
$549$	21.6569	0.924292
$550$	0	0
$551$	19.3137	0.822792
$552$	−8.82843	−0.375763
$553$	−17.4853	−0.743550
$554$	12.3431	0.524410
$555$	0	0
$556$	15.3848	0.652460
$557$	−25.6274	−1.08587	−0.542934	−	0.839775i	$-0.682687\pi$
$557$	−25.6274	−1.08587	−0.542934	+	0.839775i	$0.682687\pi$
$558$	12.4853	0.528544
$559$	−52.6863	−2.22839
$560$	0	0
$561$	4.82843	0.203856
$562$	−4.65685	−0.196438
$563$	−15.4558	−0.651386	−0.325693	−	0.945476i	$-0.605598\pi$
$563$	−15.4558	−0.651386	−0.325693	+	0.945476i	$0.605598\pi$
$564$	21.3137	0.897469
$565$	0	0
$566$	14.9706	0.629260
$567$	−22.8995	−0.961688
$568$	−4.75736	−0.199614
$569$	16.6274	0.697058	0.348529	−	0.937298i	$-0.386681\pi$
$569$	16.6274	0.697058	0.348529	+	0.937298i	$0.386681\pi$
$570$	0	0
$571$	−4.07107	−0.170369	−0.0851844	−	0.996365i	$-0.527148\pi$
$571$	−4.07107	−0.170369	−0.0851844	+	0.996365i	$0.527148\pi$
$572$	−9.31371	−0.389426
$573$	6.00000	0.250654
$574$	−8.82843	−0.368491
$575$	0	0
$576$	2.82843	0.117851
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	1.00000	0.0415945
$579$	4.00000	0.166234
$580$	0	0
$581$	32.9706	1.36785
$582$	0	0
$583$	0.343146	0.0142116
$584$	−9.65685	−0.399603
$585$	0	0
$586$	2.68629	0.110970
$587$	−7.31371	−0.301869	−0.150935	−	0.988544i	$-0.548228\pi$
$587$	−7.31371	−0.301869	−0.150935	+	0.988544i	$0.548228\pi$
$588$	−2.82843	−0.116642
$589$	−21.3137	−0.878216
$590$	0	0
$591$	−4.82843	−0.198615
$592$	−5.65685	−0.232495
$593$	14.6569	0.601885	0.300942	−	0.953642i	$-0.402699\pi$
$593$	14.6569	0.601885	0.300942	+	0.953642i	$0.402699\pi$
$594$	−0.828427	−0.0339908
$595$	0	0
$596$	19.1421	0.784092
$597$	−37.7990	−1.54701
$598$	17.0294	0.696385
$599$	25.6569	1.04831	0.524155	−	0.851623i	$-0.324381\pi$
$599$	25.6569	1.04831	0.524155	+	0.851623i	$0.324381\pi$
$600$	0	0
$601$	−14.3431	−0.585069	−0.292535	−	0.956255i	$-0.594499\pi$
$601$	−14.3431	−0.585069	−0.292535	+	0.956255i	$0.594499\pi$
$602$	27.3137	1.11322
$603$	43.3137	1.76387
$604$	−14.4853	−0.589398
$605$	0	0
$606$	−30.5563	−1.24127
$607$	−25.9289	−1.05242	−0.526211	−	0.850354i	$-0.676388\pi$
$607$	−25.9289	−1.05242	−0.526211	+	0.850354i	$0.676388\pi$
$608$	−4.82843	−0.195819
$609$	−23.3137	−0.944719
$610$	0	0
$611$	−41.1127	−1.66324
$612$	2.82843	0.114332
$613$	15.1421	0.611585	0.305793	−	0.952098i	$-0.401079\pi$
$613$	15.1421	0.611585	0.305793	+	0.952098i	$0.401079\pi$
$614$	−1.65685	−0.0668652
$615$	0	0
$616$	4.82843	0.194543
$617$	24.9706	1.00528	0.502639	−	0.864497i	$-0.332363\pi$
$617$	24.9706	1.00528	0.502639	+	0.864497i	$0.332363\pi$
$618$	−44.6274	−1.79518
$619$	14.6863	0.590292	0.295146	−	0.955452i	$-0.404632\pi$
$619$	14.6863	0.590292	0.295146	+	0.955452i	$0.404632\pi$
$620$	0	0
$621$	1.51472	0.0607836
$622$	−22.2132	−0.890668
$623$	−4.82843	−0.193447
$624$	−11.2426	−0.450066
$625$	0	0
$626$	10.9706	0.438472
$627$	−23.3137	−0.931060
$628$	−21.8284	−0.871049
$629$	−5.65685	−0.225554
$630$	0	0
$631$	0.970563	0.0386375	0.0193187	−	0.999813i	$-0.493850\pi$
$631$	0.970563	0.0386375	0.0193187	+	0.999813i	$0.493850\pi$
$632$	−7.24264	−0.288097
$633$	37.9706	1.50919
$634$	25.3137	1.00534
$635$	0	0
$636$	0.414214	0.0164246
$637$	5.45584	0.216168
$638$	−8.00000	−0.316723
$639$	−13.4558	−0.532305
$640$	0	0
$641$	−20.2843	−0.801181	−0.400590	−	0.916257i	$-0.631195\pi$
$641$	−20.2843	−0.801181	−0.400590	+	0.916257i	$0.631195\pi$
$642$	4.65685	0.183791
$643$	23.8701	0.941343	0.470672	−	0.882308i	$-0.344012\pi$
$643$	23.8701	0.941343	0.470672	+	0.882308i	$0.344012\pi$
$644$	−8.82843	−0.347889
$645$	0	0
$646$	−4.82843	−0.189972
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−9.48528	−0.372617
$649$	−16.0000	−0.628055
$650$	0	0
$651$	25.7279	1.00836
$652$	−22.4142	−0.877808
$653$	48.2843	1.88951	0.944755	−	0.327778i	$-0.106300\pi$
$653$	48.2843	1.88951	0.944755	+	0.327778i	$0.106300\pi$
$654$	−17.6569	−0.690438
$655$	0	0
$656$	−3.65685	−0.142776
$657$	−27.3137	−1.06561
$658$	21.3137	0.830895
$659$	24.1421	0.940444	0.470222	−	0.882548i	$-0.344174\pi$
$659$	24.1421	0.940444	0.470222	+	0.882548i	$0.344174\pi$
$660$	0	0
$661$	−31.8284	−1.23798	−0.618991	−	0.785398i	$-0.712458\pi$
$661$	−31.8284	−1.23798	−0.618991	+	0.785398i	$0.712458\pi$
$662$	2.48528	0.0965932
$663$	−11.2426	−0.436628
$664$	13.6569	0.529989
$665$	0	0
$666$	−16.0000	−0.619987
$667$	14.6274	0.566376
$668$	3.65685	0.141488
$669$	−3.65685	−0.141382
$670$	0	0
$671$	15.3137	0.591179
$672$	5.82843	0.224836
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	−4.97056	−0.191459
$675$	0	0
$676$	8.68629	0.334088
$677$	−49.9411	−1.91939	−0.959697	−	0.281038i	$-0.909321\pi$
$677$	−49.9411	−1.91939	−0.959697	+	0.281038i	$0.909321\pi$
$678$	42.6274	1.63710
$679$	0	0
$680$	0	0
$681$	−11.4853	−0.440117
$682$	8.82843	0.338058
$683$	−26.0711	−0.997582	−0.498791	−	0.866722i	$-0.666222\pi$
$683$	−26.0711	−0.997582	−0.498791	+	0.866722i	$0.666222\pi$
$684$	−13.6569	−0.522183
$685$	0	0
$686$	−19.7279	−0.753216
$687$	24.8995	0.949975
$688$	11.3137	0.431331
$689$	−0.798990	−0.0304391
$690$	0	0
$691$	50.3553	1.91561	0.957804	−	0.287423i	$-0.0927986\pi$
$691$	50.3553	1.91561	0.957804	+	0.287423i	$0.0927986\pi$
$692$	22.9706	0.873210
$693$	13.6569	0.518781
$694$	36.4142	1.38226
$695$	0	0
$696$	−9.65685	−0.366042
$697$	−3.65685	−0.138513
$698$	21.8284	0.826218
$699$	48.2843	1.82628
$700$	0	0
$701$	−36.6274	−1.38340	−0.691699	−	0.722186i	$-0.743138\pi$
$701$	−36.6274	−1.38340	−0.691699	+	0.722186i	$0.743138\pi$
$702$	1.92893	0.0728029
$703$	27.3137	1.03016
$704$	2.00000	0.0753778
$705$	0	0
$706$	32.6569	1.22906
$707$	−30.5563	−1.14919
$708$	−19.3137	−0.725854
$709$	−20.3431	−0.764003	−0.382001	−	0.924162i	$-0.624765\pi$
$709$	−20.3431	−0.764003	−0.382001	+	0.924162i	$0.624765\pi$
$710$	0	0
$711$	−20.4853	−0.768258
$712$	−2.00000	−0.0749532
$713$	−16.1421	−0.604528
$714$	5.82843	0.218123
$715$	0	0
$716$	19.1716	0.716475
$717$	−60.2843	−2.25136
$718$	8.14214	0.303862
$719$	−6.89949	−0.257308	−0.128654	−	0.991690i	$-0.541066\pi$
$719$	−6.89949	−0.257308	−0.128654	+	0.991690i	$0.541066\pi$
$720$	0	0
$721$	−44.6274	−1.66201
$722$	4.31371	0.160540
$723$	−59.4558	−2.21119
$724$	8.34315	0.310071
$725$	0	0
$726$	−16.8995	−0.627199
$727$	−35.1716	−1.30444	−0.652221	−	0.758029i	$-0.726162\pi$
$727$	−35.1716	−1.30444	−0.652221	+	0.758029i	$0.726162\pi$
$728$	−11.2426	−0.416680
$729$	−23.8284	−0.882534
$730$	0	0
$731$	11.3137	0.418453
$732$	18.4853	0.683236
$733$	5.82843	0.215278	0.107639	−	0.994190i	$-0.465671\pi$
$733$	5.82843	0.215278	0.107639	+	0.994190i	$0.465671\pi$
$734$	−19.0416	−0.702839
$735$	0	0
$736$	−3.65685	−0.134793
$737$	30.6274	1.12818
$738$	−10.3431	−0.380736
$739$	0.828427	0.0304742	0.0152371	−	0.999884i	$-0.495150\pi$
$739$	0.828427	0.0304742	0.0152371	+	0.999884i	$0.495150\pi$
$740$	0	0
$741$	54.2843	1.99418
$742$	0.414214	0.0152063
$743$	38.0711	1.39669	0.698346	−	0.715760i	$-0.253920\pi$
$743$	38.0711	1.39669	0.698346	+	0.715760i	$0.253920\pi$
$744$	10.6569	0.390699
$745$	0	0
$746$	−27.0000	−0.988540
$747$	38.6274	1.41330
$748$	2.00000	0.0731272
$749$	4.65685	0.170158
$750$	0	0
$751$	34.2843	1.25105	0.625525	−	0.780204i	$-0.284885\pi$
$751$	34.2843	1.25105	0.625525	+	0.780204i	$0.284885\pi$
$752$	8.82843	0.321940
$753$	−9.65685	−0.351915
$754$	18.6274	0.678371
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−10.0711	−0.365798
$759$	−17.6569	−0.640903
$760$	0	0
$761$	−25.4853	−0.923841	−0.461920	−	0.886921i	$-0.652839\pi$
$761$	−25.4853	−0.923841	−0.461920	+	0.886921i	$0.652839\pi$
$762$	23.3137	0.844567
$763$	−17.6569	−0.639221
$764$	2.48528	0.0899143
$765$	0	0
$766$	17.7990	0.643104
$767$	37.2548	1.34519
$768$	2.41421	0.0871154
$769$	−18.8579	−0.680032	−0.340016	−	0.940420i	$-0.610432\pi$
$769$	−18.8579	−0.680032	−0.340016	+	0.940420i	$0.610432\pi$
$770$	0	0
$771$	−66.3553	−2.38973
$772$	1.65685	0.0596315
$773$	−46.9411	−1.68835	−0.844177	−	0.536064i	$-0.819911\pi$
$773$	−46.9411	−1.68835	−0.844177	+	0.536064i	$0.819911\pi$
$774$	32.0000	1.15022
$775$	0	0
$776$	0	0
$777$	−32.9706	−1.18281
$778$	−14.0000	−0.501924
$779$	17.6569	0.632622
$780$	0	0
$781$	−9.51472	−0.340463
$782$	−3.65685	−0.130769
$783$	1.65685	0.0592111
$784$	−1.17157	−0.0418419
$785$	0	0
$786$	16.6569	0.594130
$787$	−24.8995	−0.887571	−0.443786	−	0.896133i	$-0.646365\pi$
$787$	−24.8995	−0.887571	−0.443786	+	0.896133i	$0.646365\pi$
$788$	−2.00000	−0.0712470
$789$	52.2843	1.86137
$790$	0	0
$791$	42.6274	1.51566
$792$	5.65685	0.201008
$793$	−35.6569	−1.26621
$794$	28.9706	1.02813
$795$	0	0
$796$	−15.6569	−0.554942
$797$	−27.8284	−0.985733	−0.492867	−	0.870105i	$-0.664051\pi$
$797$	−27.8284	−0.985733	−0.492867	+	0.870105i	$0.664051\pi$
$798$	−28.1421	−0.996221
$799$	8.82843	0.312327
$800$	0	0
$801$	−5.65685	−0.199875
$802$	26.0000	0.918092
$803$	−19.3137	−0.681566
$804$	36.9706	1.30385
$805$	0	0
$806$	−20.5563	−0.724067
$807$	−40.1421	−1.41307
$808$	−12.6569	−0.445267
$809$	33.3137	1.17125	0.585624	−	0.810583i	$-0.300850\pi$
$809$	33.3137	1.17125	0.585624	+	0.810583i	$0.300850\pi$
$810$	0	0
$811$	14.6985	0.516134	0.258067	−	0.966127i	$-0.416915\pi$
$811$	14.6985	0.516134	0.258067	+	0.966127i	$0.416915\pi$
$812$	−9.65685	−0.338889
$813$	−13.3137	−0.466932
$814$	−11.3137	−0.396545
$815$	0	0
$816$	2.41421	0.0845144
$817$	−54.6274	−1.91117
$818$	3.97056	0.138827
$819$	−31.7990	−1.11115
$820$	0	0
$821$	−8.34315	−0.291178	−0.145589	−	0.989345i	$-0.546508\pi$
$821$	−8.34315	−0.291178	−0.145589	+	0.989345i	$0.546508\pi$
$822$	8.41421	0.293479
$823$	−49.1838	−1.71444	−0.857219	−	0.514952i	$-0.827810\pi$
$823$	−49.1838	−1.71444	−0.857219	+	0.514952i	$0.827810\pi$
$824$	−18.4853	−0.643965
$825$	0	0
$826$	−19.3137	−0.672010
$827$	48.6274	1.69094	0.845470	−	0.534022i	$-0.179320\pi$
$827$	48.6274	1.69094	0.845470	+	0.534022i	$0.179320\pi$
$828$	−10.3431	−0.359449
$829$	−16.4558	−0.571535	−0.285768	−	0.958299i	$-0.592249\pi$
$829$	−16.4558	−0.571535	−0.285768	+	0.958299i	$0.592249\pi$
$830$	0	0
$831$	29.7990	1.03372
$832$	−4.65685	−0.161447
$833$	−1.17157	−0.0405926
$834$	37.1421	1.28613
$835$	0	0
$836$	−9.65685	−0.333989
$837$	−1.82843	−0.0631997
$838$	29.3137	1.01263
$839$	−22.7574	−0.785671	−0.392836	−	0.919609i	$-0.628506\pi$
$839$	−22.7574	−0.785671	−0.392836	+	0.919609i	$0.628506\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	4.31371	0.148660
$843$	−11.2426	−0.387217
$844$	15.7279	0.541377
$845$	0	0
$846$	24.9706	0.858506
$847$	−16.8995	−0.580674
$848$	0.171573	0.00589184
$849$	36.1421	1.24039
$850$	0	0
$851$	20.6863	0.709117
$852$	−11.4853	−0.393479
$853$	−14.6863	−0.502849	−0.251425	−	0.967877i	$-0.580899\pi$
$853$	−14.6863	−0.502849	−0.251425	+	0.967877i	$0.580899\pi$
$854$	18.4853	0.632553
$855$	0	0
$856$	1.92893	0.0659295
$857$	−9.65685	−0.329872	−0.164936	−	0.986304i	$-0.552742\pi$
$857$	−9.65685	−0.329872	−0.164936	+	0.986304i	$0.552742\pi$
$858$	−22.4853	−0.767635
$859$	−16.1421	−0.550763	−0.275381	−	0.961335i	$-0.588804\pi$
$859$	−16.1421	−0.550763	−0.275381	+	0.961335i	$0.588804\pi$
$860$	0	0
$861$	−21.3137	−0.726369
$862$	−3.44365	−0.117291
$863$	−35.5980	−1.21177	−0.605885	−	0.795552i	$-0.707181\pi$
$863$	−35.5980	−1.21177	−0.605885	+	0.795552i	$0.707181\pi$
$864$	−0.414214	−0.0140918
$865$	0	0
$866$	37.3137	1.26797
$867$	2.41421	0.0819910
$868$	10.6569	0.361717
$869$	−14.4853	−0.491380
$870$	0	0
$871$	−71.3137	−2.41637
$872$	−7.31371	−0.247673
$873$	0	0
$874$	17.6569	0.597252
$875$	0	0
$876$	−23.3137	−0.787697
$877$	8.68629	0.293315	0.146658	−	0.989187i	$-0.453148\pi$
$877$	8.68629	0.293315	0.146658	+	0.989187i	$0.453148\pi$
$878$	17.8701	0.603085
$879$	6.48528	0.218743
$880$	0	0
$881$	−5.02944	−0.169446	−0.0847230	−	0.996405i	$-0.527001\pi$
$881$	−5.02944	−0.169446	−0.0847230	+	0.996405i	$0.527001\pi$
$882$	−3.31371	−0.111578
$883$	−19.4558	−0.654741	−0.327371	−	0.944896i	$-0.606163\pi$
$883$	−19.4558	−0.654741	−0.327371	+	0.944896i	$0.606163\pi$
$884$	−4.65685	−0.156627
$885$	0	0
$886$	−22.4853	−0.755408
$887$	−31.5858	−1.06055	−0.530273	−	0.847827i	$-0.677911\pi$
$887$	−31.5858	−1.06055	−0.530273	+	0.847827i	$0.677911\pi$
$888$	−13.6569	−0.458294
$889$	23.3137	0.781917
$890$	0	0
$891$	−18.9706	−0.635538
$892$	−1.51472	−0.0507165
$893$	−42.6274	−1.42647
$894$	46.2132	1.54560
$895$	0	0
$896$	2.41421	0.0806532
$897$	41.1127	1.37271
$898$	29.6569	0.989662
$899$	−17.6569	−0.588889
$900$	0	0
$901$	0.171573	0.00571592
$902$	−7.31371	−0.243520
$903$	65.9411	2.19438
$904$	17.6569	0.587258
$905$	0	0
$906$	−34.9706	−1.16182
$907$	5.31371	0.176439	0.0882194	−	0.996101i	$-0.471882\pi$
$907$	5.31371	0.176439	0.0882194	+	0.996101i	$0.471882\pi$
$908$	−4.75736	−0.157879
$909$	−35.7990	−1.18738
$910$	0	0
$911$	21.7279	0.719878	0.359939	−	0.932976i	$-0.382797\pi$
$911$	21.7279	0.719878	0.359939	+	0.932976i	$0.382797\pi$
$912$	−11.6569	−0.385997
$913$	27.3137	0.903952
$914$	3.14214	0.103933
$915$	0	0
$916$	10.3137	0.340775
$917$	16.6569	0.550058
$918$	−0.414214	−0.0136711
$919$	−49.6569	−1.63803	−0.819014	−	0.573773i	$-0.805479\pi$
$919$	−49.6569	−1.63803	−0.819014	+	0.573773i	$0.805479\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	32.6274	1.07453
$923$	22.1543	0.729219
$924$	11.6569	0.383482
$925$	0	0
$926$	13.6569	0.448792
$927$	−52.2843	−1.71724
$928$	−4.00000	−0.131306
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	5.65685	0.185396
$932$	20.0000	0.655122
$933$	−53.6274	−1.75568
$934$	−29.1127	−0.952597
$935$	0	0
$936$	−13.1716	−0.430526
$937$	17.0000	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$938$	36.9706	1.20713
$939$	26.4853	0.864314
$940$	0	0
$941$	−3.65685	−0.119210	−0.0596050	−	0.998222i	$-0.518984\pi$
$941$	−3.65685	−0.119210	−0.0596050	+	0.998222i	$0.518984\pi$
$942$	−52.6985	−1.71701
$943$	13.3726	0.435471
$944$	−8.00000	−0.260378
$945$	0	0
$946$	22.6274	0.735681
$947$	−21.3137	−0.692602	−0.346301	−	0.938123i	$-0.612562\pi$
$947$	−21.3137	−0.692602	−0.346301	+	0.938123i	$0.612562\pi$
$948$	−17.4853	−0.567896
$949$	44.9706	1.45981
$950$	0	0
$951$	61.1127	1.98172
$952$	2.41421	0.0782451
$953$	30.1127	0.975446	0.487723	−	0.872998i	$-0.337828\pi$
$953$	30.1127	0.975446	0.487723	+	0.872998i	$0.337828\pi$
$954$	0.485281	0.0157116
$955$	0	0
$956$	−24.9706	−0.807606
$957$	−19.3137	−0.624324
$958$	−10.6863	−0.345258
$959$	8.41421	0.271709
$960$	0	0
$961$	−11.5147	−0.371443
$962$	26.3431	0.849337
$963$	5.45584	0.175812
$964$	−24.6274	−0.793196
$965$	0	0
$966$	−21.3137	−0.685757
$967$	−34.6274	−1.11354	−0.556771	−	0.830666i	$-0.687960\pi$
$967$	−34.6274	−1.11354	−0.556771	+	0.830666i	$0.687960\pi$
$968$	−7.00000	−0.224989
$969$	−11.6569	−0.374472
$970$	0	0
$971$	−13.9411	−0.447392	−0.223696	−	0.974659i	$-0.571812\pi$
$971$	−13.9411	−0.447392	−0.223696	+	0.974659i	$0.571812\pi$
$972$	−21.6569	−0.694644
$973$	37.1421	1.19072
$974$	−8.62742	−0.276440
$975$	0	0
$976$	7.65685	0.245090
$977$	0.313708	0.0100364	0.00501821	−	0.999987i	$-0.498403\pi$
$977$	0.313708	0.0100364	0.00501821	+	0.999987i	$0.498403\pi$
$978$	−54.1127	−1.73033
$979$	−4.00000	−0.127841
$980$	0	0
$981$	−20.6863	−0.660462
$982$	−9.65685	−0.308163
$983$	1.58579	0.0505787	0.0252894	−	0.999680i	$-0.491949\pi$
$983$	1.58579	0.0505787	0.0252894	+	0.999680i	$0.491949\pi$
$984$	−8.82843	−0.281440
$985$	0	0
$986$	−4.00000	−0.127386
$987$	51.4558	1.63786
$988$	22.4853	0.715352
$989$	−41.3726	−1.31557
$990$	0	0
$991$	−39.1838	−1.24471	−0.622357	−	0.782734i	$-0.713825\pi$
$991$	−39.1838	−1.24471	−0.622357	+	0.782734i	$0.713825\pi$
$992$	4.41421	0.140151
$993$	6.00000	0.190404
$994$	−11.4853	−0.364291
$995$	0	0
$996$	32.9706	1.04471
$997$	6.62742	0.209892	0.104946	−	0.994478i	$-0.466533\pi$
$997$	6.62742	0.209892	0.104946	+	0.994478i	$0.466533\pi$
$998$	−40.3553	−1.27743
$999$	2.34315	0.0741339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.a.o.1.2	yes	2
3.2	odd	2		7650.2.a.cw.1.2		2
4.3	odd	2		6800.2.a.bc.1.1		2
5.2	odd	4		850.2.c.j.749.3		4
5.3	odd	4		850.2.c.j.749.2		4
5.4	even	2		850.2.a.m.1.1	✓	2
15.14	odd	2		7650.2.a.dc.1.1		2
20.19	odd	2		6800.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
850.2.a.m.1.1	✓	2	5.4	even	2
850.2.a.o.1.2	yes	2	1.1	even	1	trivial
850.2.c.j.749.2		4	5.3	odd	4
850.2.c.j.749.3		4	5.2	odd	4
6800.2.a.bc.1.1		2	4.3	odd	2
6800.2.a.bi.1.2		2	20.19	odd	2
7650.2.a.cw.1.2		2	3.2	odd	2
7650.2.a.dc.1.1		2	15.14	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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +2.41421 q^{6} +2.41421 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.41421 q^{3} +1.00000 q^{4} +2.41421 q^{6} +2.41421 q^{7} +1.00000 q^{8} +2.82843 q^{9} +2.00000 q^{11} +2.41421 q^{12} -4.65685 q^{13} +2.41421 q^{14} +1.00000 q^{16} +1.00000 q^{17} +2.82843 q^{18} -4.82843 q^{19} +5.82843 q^{21} +2.00000 q^{22} -3.65685 q^{23} +2.41421 q^{24} -4.65685 q^{26} -0.414214 q^{27} +2.41421 q^{28} -4.00000 q^{29} +4.41421 q^{31} +1.00000 q^{32} +4.82843 q^{33} +1.00000 q^{34} +2.82843 q^{36} -5.65685 q^{37} -4.82843 q^{38} -11.2426 q^{39} -3.65685 q^{41} +5.82843 q^{42} +11.3137 q^{43} +2.00000 q^{44} -3.65685 q^{46} +8.82843 q^{47} +2.41421 q^{48} -1.17157 q^{49} +2.41421 q^{51} -4.65685 q^{52} +0.171573 q^{53} -0.414214 q^{54} +2.41421 q^{56} -11.6569 q^{57} -4.00000 q^{58} -8.00000 q^{59} +7.65685 q^{61} +4.41421 q^{62} +6.82843 q^{63} +1.00000 q^{64} +4.82843 q^{66} +15.3137 q^{67} +1.00000 q^{68} -8.82843 q^{69} -4.75736 q^{71} +2.82843 q^{72} -9.65685 q^{73} -5.65685 q^{74} -4.82843 q^{76} +4.82843 q^{77} -11.2426 q^{78} -7.24264 q^{79} -9.48528 q^{81} -3.65685 q^{82} +13.6569 q^{83} +5.82843 q^{84} +11.3137 q^{86} -9.65685 q^{87} +2.00000 q^{88} -2.00000 q^{89} -11.2426 q^{91} -3.65685 q^{92} +10.6569 q^{93} +8.82843 q^{94} +2.41421 q^{96} -1.17157 q^{98} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 6 q^{21} + 4 q^{22} + 4 q^{23} + 2 q^{24} + 2 q^{26} + 2 q^{27} + 2 q^{28} - 8 q^{29} + 6 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} - 4 q^{38} - 14 q^{39} + 4 q^{41} + 6 q^{42} + 4 q^{44} + 4 q^{46} + 12 q^{47} + 2 q^{48} - 8 q^{49} + 2 q^{51} + 2 q^{52} + 6 q^{53} + 2 q^{54} + 2 q^{56} - 12 q^{57} - 8 q^{58} - 16 q^{59} + 4 q^{61} + 6 q^{62} + 8 q^{63} + 2 q^{64} + 4 q^{66} + 8 q^{67} + 2 q^{68} - 12 q^{69} - 18 q^{71} - 8 q^{73} - 4 q^{76} + 4 q^{77} - 14 q^{78} - 6 q^{79} - 2 q^{81} + 4 q^{82} + 16 q^{83} + 6 q^{84} - 8 q^{87} + 4 q^{88} - 4 q^{89} - 14 q^{91} + 4 q^{92} + 10 q^{93} + 12 q^{94} + 2 q^{96} - 8 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 2 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 6 * q^21 + 4 * q^22 + 4 * q^23 + 2 * q^24 + 2 * q^26 + 2 * q^27 + 2 * q^28 - 8 * q^29 + 6 * q^31 + 2 * q^32 + 4 * q^33 + 2 * q^34 - 4 * q^38 - 14 * q^39 + 4 * q^41 + 6 * q^42 + 4 * q^44 + 4 * q^46 + 12 * q^47 + 2 * q^48 - 8 * q^49 + 2 * q^51 + 2 * q^52 + 6 * q^53 + 2 * q^54 + 2 * q^56 - 12 * q^57 - 8 * q^58 - 16 * q^59 + 4 * q^61 + 6 * q^62 + 8 * q^63 + 2 * q^64 + 4 * q^66 + 8 * q^67 + 2 * q^68 - 12 * q^69 - 18 * q^71 - 8 * q^73 - 4 * q^76 + 4 * q^77 - 14 * q^78 - 6 * q^79 - 2 * q^81 + 4 * q^82 + 16 * q^83 + 6 * q^84 - 8 * q^87 + 4 * q^88 - 4 * q^89 - 14 * q^91 + 4 * q^92 + 10 * q^93 + 12 * q^94 + 2 * q^96 - 8 * q^98

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