LMFDB - Embedded newform 85.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(1,85)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, 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("85.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.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:	$0.678728417181$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
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	$\chi$	$=$	85.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{2} -3.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.41421 q^{6} -0.585786 q^{7} -1.58579 q^{8} +8.65685 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} -3.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.41421 q^{6} -0.585786 q^{7} -1.58579 q^{8} +8.65685 q^{9} -0.414214 q^{10} -2.58579 q^{11} +6.24264 q^{12} -2.82843 q^{13} -0.242641 q^{14} +3.41421 q^{15} +3.00000 q^{16} -1.00000 q^{17} +3.58579 q^{18} -2.82843 q^{19} +1.82843 q^{20} +2.00000 q^{21} -1.07107 q^{22} -3.41421 q^{23} +5.41421 q^{24} +1.00000 q^{25} -1.17157 q^{26} -19.3137 q^{27} +1.07107 q^{28} -4.82843 q^{29} +1.41421 q^{30} +4.24264 q^{31} +4.41421 q^{32} +8.82843 q^{33} -0.414214 q^{34} +0.585786 q^{35} -15.8284 q^{36} +6.48528 q^{37} -1.17157 q^{38} +9.65685 q^{39} +1.58579 q^{40} -6.48528 q^{41} +0.828427 q^{42} +7.65685 q^{43} +4.72792 q^{44} -8.65685 q^{45} -1.41421 q^{46} -4.82843 q^{47} -10.2426 q^{48} -6.65685 q^{49} +0.414214 q^{50} +3.41421 q^{51} +5.17157 q^{52} +0.343146 q^{53} -8.00000 q^{54} +2.58579 q^{55} +0.928932 q^{56} +9.65685 q^{57} -2.00000 q^{58} -9.17157 q^{59} -6.24264 q^{60} +7.65685 q^{61} +1.75736 q^{62} -5.07107 q^{63} -4.17157 q^{64} +2.82843 q^{65} +3.65685 q^{66} -3.17157 q^{67} +1.82843 q^{68} +11.6569 q^{69} +0.242641 q^{70} -4.24264 q^{71} -13.7279 q^{72} -4.82843 q^{73} +2.68629 q^{74} -3.41421 q^{75} +5.17157 q^{76} +1.51472 q^{77} +4.00000 q^{78} +5.41421 q^{79} -3.00000 q^{80} +39.9706 q^{81} -2.68629 q^{82} +9.31371 q^{83} -3.65685 q^{84} +1.00000 q^{85} +3.17157 q^{86} +16.4853 q^{87} +4.10051 q^{88} -2.34315 q^{89} -3.58579 q^{90} +1.65685 q^{91} +6.24264 q^{92} -14.4853 q^{93} -2.00000 q^{94} +2.82843 q^{95} -15.0711 q^{96} +3.65685 q^{97} -2.75736 q^{98} -22.3848 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^8 + 6 * q^9	$ 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23} + 8 q^{24} + 2 q^{25} - 8 q^{26} - 16 q^{27} - 12 q^{28} - 4 q^{29} + 6 q^{32} + 12 q^{33} + 2 q^{34} + 4 q^{35} - 26 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{39} + 6 q^{40} + 4 q^{41} - 4 q^{42} + 4 q^{43} - 16 q^{44} - 6 q^{45} - 4 q^{47} - 12 q^{48} - 2 q^{49} - 2 q^{50} + 4 q^{51} + 16 q^{52} + 12 q^{53} - 16 q^{54} + 8 q^{55} + 16 q^{56} + 8 q^{57} - 4 q^{58} - 24 q^{59} - 4 q^{60} + 4 q^{61} + 12 q^{62} + 4 q^{63} - 14 q^{64} - 4 q^{66} - 12 q^{67} - 2 q^{68} + 12 q^{69} - 8 q^{70} - 2 q^{72} - 4 q^{73} + 28 q^{74} - 4 q^{75} + 16 q^{76} + 20 q^{77} + 8 q^{78} + 8 q^{79} - 6 q^{80} + 46 q^{81} - 28 q^{82} - 4 q^{83} + 4 q^{84} + 2 q^{85} + 12 q^{86} + 16 q^{87} + 28 q^{88} - 16 q^{89} - 10 q^{90} - 8 q^{91} + 4 q^{92} - 12 q^{93} - 4 q^{94} - 16 q^{96} - 4 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 2 * q^10 - 8 * q^11 + 4 * q^12 + 8 * q^14 + 4 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 4 * q^23 + 8 * q^24 + 2 * q^25 - 8 * q^26 - 16 * q^27 - 12 * q^28 - 4 * q^29 + 6 * q^32 + 12 * q^33 + 2 * q^34 + 4 * q^35 - 26 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^39 + 6 * q^40 + 4 * q^41 - 4 * q^42 + 4 * q^43 - 16 * q^44 - 6 * q^45 - 4 * q^47 - 12 * q^48 - 2 * q^49 - 2 * q^50 + 4 * q^51 + 16 * q^52 + 12 * q^53 - 16 * q^54 + 8 * q^55 + 16 * q^56 + 8 * q^57 - 4 * q^58 - 24 * q^59 - 4 * q^60 + 4 * q^61 + 12 * q^62 + 4 * q^63 - 14 * q^64 - 4 * q^66 - 12 * q^67 - 2 * q^68 + 12 * q^69 - 8 * q^70 - 2 * q^72 - 4 * q^73 + 28 * q^74 - 4 * q^75 + 16 * q^76 + 20 * q^77 + 8 * q^78 + 8 * q^79 - 6 * q^80 + 46 * q^81 - 28 * q^82 - 4 * q^83 + 4 * q^84 + 2 * q^85 + 12 * q^86 + 16 * q^87 + 28 * q^88 - 16 * q^89 - 10 * q^90 - 8 * q^91 + 4 * q^92 - 12 * q^93 - 4 * q^94 - 16 * q^96 - 4 * q^97 - 14 * q^98 - 8 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	−3.41421	−1.97120	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−3.41421	−1.97120	−0.985599	+	0.169102i	$0.945913\pi$
$4$	−1.82843	−0.914214
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.41421	−0.577350
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	−1.58579	−0.560660
$9$	8.65685	2.88562
$10$	−0.414214	−0.130986
$11$	−2.58579	−0.779644	−0.389822	−	0.920890i	$-0.627463\pi$
$11$	−2.58579	−0.779644	−0.389822	+	0.920890i	$0.627463\pi$
$12$	6.24264	1.80210
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	−0.242641	−0.0648485
$15$	3.41421	0.881546
$16$	3.00000	0.750000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	3.58579	0.845178
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.82843	0.408849
$21$	2.00000	0.436436
$22$	−1.07107	−0.228352
$23$	−3.41421	−0.711913	−0.355956	−	0.934503i	$-0.615845\pi$
$23$	−3.41421	−0.711913	−0.355956	+	0.934503i	$0.615845\pi$
$24$	5.41421	1.10517
$25$	1.00000	0.200000
$26$	−1.17157	−0.229764
$27$	−19.3137	−3.71692
$28$	1.07107	0.202413
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	1.41421	0.258199
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	4.41421	0.780330
$33$	8.82843	1.53683
$34$	−0.414214	−0.0710370
$35$	0.585786	0.0990160
$36$	−15.8284	−2.63807
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	−1.17157	−0.190054
$39$	9.65685	1.54633
$40$	1.58579	0.250735
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0.828427	0.127829
$43$	7.65685	1.16766	0.583830	−	0.811876i	$-0.301554\pi$
$43$	7.65685	1.16766	0.583830	+	0.811876i	$0.301554\pi$
$44$	4.72792	0.712761
$45$	−8.65685	−1.29049
$46$	−1.41421	−0.208514
$47$	−4.82843	−0.704298	−0.352149	−	0.935944i	$-0.614549\pi$
$47$	−4.82843	−0.704298	−0.352149	+	0.935944i	$0.614549\pi$
$48$	−10.2426	−1.47840
$49$	−6.65685	−0.950979
$50$	0.414214	0.0585786
$51$	3.41421	0.478086
$52$	5.17157	0.717168
$53$	0.343146	0.0471347	0.0235673	−	0.999722i	$-0.492498\pi$
$53$	0.343146	0.0471347	0.0235673	+	0.999722i	$0.492498\pi$
$54$	−8.00000	−1.08866
$55$	2.58579	0.348667
$56$	0.928932	0.124134
$57$	9.65685	1.27908
$58$	−2.00000	−0.262613
$59$	−9.17157	−1.19404	−0.597019	−	0.802227i	$-0.703648\pi$
$59$	−9.17157	−1.19404	−0.597019	+	0.802227i	$0.703648\pi$
$60$	−6.24264	−0.805921
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	1.75736	0.223185
$63$	−5.07107	−0.638894
$64$	−4.17157	−0.521447
$65$	2.82843	0.350823
$66$	3.65685	0.450128
$67$	−3.17157	−0.387469	−0.193735	−	0.981054i	$-0.562060\pi$
$67$	−3.17157	−0.387469	−0.193735	+	0.981054i	$0.562060\pi$
$68$	1.82843	0.221729
$69$	11.6569	1.40332
$70$	0.242641	0.0290011
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	−13.7279	−1.61785
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	2.68629	0.312275
$75$	−3.41421	−0.394239
$76$	5.17157	0.593220
$77$	1.51472	0.172618
$78$	4.00000	0.452911
$79$	5.41421	0.609147	0.304573	−	0.952489i	$-0.401486\pi$
$79$	5.41421	0.609147	0.304573	+	0.952489i	$0.401486\pi$
$80$	−3.00000	−0.335410
$81$	39.9706	4.44117
$82$	−2.68629	−0.296651
$83$	9.31371	1.02231	0.511156	−	0.859488i	$-0.329217\pi$
$83$	9.31371	1.02231	0.511156	+	0.859488i	$0.329217\pi$
$84$	−3.65685	−0.398996
$85$	1.00000	0.108465
$86$	3.17157	0.341999
$87$	16.4853	1.76741
$88$	4.10051	0.437115
$89$	−2.34315	−0.248373	−0.124186	−	0.992259i	$-0.539632\pi$
$89$	−2.34315	−0.248373	−0.124186	+	0.992259i	$0.539632\pi$
$90$	−3.58579	−0.377975
$91$	1.65685	0.173686
$92$	6.24264	0.650840
$93$	−14.4853	−1.50205
$94$	−2.00000	−0.206284
$95$	2.82843	0.290191
$96$	−15.0711	−1.53818
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	−2.75736	−0.278535
$99$	−22.3848	−2.24975
$100$	−1.82843	−0.182843
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	1.41421	0.140028
$103$	−0.828427	−0.0816274	−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427	−0.0816274	−0.0408137	+	0.999167i	$0.512995\pi$
$104$	4.48528	0.439818
$105$	−2.00000	−0.195180
$106$	0.142136	0.0138054
$107$	11.8995	1.15037	0.575184	−	0.818024i	$-0.304931\pi$
$107$	11.8995	1.15037	0.575184	+	0.818024i	$0.304931\pi$
$108$	35.3137	3.39806
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	1.07107	0.102122
$111$	−22.1421	−2.10164
$112$	−1.75736	−0.166055
$113$	−3.17157	−0.298356	−0.149178	−	0.988810i	$-0.547663\pi$
$113$	−3.17157	−0.298356	−0.149178	+	0.988810i	$0.547663\pi$
$114$	4.00000	0.374634
$115$	3.41421	0.318377
$116$	8.82843	0.819699
$117$	−24.4853	−2.26367
$118$	−3.79899	−0.349725
$119$	0.585786	0.0536990
$120$	−5.41421	−0.494248
$121$	−4.31371	−0.392155
$122$	3.17157	0.287141
$123$	22.1421	1.99649
$124$	−7.75736	−0.696631
$125$	−1.00000	−0.0894427
$126$	−2.10051	−0.187128
$127$	−17.3137	−1.53634	−0.768172	−	0.640244i	$-0.778833\pi$
$127$	−17.3137	−1.53634	−0.768172	+	0.640244i	$0.778833\pi$
$128$	−10.5563	−0.933058
$129$	−26.1421	−2.30169
$130$	1.17157	0.102754
$131$	−13.8995	−1.21440	−0.607202	−	0.794547i	$-0.707708\pi$
$131$	−13.8995	−1.21440	−0.607202	+	0.794547i	$0.707708\pi$
$132$	−16.1421	−1.40499
$133$	1.65685	0.143667
$134$	−1.31371	−0.113487
$135$	19.3137	1.66226
$136$	1.58579	0.135980
$137$	1.17157	0.100094	0.0500471	−	0.998747i	$-0.484063\pi$
$137$	1.17157	0.100094	0.0500471	+	0.998747i	$0.484063\pi$
$138$	4.82843	0.411023
$139$	17.8995	1.51822	0.759108	−	0.650965i	$-0.225636\pi$
$139$	17.8995	1.51822	0.759108	+	0.650965i	$0.225636\pi$
$140$	−1.07107	−0.0905218
$141$	16.4853	1.38831
$142$	−1.75736	−0.147474
$143$	7.31371	0.611603
$144$	25.9706	2.16421
$145$	4.82843	0.400979
$146$	−2.00000	−0.165521
$147$	22.7279	1.87457
$148$	−11.8579	−0.974710
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−1.41421	−0.115470
$151$	7.51472	0.611539	0.305770	−	0.952106i	$-0.401086\pi$
$151$	7.51472	0.611539	0.305770	+	0.952106i	$0.401086\pi$
$152$	4.48528	0.363804
$153$	−8.65685	−0.699865
$154$	0.627417	0.0505587
$155$	−4.24264	−0.340777
$156$	−17.6569	−1.41368
$157$	−21.3137	−1.70102	−0.850510	−	0.525960i	$-0.823706\pi$
$157$	−21.3137	−1.70102	−0.850510	+	0.525960i	$0.823706\pi$
$158$	2.24264	0.178415
$159$	−1.17157	−0.0929118
$160$	−4.41421	−0.348974
$161$	2.00000	0.157622
$162$	16.5563	1.30079
$163$	0.585786	0.0458823	0.0229412	−	0.999737i	$-0.492697\pi$
$163$	0.585786	0.0458823	0.0229412	+	0.999737i	$0.492697\pi$
$164$	11.8579	0.925944
$165$	−8.82843	−0.687292
$166$	3.85786	0.299428
$167$	2.24264	0.173541	0.0867704	−	0.996228i	$-0.472345\pi$
$167$	2.24264	0.173541	0.0867704	+	0.996228i	$0.472345\pi$
$168$	−3.17157	−0.244692
$169$	−5.00000	−0.384615
$170$	0.414214	0.0317687
$171$	−24.4853	−1.87244
$172$	−14.0000	−1.06749
$173$	12.8284	0.975327	0.487664	−	0.873032i	$-0.337849\pi$
$173$	12.8284	0.975327	0.487664	+	0.873032i	$0.337849\pi$
$174$	6.82843	0.517662
$175$	−0.585786	−0.0442813
$176$	−7.75736	−0.584733
$177$	31.3137	2.35368
$178$	−0.970563	−0.0727468
$179$	−6.82843	−0.510381	−0.255190	−	0.966891i	$-0.582138\pi$
$179$	−6.82843	−0.510381	−0.255190	+	0.966891i	$0.582138\pi$
$180$	15.8284	1.17978
$181$	2.48528	0.184730	0.0923648	−	0.995725i	$-0.470557\pi$
$181$	2.48528	0.184730	0.0923648	+	0.995725i	$0.470557\pi$
$182$	0.686292	0.0508713
$183$	−26.1421	−1.93248
$184$	5.41421	0.399141
$185$	−6.48528	−0.476807
$186$	−6.00000	−0.439941
$187$	2.58579	0.189091
$188$	8.82843	0.643879
$189$	11.3137	0.822951
$190$	1.17157	0.0849948
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	14.2426	1.02787
$193$	−20.8284	−1.49926	−0.749631	−	0.661855i	$-0.769769\pi$
$193$	−20.8284	−1.49926	−0.749631	+	0.661855i	$0.769769\pi$
$194$	1.51472	0.108750
$195$	−9.65685	−0.691542
$196$	12.1716	0.869398
$197$	12.8284	0.913988	0.456994	−	0.889470i	$-0.348926\pi$
$197$	12.8284	0.913988	0.456994	+	0.889470i	$0.348926\pi$
$198$	−9.27208	−0.658938
$199$	24.2426	1.71852	0.859258	−	0.511543i	$-0.170926\pi$
$199$	24.2426	1.71852	0.859258	+	0.511543i	$0.170926\pi$
$200$	−1.58579	−0.112132
$201$	10.8284	0.763778
$202$	−3.31371	−0.233152
$203$	2.82843	0.198517
$204$	−6.24264	−0.437072
$205$	6.48528	0.452952
$206$	−0.343146	−0.0239081
$207$	−29.5563	−2.05431
$208$	−8.48528	−0.588348
$209$	7.31371	0.505900
$210$	−0.828427	−0.0571669
$211$	−2.10051	−0.144605	−0.0723024	−	0.997383i	$-0.523035\pi$
$211$	−2.10051	−0.144605	−0.0723024	+	0.997383i	$0.523035\pi$
$212$	−0.627417	−0.0430912
$213$	14.4853	0.992515
$214$	4.92893	0.336935
$215$	−7.65685	−0.522193
$216$	30.6274	2.08393
$217$	−2.48528	−0.168712
$218$	−7.17157	−0.485720
$219$	16.4853	1.11397
$220$	−4.72792	−0.318756
$221$	2.82843	0.190261
$222$	−9.17157	−0.615556
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	−2.58579	−0.172770
$225$	8.65685	0.577124
$226$	−1.31371	−0.0873866
$227$	−22.7279	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$227$	−22.7279	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$228$	−17.6569	−1.16935
$229$	−0.686292	−0.0453514	−0.0226757	−	0.999743i	$-0.507219\pi$
$229$	−0.686292	−0.0453514	−0.0226757	+	0.999743i	$0.507219\pi$
$230$	1.41421	0.0932505
$231$	−5.17157	−0.340265
$232$	7.65685	0.502697
$233$	9.31371	0.610161	0.305081	−	0.952327i	$-0.401317\pi$
$233$	9.31371	0.610161	0.305081	+	0.952327i	$0.401317\pi$
$234$	−10.1421	−0.663012
$235$	4.82843	0.314972
$236$	16.7696	1.09160
$237$	−18.4853	−1.20075
$238$	0.242641	0.0157281
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	10.2426	0.661160
$241$	12.8284	0.826352	0.413176	−	0.910651i	$-0.364419\pi$
$241$	12.8284	0.826352	0.413176	+	0.910651i	$0.364419\pi$
$242$	−1.78680	−0.114860
$243$	−78.5269	−5.03750
$244$	−14.0000	−0.896258
$245$	6.65685	0.425291
$246$	9.17157	0.584758
$247$	8.00000	0.509028
$248$	−6.72792	−0.427223
$249$	−31.7990	−2.01518
$250$	−0.414214	−0.0261972
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	9.27208	0.584086
$253$	8.82843	0.555038
$254$	−7.17157	−0.449985
$255$	−3.41421	−0.213806
$256$	3.97056	0.248160
$257$	−2.82843	−0.176432	−0.0882162	−	0.996101i	$-0.528117\pi$
$257$	−2.82843	−0.176432	−0.0882162	+	0.996101i	$0.528117\pi$
$258$	−10.8284	−0.674148
$259$	−3.79899	−0.236058
$260$	−5.17157	−0.320727
$261$	−41.7990	−2.58729
$262$	−5.75736	−0.355691
$263$	9.31371	0.574308	0.287154	−	0.957884i	$-0.407291\pi$
$263$	9.31371	0.574308	0.287154	+	0.957884i	$0.407291\pi$
$264$	−14.0000	−0.861640
$265$	−0.343146	−0.0210793
$266$	0.686292	0.0420792
$267$	8.00000	0.489592
$268$	5.79899	0.354230
$269$	18.9706	1.15666	0.578328	−	0.815804i	$-0.303705\pi$
$269$	18.9706	1.15666	0.578328	+	0.815804i	$0.303705\pi$
$270$	8.00000	0.486864
$271$	−13.6569	−0.829595	−0.414797	−	0.909914i	$-0.636148\pi$
$271$	−13.6569	−0.829595	−0.414797	+	0.909914i	$0.636148\pi$
$272$	−3.00000	−0.181902
$273$	−5.65685	−0.342368
$274$	0.485281	0.0293169
$275$	−2.58579	−0.155929
$276$	−21.3137	−1.28293
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	7.41421	0.444675
$279$	36.7279	2.19884
$280$	−0.928932	−0.0555143
$281$	−4.34315	−0.259090	−0.129545	−	0.991574i	$-0.541352\pi$
$281$	−4.34315	−0.259090	−0.129545	+	0.991574i	$0.541352\pi$
$282$	6.82843	0.406627
$283$	14.2426	0.846637	0.423319	−	0.905981i	$-0.360865\pi$
$283$	14.2426	0.846637	0.423319	+	0.905981i	$0.360865\pi$
$284$	7.75736	0.460315
$285$	−9.65685	−0.572023
$286$	3.02944	0.179134
$287$	3.79899	0.224247
$288$	38.2132	2.25173
$289$	1.00000	0.0588235
$290$	2.00000	0.117444
$291$	−12.4853	−0.731900
$292$	8.82843	0.516645
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	9.41421	0.549048
$295$	9.17157	0.533990
$296$	−10.2843	−0.597761
$297$	49.9411	2.89788
$298$	0.828427	0.0479895
$299$	9.65685	0.558470
$300$	6.24264	0.360419
$301$	−4.48528	−0.258527
$302$	3.11270	0.179116
$303$	27.3137	1.56913
$304$	−8.48528	−0.486664
$305$	−7.65685	−0.438430
$306$	−3.58579	−0.204986
$307$	−16.8284	−0.960449	−0.480225	−	0.877146i	$-0.659445\pi$
$307$	−16.8284	−0.960449	−0.480225	+	0.877146i	$0.659445\pi$
$308$	−2.76955	−0.157810
$309$	2.82843	0.160904
$310$	−1.75736	−0.0998113
$311$	15.0711	0.854602	0.427301	−	0.904109i	$-0.359464\pi$
$311$	15.0711	0.854602	0.427301	+	0.904109i	$0.359464\pi$
$312$	−15.3137	−0.866968
$313$	−5.79899	−0.327778	−0.163889	−	0.986479i	$-0.552404\pi$
$313$	−5.79899	−0.327778	−0.163889	+	0.986479i	$0.552404\pi$
$314$	−8.82843	−0.498217
$315$	5.07107	0.285722
$316$	−9.89949	−0.556890
$317$	−21.3137	−1.19710	−0.598549	−	0.801087i	$-0.704256\pi$
$317$	−21.3137	−1.19710	−0.598549	+	0.801087i	$0.704256\pi$
$318$	−0.485281	−0.0272132
$319$	12.4853	0.699042
$320$	4.17157	0.233198
$321$	−40.6274	−2.26760
$322$	0.828427	0.0461664
$323$	2.82843	0.157378
$324$	−73.0833	−4.06018
$325$	−2.82843	−0.156893
$326$	0.242641	0.0134386
$327$	59.1127	3.26894
$328$	10.2843	0.567854
$329$	2.82843	0.155936
$330$	−3.65685	−0.201303
$331$	26.8284	1.47462	0.737312	−	0.675553i	$-0.236095\pi$
$331$	26.8284	1.47462	0.737312	+	0.675553i	$0.236095\pi$
$332$	−17.0294	−0.934612
$333$	56.1421	3.07657
$334$	0.928932	0.0508289
$335$	3.17157	0.173282
$336$	6.00000	0.327327
$337$	−20.6274	−1.12365	−0.561824	−	0.827257i	$-0.689900\pi$
$337$	−20.6274	−1.12365	−0.561824	+	0.827257i	$0.689900\pi$
$338$	−2.07107	−0.112651
$339$	10.8284	0.588119
$340$	−1.82843	−0.0991604
$341$	−10.9706	−0.594089
$342$	−10.1421	−0.548424
$343$	8.00000	0.431959
$344$	−12.1421	−0.654660
$345$	−11.6569	−0.627584
$346$	5.31371	0.285667
$347$	−23.6985	−1.27220	−0.636101	−	0.771606i	$-0.719454\pi$
$347$	−23.6985	−1.27220	−0.636101	+	0.771606i	$0.719454\pi$
$348$	−30.1421	−1.61579
$349$	31.6569	1.69455	0.847276	−	0.531152i	$-0.178241\pi$
$349$	31.6569	1.69455	0.847276	+	0.531152i	$0.178241\pi$
$350$	−0.242641	−0.0129697
$351$	54.6274	2.91580
$352$	−11.4142	−0.608380
$353$	−27.6569	−1.47203	−0.736013	−	0.676967i	$-0.763294\pi$
$353$	−27.6569	−1.47203	−0.736013	+	0.676967i	$0.763294\pi$
$354$	12.9706	0.689378
$355$	4.24264	0.225176
$356$	4.28427	0.227066
$357$	−2.00000	−0.105851
$358$	−2.82843	−0.149487
$359$	−31.7990	−1.67829	−0.839143	−	0.543910i	$-0.816943\pi$
$359$	−31.7990	−1.67829	−0.839143	+	0.543910i	$0.816943\pi$
$360$	13.7279	0.723525
$361$	−11.0000	−0.578947
$362$	1.02944	0.0541060
$363$	14.7279	0.773015
$364$	−3.02944	−0.158786
$365$	4.82843	0.252731
$366$	−10.8284	−0.566011
$367$	−14.2426	−0.743460	−0.371730	−	0.928341i	$-0.621235\pi$
$367$	−14.2426	−0.743460	−0.371730	+	0.928341i	$0.621235\pi$
$368$	−10.2426	−0.533935
$369$	−56.1421	−2.92264
$370$	−2.68629	−0.139654
$371$	−0.201010	−0.0104359
$372$	26.4853	1.37320
$373$	11.7990	0.610929	0.305464	−	0.952204i	$-0.401188\pi$
$373$	11.7990	0.610929	0.305464	+	0.952204i	$0.401188\pi$
$374$	1.07107	0.0553836
$375$	3.41421	0.176309
$376$	7.65685	0.394872
$377$	13.6569	0.703364
$378$	4.68629	0.241037
$379$	−26.8701	−1.38022	−0.690111	−	0.723703i	$-0.742438\pi$
$379$	−26.8701	−1.38022	−0.690111	+	0.723703i	$0.742438\pi$
$380$	−5.17157	−0.265296
$381$	59.1127	3.02844
$382$	4.97056	0.254316
$383$	−34.2843	−1.75184	−0.875922	−	0.482452i	$-0.839746\pi$
$383$	−34.2843	−1.75184	−0.875922	+	0.482452i	$0.839746\pi$
$384$	36.0416	1.83924
$385$	−1.51472	−0.0771972
$386$	−8.62742	−0.439124
$387$	66.2843	3.36942
$388$	−6.68629	−0.339445
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	−4.00000	−0.202548
$391$	3.41421	0.172664
$392$	10.5563	0.533176
$393$	47.4558	2.39383
$394$	5.31371	0.267701
$395$	−5.41421	−0.272419
$396$	40.9289	2.05676
$397$	13.3137	0.668196	0.334098	−	0.942538i	$-0.391568\pi$
$397$	13.3137	0.668196	0.334098	+	0.942538i	$0.391568\pi$
$398$	10.0416	0.503341
$399$	−5.65685	−0.283197
$400$	3.00000	0.150000
$401$	−16.3431	−0.816138	−0.408069	−	0.912951i	$-0.633798\pi$
$401$	−16.3431	−0.816138	−0.408069	+	0.912951i	$0.633798\pi$
$402$	4.48528	0.223706
$403$	−12.0000	−0.597763
$404$	14.6274	0.727741
$405$	−39.9706	−1.98615
$406$	1.17157	0.0581442
$407$	−16.7696	−0.831236
$408$	−5.41421	−0.268044
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	2.68629	0.132666
$411$	−4.00000	−0.197305
$412$	1.51472	0.0746248
$413$	5.37258	0.264368
$414$	−12.2426	−0.601693
$415$	−9.31371	−0.457192
$416$	−12.4853	−0.612141
$417$	−61.1127	−2.99270
$418$	3.02944	0.148175
$419$	12.2426	0.598092	0.299046	−	0.954239i	$-0.403332\pi$
$419$	12.2426	0.598092	0.299046	+	0.954239i	$0.403332\pi$
$420$	3.65685	0.178436
$421$	−28.9706	−1.41194	−0.705969	−	0.708242i	$-0.749488\pi$
$421$	−28.9706	−1.41194	−0.705969	+	0.708242i	$0.749488\pi$
$422$	−0.870058	−0.0423537
$423$	−41.7990	−2.03234
$424$	−0.544156	−0.0264265
$425$	−1.00000	−0.0485071
$426$	6.00000	0.290701
$427$	−4.48528	−0.217058
$428$	−21.7574	−1.05168
$429$	−24.9706	−1.20559
$430$	−3.17157	−0.152947
$431$	1.41421	0.0681203	0.0340601	−	0.999420i	$-0.489156\pi$
$431$	1.41421	0.0681203	0.0340601	+	0.999420i	$0.489156\pi$
$432$	−57.9411	−2.78769
$433$	2.82843	0.135926	0.0679628	−	0.997688i	$-0.478350\pi$
$433$	2.82843	0.135926	0.0679628	+	0.997688i	$0.478350\pi$
$434$	−1.02944	−0.0494146
$435$	−16.4853	−0.790409
$436$	31.6569	1.51609
$437$	9.65685	0.461950
$438$	6.82843	0.326275
$439$	5.41421	0.258406	0.129203	−	0.991618i	$-0.458758\pi$
$439$	5.41421	0.258406	0.129203	+	0.991618i	$0.458758\pi$
$440$	−4.10051	−0.195484
$441$	−57.6274	−2.74416
$442$	1.17157	0.0557260
$443$	−14.4853	−0.688216	−0.344108	−	0.938930i	$-0.611819\pi$
$443$	−14.4853	−0.688216	−0.344108	+	0.938930i	$0.611819\pi$
$444$	40.4853	1.92135
$445$	2.34315	0.111076
$446$	2.48528	0.117681
$447$	−6.82843	−0.322974
$448$	2.44365	0.115452
$449$	−26.4853	−1.24992	−0.624959	−	0.780658i	$-0.714884\pi$
$449$	−26.4853	−1.24992	−0.624959	+	0.780658i	$0.714884\pi$
$450$	3.58579	0.169036
$451$	16.7696	0.789647
$452$	5.79899	0.272762
$453$	−25.6569	−1.20546
$454$	−9.41421	−0.441831
$455$	−1.65685	−0.0776745
$456$	−15.3137	−0.717130
$457$	39.1127	1.82961	0.914807	−	0.403890i	$-0.132342\pi$
$457$	39.1127	1.82961	0.914807	+	0.403890i	$0.132342\pi$
$458$	−0.284271	−0.0132831
$459$	19.3137	0.901487
$460$	−6.24264	−0.291065
$461$	41.5980	1.93741	0.968706	−	0.248213i	$-0.0798432\pi$
$461$	41.5980	1.93741	0.968706	+	0.248213i	$0.0798432\pi$
$462$	−2.14214	−0.0996612
$463$	−3.17157	−0.147395	−0.0736977	−	0.997281i	$-0.523480\pi$
$463$	−3.17157	−0.147395	−0.0736977	+	0.997281i	$0.523480\pi$
$464$	−14.4853	−0.672462
$465$	14.4853	0.671739
$466$	3.85786	0.178712
$467$	−0.343146	−0.0158789	−0.00793945	−	0.999968i	$-0.502527\pi$
$467$	−0.343146	−0.0158789	−0.00793945	+	0.999968i	$0.502527\pi$
$468$	44.7696	2.06947
$469$	1.85786	0.0857882
$470$	2.00000	0.0922531
$471$	72.7696	3.35304
$472$	14.5442	0.669449
$473$	−19.7990	−0.910359
$474$	−7.65685	−0.351691
$475$	−2.82843	−0.129777
$476$	−1.07107	−0.0490923
$477$	2.97056	0.136013
$478$	8.28427	0.378914
$479$	−15.7574	−0.719972	−0.359986	−	0.932958i	$-0.617219\pi$
$479$	−15.7574	−0.719972	−0.359986	+	0.932958i	$0.617219\pi$
$480$	15.0711	0.687897
$481$	−18.3431	−0.836375
$482$	5.31371	0.242033
$483$	−6.82843	−0.310704
$484$	7.88730	0.358514
$485$	−3.65685	−0.166049
$486$	−32.5269	−1.47545
$487$	−3.89949	−0.176703	−0.0883515	−	0.996089i	$-0.528160\pi$
$487$	−3.89949	−0.176703	−0.0883515	+	0.996089i	$0.528160\pi$
$488$	−12.1421	−0.549649
$489$	−2.00000	−0.0904431
$490$	2.75736	0.124565
$491$	−16.4853	−0.743970	−0.371985	−	0.928239i	$-0.621323\pi$
$491$	−16.4853	−0.743970	−0.371985	+	0.928239i	$0.621323\pi$
$492$	−40.4853	−1.82522
$493$	4.82843	0.217461
$494$	3.31371	0.149091
$495$	22.3848	1.00612
$496$	12.7279	0.571501
$497$	2.48528	0.111480
$498$	−13.1716	−0.590232
$499$	12.2426	0.548056	0.274028	−	0.961722i	$-0.411644\pi$
$499$	12.2426	0.548056	0.274028	+	0.961722i	$0.411644\pi$
$500$	1.82843	0.0817697
$501$	−7.65685	−0.342083
$502$	−4.97056	−0.221847
$503$	31.6985	1.41337	0.706683	−	0.707531i	$-0.250191\pi$
$503$	31.6985	1.41337	0.706683	+	0.707531i	$0.250191\pi$
$504$	8.04163	0.358203
$505$	8.00000	0.355995
$506$	3.65685	0.162567
$507$	17.0711	0.758153
$508$	31.6569	1.40455
$509$	20.6274	0.914294	0.457147	−	0.889391i	$-0.348871\pi$
$509$	20.6274	0.914294	0.457147	+	0.889391i	$0.348871\pi$
$510$	−1.41421	−0.0626224
$511$	2.82843	0.125122
$512$	22.7574	1.00574
$513$	54.6274	2.41186
$514$	−1.17157	−0.0516759
$515$	0.828427	0.0365049
$516$	47.7990	2.10423
$517$	12.4853	0.549102
$518$	−1.57359	−0.0691397
$519$	−43.7990	−1.92256
$520$	−4.48528	−0.196693
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−17.3137	−0.757800
$523$	−0.142136	−0.00621516	−0.00310758	−	0.999995i	$-0.500989\pi$
$523$	−0.142136	−0.00621516	−0.00310758	+	0.999995i	$0.500989\pi$
$524$	25.4142	1.11023
$525$	2.00000	0.0872872
$526$	3.85786	0.168211
$527$	−4.24264	−0.184812
$528$	26.4853	1.15262
$529$	−11.3431	−0.493180
$530$	−0.142136	−0.00617398
$531$	−79.3970	−3.44554
$532$	−3.02944	−0.131343
$533$	18.3431	0.794530
$534$	3.31371	0.143398
$535$	−11.8995	−0.514460
$536$	5.02944	0.217239
$537$	23.3137	1.00606
$538$	7.85786	0.338777
$539$	17.2132	0.741425
$540$	−35.3137	−1.51966
$541$	−42.7696	−1.83881	−0.919403	−	0.393316i	$-0.871328\pi$
$541$	−42.7696	−1.83881	−0.919403	+	0.393316i	$0.871328\pi$
$542$	−5.65685	−0.242983
$543$	−8.48528	−0.364138
$544$	−4.41421	−0.189258
$545$	17.3137	0.741638
$546$	−2.34315	−0.100277
$547$	7.21320	0.308414	0.154207	−	0.988039i	$-0.450718\pi$
$547$	7.21320	0.308414	0.154207	+	0.988039i	$0.450718\pi$
$548$	−2.14214	−0.0915075
$549$	66.2843	2.82894
$550$	−1.07107	−0.0456705
$551$	13.6569	0.581802
$552$	−18.4853	−0.786786
$553$	−3.17157	−0.134869
$554$	−4.14214	−0.175982
$555$	22.1421	0.939881
$556$	−32.7279	−1.38797
$557$	26.8284	1.13676	0.568378	−	0.822767i	$-0.307571\pi$
$557$	26.8284	1.13676	0.568378	+	0.822767i	$0.307571\pi$
$558$	15.2132	0.644026
$559$	−21.6569	−0.915987
$560$	1.75736	0.0742620
$561$	−8.82843	−0.372736
$562$	−1.79899	−0.0758858
$563$	−20.3431	−0.857361	−0.428681	−	0.903456i	$-0.641021\pi$
$563$	−20.3431	−0.857361	−0.428681	+	0.903456i	$0.641021\pi$
$564$	−30.1421	−1.26921
$565$	3.17157	0.133429
$566$	5.89949	0.247974
$567$	−23.4142	−0.983305
$568$	6.72792	0.282297
$569$	−26.2843	−1.10189	−0.550947	−	0.834540i	$-0.685733\pi$
$569$	−26.2843	−1.10189	−0.550947	+	0.834540i	$0.685733\pi$
$570$	−4.00000	−0.167542
$571$	−4.44365	−0.185961	−0.0929805	−	0.995668i	$-0.529639\pi$
$571$	−4.44365	−0.185961	−0.0929805	+	0.995668i	$0.529639\pi$
$572$	−13.3726	−0.559136
$573$	−40.9706	−1.71157
$574$	1.57359	0.0656805
$575$	−3.41421	−0.142383
$576$	−36.1127	−1.50470
$577$	22.8284	0.950360	0.475180	−	0.879889i	$-0.342383\pi$
$577$	22.8284	0.950360	0.475180	+	0.879889i	$0.342383\pi$
$578$	0.414214	0.0172290
$579$	71.1127	2.95534
$580$	−8.82843	−0.366580
$581$	−5.45584	−0.226347
$582$	−5.17157	−0.214369
$583$	−0.887302	−0.0367483
$584$	7.65685	0.316843
$585$	24.4853	1.01234
$586$	7.45584	0.307998
$587$	28.6274	1.18158	0.590790	−	0.806825i	$-0.298816\pi$
$587$	28.6274	1.18158	0.590790	+	0.806825i	$0.298816\pi$
$588$	−41.5563	−1.71375
$589$	−12.0000	−0.494451
$590$	3.79899	0.156402
$591$	−43.7990	−1.80165
$592$	19.4558	0.799630
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	20.6863	0.848769
$595$	−0.585786	−0.0240149
$596$	−3.65685	−0.149791
$597$	−82.7696	−3.38753
$598$	4.00000	0.163572
$599$	45.9411	1.87710	0.938552	−	0.345139i	$-0.112168\pi$
$599$	45.9411	1.87710	0.938552	+	0.345139i	$0.112168\pi$
$600$	5.41421	0.221034
$601$	−25.7990	−1.05236	−0.526181	−	0.850372i	$-0.676377\pi$
$601$	−25.7990	−1.05236	−0.526181	+	0.850372i	$0.676377\pi$
$602$	−1.85786	−0.0757209
$603$	−27.4558	−1.11809
$604$	−13.7401	−0.559077
$605$	4.31371	0.175377
$606$	11.3137	0.459588
$607$	−4.58579	−0.186131	−0.0930657	−	0.995660i	$-0.529667\pi$
$607$	−4.58579	−0.186131	−0.0930657	+	0.995660i	$0.529667\pi$
$608$	−12.4853	−0.506345
$609$	−9.65685	−0.391315
$610$	−3.17157	−0.128413
$611$	13.6569	0.552497
$612$	15.8284	0.639826
$613$	−46.9706	−1.89712	−0.948562	−	0.316593i	$-0.897461\pi$
$613$	−46.9706	−1.89712	−0.948562	+	0.316593i	$0.897461\pi$
$614$	−6.97056	−0.281309
$615$	−22.1421	−0.892857
$616$	−2.40202	−0.0967802
$617$	25.5147	1.02718	0.513592	−	0.858035i	$-0.328315\pi$
$617$	25.5147	1.02718	0.513592	+	0.858035i	$0.328315\pi$
$618$	1.17157	0.0471276
$619$	−16.9289	−0.680431	−0.340216	−	0.940347i	$-0.610500\pi$
$619$	−16.9289	−0.680431	−0.340216	+	0.940347i	$0.610500\pi$
$620$	7.75736	0.311543
$621$	65.9411	2.64613
$622$	6.24264	0.250307
$623$	1.37258	0.0549914
$624$	28.9706	1.15975
$625$	1.00000	0.0400000
$626$	−2.40202	−0.0960040
$627$	−24.9706	−0.997228
$628$	38.9706	1.55509
$629$	−6.48528	−0.258585
$630$	2.10051	0.0836861
$631$	−15.7990	−0.628948	−0.314474	−	0.949266i	$-0.601828\pi$
$631$	−15.7990	−0.628948	−0.314474	+	0.949266i	$0.601828\pi$
$632$	−8.58579	−0.341524
$633$	7.17157	0.285044
$634$	−8.82843	−0.350622
$635$	17.3137	0.687074
$636$	2.14214	0.0849412
$637$	18.8284	0.746009
$638$	5.17157	0.204745
$639$	−36.7279	−1.45293
$640$	10.5563	0.417276
$641$	28.1421	1.11155	0.555774	−	0.831334i	$-0.312422\pi$
$641$	28.1421	1.11155	0.555774	+	0.831334i	$0.312422\pi$
$642$	−16.8284	−0.664165
$643$	−47.6985	−1.88104	−0.940522	−	0.339732i	$-0.889664\pi$
$643$	−47.6985	−1.88104	−0.940522	+	0.339732i	$0.889664\pi$
$644$	−3.65685	−0.144100
$645$	26.1421	1.02935
$646$	1.17157	0.0460949
$647$	−20.8284	−0.818850	−0.409425	−	0.912344i	$-0.634271\pi$
$647$	−20.8284	−0.818850	−0.409425	+	0.912344i	$0.634271\pi$
$648$	−63.3848	−2.48999
$649$	23.7157	0.930924
$650$	−1.17157	−0.0459529
$651$	8.48528	0.332564
$652$	−1.07107	−0.0419463
$653$	18.4853	0.723385	0.361692	−	0.932297i	$-0.382199\pi$
$653$	18.4853	0.723385	0.361692	+	0.932297i	$0.382199\pi$
$654$	24.4853	0.957450
$655$	13.8995	0.543098
$656$	−19.4558	−0.759623
$657$	−41.7990	−1.63073
$658$	1.17157	0.0456727
$659$	−4.68629	−0.182552	−0.0912760	−	0.995826i	$-0.529095\pi$
$659$	−4.68629	−0.182552	−0.0912760	+	0.995826i	$0.529095\pi$
$660$	16.1421	0.628332
$661$	13.3137	0.517843	0.258922	−	0.965898i	$-0.416633\pi$
$661$	13.3137	0.517843	0.258922	+	0.965898i	$0.416633\pi$
$662$	11.1127	0.431907
$663$	−9.65685	−0.375041
$664$	−14.7696	−0.573170
$665$	−1.65685	−0.0642501
$666$	23.2548	0.901107
$667$	16.4853	0.638313
$668$	−4.10051	−0.158653
$669$	−20.4853	−0.792007
$670$	1.31371	0.0507530
$671$	−19.7990	−0.764332
$672$	8.82843	0.340564
$673$	−28.1421	−1.08480	−0.542400	−	0.840120i	$-0.682484\pi$
$673$	−28.1421	−1.08480	−0.542400	+	0.840120i	$0.682484\pi$
$674$	−8.54416	−0.329109
$675$	−19.3137	−0.743385
$676$	9.14214	0.351621
$677$	4.62742	0.177846	0.0889230	−	0.996038i	$-0.471657\pi$
$677$	4.62742	0.177846	0.0889230	+	0.996038i	$0.471657\pi$
$678$	4.48528	0.172256
$679$	−2.14214	−0.0822076
$680$	−1.58579	−0.0608121
$681$	77.5980	2.97356
$682$	−4.54416	−0.174005
$683$	−14.7279	−0.563548	−0.281774	−	0.959481i	$-0.590923\pi$
$683$	−14.7279	−0.563548	−0.281774	+	0.959481i	$0.590923\pi$
$684$	44.7696	1.71181
$685$	−1.17157	−0.0447635
$686$	3.31371	0.126518
$687$	2.34315	0.0893966
$688$	22.9706	0.875744
$689$	−0.970563	−0.0369755
$690$	−4.82843	−0.183815
$691$	−17.2132	−0.654821	−0.327411	−	0.944882i	$-0.606176\pi$
$691$	−17.2132	−0.654821	−0.327411	+	0.944882i	$0.606176\pi$
$692$	−23.4558	−0.891657
$693$	13.1127	0.498110
$694$	−9.81623	−0.372619
$695$	−17.8995	−0.678967
$696$	−26.1421	−0.990915
$697$	6.48528	0.245648
$698$	13.1127	0.496323
$699$	−31.7990	−1.20275
$700$	1.07107	0.0404826
$701$	30.3431	1.14604	0.573022	−	0.819540i	$-0.305771\pi$
$701$	30.3431	1.14604	0.573022	+	0.819540i	$0.305771\pi$
$702$	22.6274	0.854017
$703$	−18.3431	−0.691825
$704$	10.7868	0.406543
$705$	−16.4853	−0.620872
$706$	−11.4558	−0.431146
$707$	4.68629	0.176246
$708$	−57.2548	−2.15177
$709$	25.7990	0.968901	0.484451	−	0.874819i	$-0.339019\pi$
$709$	25.7990	0.968901	0.484451	+	0.874819i	$0.339019\pi$
$710$	1.75736	0.0659525
$711$	46.8701	1.75776
$712$	3.71573	0.139253
$713$	−14.4853	−0.542478
$714$	−0.828427	−0.0310031
$715$	−7.31371	−0.273517
$716$	12.4853	0.466597
$717$	−68.2843	−2.55012
$718$	−13.1716	−0.491559
$719$	−53.4975	−1.99512	−0.997560	−	0.0698205i	$-0.977757\pi$
$719$	−53.4975	−1.99512	−0.997560	+	0.0698205i	$0.977757\pi$
$720$	−25.9706	−0.967866
$721$	0.485281	0.0180728
$722$	−4.55635	−0.169570
$723$	−43.7990	−1.62890
$724$	−4.54416	−0.168882
$725$	−4.82843	−0.179323
$726$	6.10051	0.226411
$727$	11.4558	0.424874	0.212437	−	0.977175i	$-0.431860\pi$
$727$	11.4558	0.424874	0.212437	+	0.977175i	$0.431860\pi$
$728$	−2.62742	−0.0973786
$729$	148.196	5.48874
$730$	2.00000	0.0740233
$731$	−7.65685	−0.283199
$732$	47.7990	1.76670
$733$	46.2843	1.70955	0.854774	−	0.519000i	$-0.173696\pi$
$733$	46.2843	1.70955	0.854774	+	0.519000i	$0.173696\pi$
$734$	−5.89949	−0.217754
$735$	−22.7279	−0.838332
$736$	−15.0711	−0.555527
$737$	8.20101	0.302088
$738$	−23.2548	−0.856022
$739$	27.7990	1.02260	0.511301	−	0.859402i	$-0.329164\pi$
$739$	27.7990	1.02260	0.511301	+	0.859402i	$0.329164\pi$
$740$	11.8579	0.435904
$741$	−27.3137	−1.00339
$742$	−0.0832611	−0.00305661
$743$	14.0416	0.515137	0.257569	−	0.966260i	$-0.417079\pi$
$743$	14.0416	0.515137	0.257569	+	0.966260i	$0.417079\pi$
$744$	22.9706	0.842142
$745$	−2.00000	−0.0732743
$746$	4.88730	0.178937
$747$	80.6274	2.95000
$748$	−4.72792	−0.172870
$749$	−6.97056	−0.254699
$750$	1.41421	0.0516398
$751$	−46.1838	−1.68527	−0.842635	−	0.538485i	$-0.818997\pi$
$751$	−46.1838	−1.68527	−0.842635	+	0.538485i	$0.818997\pi$
$752$	−14.4853	−0.528224
$753$	40.9706	1.49305
$754$	5.65685	0.206010
$755$	−7.51472	−0.273489
$756$	−20.6863	−0.752353
$757$	41.1716	1.49641	0.748203	−	0.663470i	$-0.230917\pi$
$757$	41.1716	1.49641	0.748203	+	0.663470i	$0.230917\pi$
$758$	−11.1299	−0.404258
$759$	−30.1421	−1.09409
$760$	−4.48528	−0.162698
$761$	12.6863	0.459878	0.229939	−	0.973205i	$-0.426147\pi$
$761$	12.6863	0.459878	0.229939	+	0.973205i	$0.426147\pi$
$762$	24.4853	0.887008
$763$	10.1421	0.367170
$764$	−21.9411	−0.793802
$765$	8.65685	0.312989
$766$	−14.2010	−0.513103
$767$	25.9411	0.936680
$768$	−13.5563	−0.489173
$769$	−34.3431	−1.23845	−0.619223	−	0.785215i	$-0.712552\pi$
$769$	−34.3431	−1.23845	−0.619223	+	0.785215i	$0.712552\pi$
$770$	−0.627417	−0.0226105
$771$	9.65685	0.347783
$772$	38.0833	1.37065
$773$	27.1127	0.975176	0.487588	−	0.873074i	$-0.337877\pi$
$773$	27.1127	0.975176	0.487588	+	0.873074i	$0.337877\pi$
$774$	27.4558	0.986880
$775$	4.24264	0.152400
$776$	−5.79899	−0.208172
$777$	12.9706	0.465316
$778$	−6.62742	−0.237604
$779$	18.3431	0.657211
$780$	17.6569	0.632217
$781$	10.9706	0.392558
$782$	1.41421	0.0505722
$783$	93.2548	3.33266
$784$	−19.9706	−0.713234
$785$	21.3137	0.760719
$786$	19.6569	0.701137
$787$	−31.2132	−1.11263	−0.556315	−	0.830971i	$-0.687785\pi$
$787$	−31.2132	−1.11263	−0.556315	+	0.830971i	$0.687785\pi$
$788$	−23.4558	−0.835580
$789$	−31.7990	−1.13207
$790$	−2.24264	−0.0797896
$791$	1.85786	0.0660581
$792$	35.4975	1.26135
$793$	−21.6569	−0.769057
$794$	5.51472	0.195710
$795$	1.17157	0.0415514
$796$	−44.3259	−1.57109
$797$	−36.6274	−1.29741	−0.648705	−	0.761040i	$-0.724689\pi$
$797$	−36.6274	−1.29741	−0.648705	+	0.761040i	$0.724689\pi$
$798$	−2.34315	−0.0829465
$799$	4.82843	0.170817
$800$	4.41421	0.156066
$801$	−20.2843	−0.716709
$802$	−6.76955	−0.239041
$803$	12.4853	0.440596
$804$	−19.7990	−0.698257
$805$	−2.00000	−0.0704907
$806$	−4.97056	−0.175081
$807$	−64.7696	−2.28000
$808$	12.6863	0.446302
$809$	−31.6569	−1.11300	−0.556498	−	0.830849i	$-0.687855\pi$
$809$	−31.6569	−1.11300	−0.556498	+	0.830849i	$0.687855\pi$
$810$	−16.5563	−0.581731
$811$	18.5858	0.652635	0.326318	−	0.945260i	$-0.394192\pi$
$811$	18.5858	0.652635	0.326318	+	0.945260i	$0.394192\pi$
$812$	−5.17157	−0.181487
$813$	46.6274	1.63529
$814$	−6.94618	−0.243463
$815$	−0.585786	−0.0205192
$816$	10.2426	0.358564
$817$	−21.6569	−0.757677
$818$	−2.48528	−0.0868958
$819$	14.3431	0.501190
$820$	−11.8579	−0.414095
$821$	36.6274	1.27831	0.639153	−	0.769080i	$-0.279285\pi$
$821$	36.6274	1.27831	0.639153	+	0.769080i	$0.279285\pi$
$822$	−1.65685	−0.0577894
$823$	−25.0711	−0.873922	−0.436961	−	0.899480i	$-0.643945\pi$
$823$	−25.0711	−0.873922	−0.436961	+	0.899480i	$0.643945\pi$
$824$	1.31371	0.0457652
$825$	8.82843	0.307366
$826$	2.22540	0.0774315
$827$	−29.5563	−1.02777	−0.513887	−	0.857858i	$-0.671795\pi$
$827$	−29.5563	−1.02777	−0.513887	+	0.857858i	$0.671795\pi$
$828$	54.0416	1.87808
$829$	39.9411	1.38721	0.693606	−	0.720354i	$-0.256021\pi$
$829$	39.9411	1.38721	0.693606	+	0.720354i	$0.256021\pi$
$830$	−3.85786	−0.133908
$831$	34.1421	1.18438
$832$	11.7990	0.409056
$833$	6.65685	0.230646
$834$	−25.3137	−0.876542
$835$	−2.24264	−0.0776098
$836$	−13.3726	−0.462500
$837$	−81.9411	−2.83230
$838$	5.07107	0.175177
$839$	42.1838	1.45635	0.728173	−	0.685394i	$-0.240370\pi$
$839$	42.1838	1.45635	0.728173	+	0.685394i	$0.240370\pi$
$840$	3.17157	0.109430
$841$	−5.68629	−0.196079
$842$	−12.0000	−0.413547
$843$	14.8284	0.510718
$844$	3.84062	0.132200
$845$	5.00000	0.172005
$846$	−17.3137	−0.595258
$847$	2.52691	0.0868257
$848$	1.02944	0.0353510
$849$	−48.6274	−1.66889
$850$	−0.414214	−0.0142074
$851$	−22.1421	−0.759023
$852$	−26.4853	−0.907371
$853$	−31.1716	−1.06729	−0.533647	−	0.845707i	$-0.679179\pi$
$853$	−31.1716	−1.06729	−0.533647	+	0.845707i	$0.679179\pi$
$854$	−1.85786	−0.0635748
$855$	24.4853	0.837379
$856$	−18.8701	−0.644965
$857$	13.5147	0.461654	0.230827	−	0.972995i	$-0.425857\pi$
$857$	13.5147	0.461654	0.230827	+	0.972995i	$0.425857\pi$
$858$	−10.3431	−0.353109
$859$	36.7696	1.25456	0.627280	−	0.778793i	$-0.284168\pi$
$859$	36.7696	1.25456	0.627280	+	0.778793i	$0.284168\pi$
$860$	14.0000	0.477396
$861$	−12.9706	−0.442036
$862$	0.585786	0.0199520
$863$	6.48528	0.220762	0.110381	−	0.993889i	$-0.464793\pi$
$863$	6.48528	0.220762	0.110381	+	0.993889i	$0.464793\pi$
$864$	−85.2548	−2.90043
$865$	−12.8284	−0.436180
$866$	1.17157	0.0398117
$867$	−3.41421	−0.115953
$868$	4.54416	0.154239
$869$	−14.0000	−0.474917
$870$	−6.82843	−0.231505
$871$	8.97056	0.303956
$872$	27.4558	0.929772
$873$	31.6569	1.07142
$874$	4.00000	0.135302
$875$	0.585786	0.0198032
$876$	−30.1421	−1.01841
$877$	−2.28427	−0.0771344	−0.0385672	−	0.999256i	$-0.512279\pi$
$877$	−2.28427	−0.0771344	−0.0385672	+	0.999256i	$0.512279\pi$
$878$	2.24264	0.0756855
$879$	−61.4558	−2.07285
$880$	7.75736	0.261501
$881$	48.1421	1.62195	0.810975	−	0.585081i	$-0.198937\pi$
$881$	48.1421	1.62195	0.810975	+	0.585081i	$0.198937\pi$
$882$	−23.8701	−0.803747
$883$	15.1716	0.510564	0.255282	−	0.966867i	$-0.417832\pi$
$883$	15.1716	0.510564	0.255282	+	0.966867i	$0.417832\pi$
$884$	−5.17157	−0.173939
$885$	−31.3137	−1.05260
$886$	−6.00000	−0.201574
$887$	−14.9289	−0.501264	−0.250632	−	0.968082i	$-0.580638\pi$
$887$	−14.9289	−0.501264	−0.250632	+	0.968082i	$0.580638\pi$
$888$	35.1127	1.17831
$889$	10.1421	0.340156
$890$	0.970563	0.0325333
$891$	−103.355	−3.46253
$892$	−10.9706	−0.367322
$893$	13.6569	0.457009
$894$	−2.82843	−0.0945968
$895$	6.82843	0.228249
$896$	6.18377	0.206585
$897$	−32.9706	−1.10086
$898$	−10.9706	−0.366092
$899$	−20.4853	−0.683222
$900$	−15.8284	−0.527614
$901$	−0.343146	−0.0114318
$902$	6.94618	0.231282
$903$	15.3137	0.509608
$904$	5.02944	0.167277
$905$	−2.48528	−0.0826135
$906$	−10.6274	−0.353072
$907$	−6.72792	−0.223397	−0.111698	−	0.993742i	$-0.535629\pi$
$907$	−6.72792	−0.223397	−0.111698	+	0.993742i	$0.535629\pi$
$908$	41.5563	1.37910
$909$	−69.2548	−2.29704
$910$	−0.686292	−0.0227503
$911$	12.2426	0.405617	0.202808	−	0.979218i	$-0.434993\pi$
$911$	12.2426	0.405617	0.202808	+	0.979218i	$0.434993\pi$
$912$	28.9706	0.959311
$913$	−24.0833	−0.797040
$914$	16.2010	0.535882
$915$	26.1421	0.864232
$916$	1.25483	0.0414609
$917$	8.14214	0.268877
$918$	8.00000	0.264039
$919$	40.9706	1.35149	0.675747	−	0.737134i	$-0.263821\pi$
$919$	40.9706	1.35149	0.675747	+	0.737134i	$0.263821\pi$
$920$	−5.41421	−0.178501
$921$	57.4558	1.89323
$922$	17.2304	0.567455
$923$	12.0000	0.394985
$924$	9.45584	0.311074
$925$	6.48528	0.213235
$926$	−1.31371	−0.0431711
$927$	−7.17157	−0.235545
$928$	−21.3137	−0.699657
$929$	35.4558	1.16327	0.581634	−	0.813450i	$-0.302414\pi$
$929$	35.4558	1.16327	0.581634	+	0.813450i	$0.302414\pi$
$930$	6.00000	0.196748
$931$	18.8284	0.617077
$932$	−17.0294	−0.557818
$933$	−51.4558	−1.68459
$934$	−0.142136	−0.00465082
$935$	−2.58579	−0.0845643
$936$	38.8284	1.26915
$937$	34.2843	1.12002	0.560009	−	0.828486i	$-0.310798\pi$
$937$	34.2843	1.12002	0.560009	+	0.828486i	$0.310798\pi$
$938$	0.769553	0.0251268
$939$	19.7990	0.646116
$940$	−8.82843	−0.287952
$941$	−3.45584	−0.112657	−0.0563286	−	0.998412i	$-0.517939\pi$
$941$	−3.45584	−0.112657	−0.0563286	+	0.998412i	$0.517939\pi$
$942$	30.1421	0.982084
$943$	22.1421	0.721047
$944$	−27.5147	−0.895528
$945$	−11.3137	−0.368035
$946$	−8.20101	−0.266638
$947$	−35.8995	−1.16658	−0.583288	−	0.812265i	$-0.698234\pi$
$947$	−35.8995	−1.16658	−0.583288	+	0.812265i	$0.698234\pi$
$948$	33.7990	1.09774
$949$	13.6569	0.443320
$950$	−1.17157	−0.0380108
$951$	72.7696	2.35971
$952$	−0.928932	−0.0301069
$953$	−21.8579	−0.708046	−0.354023	−	0.935237i	$-0.615186\pi$
$953$	−21.8579	−0.708046	−0.354023	+	0.935237i	$0.615186\pi$
$954$	1.23045	0.0398372
$955$	−12.0000	−0.388311
$956$	−36.5685	−1.18271
$957$	−42.6274	−1.37795
$958$	−6.52691	−0.210875
$959$	−0.686292	−0.0221615
$960$	−14.2426	−0.459679
$961$	−13.0000	−0.419355
$962$	−7.59798	−0.244969
$963$	103.012	3.31952
$964$	−23.4558	−0.755462
$965$	20.8284	0.670491
$966$	−2.82843	−0.0910032
$967$	−1.02944	−0.0331045	−0.0165522	−	0.999863i	$-0.505269\pi$
$967$	−1.02944	−0.0331045	−0.0165522	+	0.999863i	$0.505269\pi$
$968$	6.84062	0.219866
$969$	−9.65685	−0.310223
$970$	−1.51472	−0.0486347
$971$	−8.20101	−0.263183	−0.131591	−	0.991304i	$-0.542009\pi$
$971$	−8.20101	−0.263183	−0.131591	+	0.991304i	$0.542009\pi$
$972$	143.581	4.60535
$973$	−10.4853	−0.336143
$974$	−1.61522	−0.0517551
$975$	9.65685	0.309267
$976$	22.9706	0.735270
$977$	59.2548	1.89573	0.947865	−	0.318672i	$-0.103237\pi$
$977$	59.2548	1.89573	0.947865	+	0.318672i	$0.103237\pi$
$978$	−0.828427	−0.0264902
$979$	6.05887	0.193642
$980$	−12.1716	−0.388807
$981$	−149.882	−4.78537
$982$	−6.82843	−0.217904
$983$	−41.3553	−1.31903	−0.659515	−	0.751691i	$-0.729238\pi$
$983$	−41.3553	−1.31903	−0.659515	+	0.751691i	$0.729238\pi$
$984$	−35.1127	−1.11935
$985$	−12.8284	−0.408748
$986$	2.00000	0.0636930
$987$	−9.65685	−0.307381
$988$	−14.6274	−0.465360
$989$	−26.1421	−0.831272
$990$	9.27208	0.294686
$991$	33.4142	1.06144	0.530719	−	0.847548i	$-0.321922\pi$
$991$	33.4142	1.06144	0.530719	+	0.847548i	$0.321922\pi$
$992$	18.7279	0.594612
$993$	−91.5980	−2.90677
$994$	1.02944	0.0326518
$995$	−24.2426	−0.768543
$996$	58.1421	1.84230
$997$	40.1421	1.27131	0.635657	−	0.771972i	$-0.280729\pi$
$997$	40.1421	1.27131	0.635657	+	0.771972i	$0.280729\pi$
$998$	5.07107	0.160522
$999$	−125.255	−3.96289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.a.b.1.2	✓	2
3.2	odd	2		765.2.a.i.1.1		2
4.3	odd	2		1360.2.a.o.1.2		2
5.2	odd	4		425.2.b.e.324.3		4
5.3	odd	4		425.2.b.e.324.2		4
5.4	even	2		425.2.a.f.1.1		2
7.6	odd	2		4165.2.a.q.1.2		2
8.3	odd	2		5440.2.a.ba.1.1		2
8.5	even	2		5440.2.a.bm.1.2		2
15.14	odd	2		3825.2.a.p.1.2		2
17.4	even	4		1445.2.d.f.866.2		4
17.13	even	4		1445.2.d.f.866.1		4
17.16	even	2		1445.2.a.f.1.2		2
20.19	odd	2		6800.2.a.ba.1.1		2
85.84	even	2		7225.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.b.1.2	✓	2	1.1	even	1	trivial
425.2.a.f.1.1		2	5.4	even	2
425.2.b.e.324.2		4	5.3	odd	4
425.2.b.e.324.3		4	5.2	odd	4
765.2.a.i.1.1		2	3.2	odd	2
1360.2.a.o.1.2		2	4.3	odd	2
1445.2.a.f.1.2		2	17.16	even	2
1445.2.d.f.866.1		4	17.13	even	4
1445.2.d.f.866.2		4	17.4	even	4
3825.2.a.p.1.2		2	15.14	odd	2
4165.2.a.q.1.2		2	7.6	odd	2
5440.2.a.ba.1.1		2	8.3	odd	2
5440.2.a.bm.1.2		2	8.5	even	2
6800.2.a.ba.1.1		2	20.19	odd	2
7225.2.a.o.1.1		2	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -3.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.41421 q^{6} -0.585786 q^{7} -1.58579 q^{8} +8.65685 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} -3.41421 q^{3} -1.82843 q^{4} -1.00000 q^{5} -1.41421 q^{6} -0.585786 q^{7} -1.58579 q^{8} +8.65685 q^{9} -0.414214 q^{10} -2.58579 q^{11} +6.24264 q^{12} -2.82843 q^{13} -0.242641 q^{14} +3.41421 q^{15} +3.00000 q^{16} -1.00000 q^{17} +3.58579 q^{18} -2.82843 q^{19} +1.82843 q^{20} +2.00000 q^{21} -1.07107 q^{22} -3.41421 q^{23} +5.41421 q^{24} +1.00000 q^{25} -1.17157 q^{26} -19.3137 q^{27} +1.07107 q^{28} -4.82843 q^{29} +1.41421 q^{30} +4.24264 q^{31} +4.41421 q^{32} +8.82843 q^{33} -0.414214 q^{34} +0.585786 q^{35} -15.8284 q^{36} +6.48528 q^{37} -1.17157 q^{38} +9.65685 q^{39} +1.58579 q^{40} -6.48528 q^{41} +0.828427 q^{42} +7.65685 q^{43} +4.72792 q^{44} -8.65685 q^{45} -1.41421 q^{46} -4.82843 q^{47} -10.2426 q^{48} -6.65685 q^{49} +0.414214 q^{50} +3.41421 q^{51} +5.17157 q^{52} +0.343146 q^{53} -8.00000 q^{54} +2.58579 q^{55} +0.928932 q^{56} +9.65685 q^{57} -2.00000 q^{58} -9.17157 q^{59} -6.24264 q^{60} +7.65685 q^{61} +1.75736 q^{62} -5.07107 q^{63} -4.17157 q^{64} +2.82843 q^{65} +3.65685 q^{66} -3.17157 q^{67} +1.82843 q^{68} +11.6569 q^{69} +0.242641 q^{70} -4.24264 q^{71} -13.7279 q^{72} -4.82843 q^{73} +2.68629 q^{74} -3.41421 q^{75} +5.17157 q^{76} +1.51472 q^{77} +4.00000 q^{78} +5.41421 q^{79} -3.00000 q^{80} +39.9706 q^{81} -2.68629 q^{82} +9.31371 q^{83} -3.65685 q^{84} +1.00000 q^{85} +3.17157 q^{86} +16.4853 q^{87} +4.10051 q^{88} -2.34315 q^{89} -3.58579 q^{90} +1.65685 q^{91} +6.24264 q^{92} -14.4853 q^{93} -2.00000 q^{94} +2.82843 q^{95} -15.0711 q^{96} +3.65685 q^{97} -2.75736 q^{98} -22.3848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{7} - 6 q^{8} + 6 q^{9} + 2 q^{10} - 8 q^{11} + 4 q^{12} + 8 q^{14} + 4 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 4 q^{23} + 8 q^{24} + 2 q^{25} - 8 q^{26} - 16 q^{27} - 12 q^{28} - 4 q^{29} + 6 q^{32} + 12 q^{33} + 2 q^{34} + 4 q^{35} - 26 q^{36} - 4 q^{37} - 8 q^{38} + 8 q^{39} + 6 q^{40} + 4 q^{41} - 4 q^{42} + 4 q^{43} - 16 q^{44} - 6 q^{45} - 4 q^{47} - 12 q^{48} - 2 q^{49} - 2 q^{50} + 4 q^{51} + 16 q^{52} + 12 q^{53} - 16 q^{54} + 8 q^{55} + 16 q^{56} + 8 q^{57} - 4 q^{58} - 24 q^{59} - 4 q^{60} + 4 q^{61} + 12 q^{62} + 4 q^{63} - 14 q^{64} - 4 q^{66} - 12 q^{67} - 2 q^{68} + 12 q^{69} - 8 q^{70} - 2 q^{72} - 4 q^{73} + 28 q^{74} - 4 q^{75} + 16 q^{76} + 20 q^{77} + 8 q^{78} + 8 q^{79} - 6 q^{80} + 46 q^{81} - 28 q^{82} - 4 q^{83} + 4 q^{84} + 2 q^{85} + 12 q^{86} + 16 q^{87} + 28 q^{88} - 16 q^{89} - 10 q^{90} - 8 q^{91} + 4 q^{92} - 12 q^{93} - 4 q^{94} - 16 q^{96} - 4 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^7 - 6 * q^8 + 6 * q^9 + 2 * q^10 - 8 * q^11 + 4 * q^12 + 8 * q^14 + 4 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 4 * q^23 + 8 * q^24 + 2 * q^25 - 8 * q^26 - 16 * q^27 - 12 * q^28 - 4 * q^29 + 6 * q^32 + 12 * q^33 + 2 * q^34 + 4 * q^35 - 26 * q^36 - 4 * q^37 - 8 * q^38 + 8 * q^39 + 6 * q^40 + 4 * q^41 - 4 * q^42 + 4 * q^43 - 16 * q^44 - 6 * q^45 - 4 * q^47 - 12 * q^48 - 2 * q^49 - 2 * q^50 + 4 * q^51 + 16 * q^52 + 12 * q^53 - 16 * q^54 + 8 * q^55 + 16 * q^56 + 8 * q^57 - 4 * q^58 - 24 * q^59 - 4 * q^60 + 4 * q^61 + 12 * q^62 + 4 * q^63 - 14 * q^64 - 4 * q^66 - 12 * q^67 - 2 * q^68 + 12 * q^69 - 8 * q^70 - 2 * q^72 - 4 * q^73 + 28 * q^74 - 4 * q^75 + 16 * q^76 + 20 * q^77 + 8 * q^78 + 8 * q^79 - 6 * q^80 + 46 * q^81 - 28 * q^82 - 4 * q^83 + 4 * q^84 + 2 * q^85 + 12 * q^86 + 16 * q^87 + 28 * q^88 - 16 * q^89 - 10 * q^90 - 8 * q^91 + 4 * q^92 - 12 * q^93 - 4 * q^94 - 16 * q^96 - 4 * q^97 - 14 * q^98 - 8 * q^99

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