LMFDB - Embedded newform 8085.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8085,2,Mod(1,8085)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8085, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8085.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8085.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:	$64.5590500342$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8085.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.46410 q^{13} -1.00000 q^{15} -5.00000 q^{16} +1.73205 q^{18} +1.46410 q^{19} +1.00000 q^{20} -1.73205 q^{22} -6.92820 q^{23} +1.73205 q^{24} +1.00000 q^{25} +2.53590 q^{26} -1.00000 q^{27} +3.46410 q^{29} -1.73205 q^{30} -2.92820 q^{31} -5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} +8.92820 q^{37} +2.53590 q^{38} -1.46410 q^{39} -1.73205 q^{40} +3.46410 q^{41} +8.92820 q^{43} -1.00000 q^{44} +1.00000 q^{45} -12.0000 q^{46} -6.92820 q^{47} +5.00000 q^{48} +1.73205 q^{50} +1.46410 q^{52} -12.9282 q^{53} -1.73205 q^{54} -1.00000 q^{55} -1.46410 q^{57} +6.00000 q^{58} -6.92820 q^{59} -1.00000 q^{60} -2.00000 q^{61} -5.07180 q^{62} +1.00000 q^{64} +1.46410 q^{65} +1.73205 q^{66} +8.00000 q^{67} +6.92820 q^{69} -13.8564 q^{71} -1.73205 q^{72} -12.3923 q^{73} +15.4641 q^{74} -1.00000 q^{75} +1.46410 q^{76} -2.53590 q^{78} -13.4641 q^{79} -5.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -15.4641 q^{83} +15.4641 q^{86} -3.46410 q^{87} +1.73205 q^{88} +12.9282 q^{89} +1.73205 q^{90} -6.92820 q^{92} +2.92820 q^{93} -12.0000 q^{94} +1.46410 q^{95} +5.19615 q^{96} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{15} - 10 q^{16} - 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 8 q^{31} + 2 q^{33} + 2 q^{36} + 4 q^{37} + 12 q^{38} + 4 q^{39} + 4 q^{43} - 2 q^{44} + 2 q^{45} - 24 q^{46} + 10 q^{48} - 4 q^{52} - 12 q^{53} - 2 q^{55} + 4 q^{57} + 12 q^{58} - 2 q^{60} - 4 q^{61} - 24 q^{62} + 2 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{73} + 24 q^{74} - 2 q^{75} - 4 q^{76} - 12 q^{78} - 20 q^{79} - 10 q^{80} + 2 q^{81} + 12 q^{82} - 24 q^{83} + 24 q^{86} + 12 q^{89} - 8 q^{93} - 24 q^{94} - 4 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 4 * q^13 - 2 * q^15 - 10 * q^16 - 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 8 * q^31 + 2 * q^33 + 2 * q^36 + 4 * q^37 + 12 * q^38 + 4 * q^39 + 4 * q^43 - 2 * q^44 + 2 * q^45 - 24 * q^46 + 10 * q^48 - 4 * q^52 - 12 * q^53 - 2 * q^55 + 4 * q^57 + 12 * q^58 - 2 * q^60 - 4 * q^61 - 24 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 - 4 * q^73 + 24 * q^74 - 2 * q^75 - 4 * q^76 - 12 * q^78 - 20 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 - 24 * q^83 + 24 * q^86 + 12 * q^89 - 8 * q^93 - 24 * q^94 - 4 * q^95 + 20 * q^97 - 2 * 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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.73205	−0.707107
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	1.73205	0.547723
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.73205	0.408248
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.73205	−0.369274
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	2.53590	0.497331
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	−1.73205	−0.316228
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	−5.19615	−0.918559
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	8.92820	1.46779	0.733894	−	0.679264i	$-0.237701\pi$
$37$	8.92820	1.46779	0.733894	+	0.679264i	$0.237701\pi$
$38$	2.53590	0.411377
$39$	−1.46410	−0.234444
$40$	−1.73205	−0.273861
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−12.0000	−1.76930
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	5.00000	0.721688
$49$	0	0
$50$	1.73205	0.244949
$51$	0	0
$52$	1.46410	0.203034
$53$	−12.9282	−1.77583	−0.887913	−	0.460012i	$-0.847845\pi$
$53$	−12.9282	−1.77583	−0.887913	+	0.460012i	$0.847845\pi$
$54$	−1.73205	−0.235702
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−1.46410	−0.193925
$58$	6.00000	0.787839
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−5.07180	−0.644119
$63$	0	0
$64$	1.00000	0.125000
$65$	1.46410	0.181599
$66$	1.73205	0.213201
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	6.92820	0.834058
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	−1.73205	−0.204124
$73$	−12.3923	−1.45041	−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923	−1.45041	−0.725205	+	0.688533i	$0.758255\pi$
$74$	15.4641	1.79767
$75$	−1.00000	−0.115470
$76$	1.46410	0.167944
$77$	0	0
$78$	−2.53590	−0.287134
$79$	−13.4641	−1.51483	−0.757415	−	0.652934i	$-0.773538\pi$
$79$	−13.4641	−1.51483	−0.757415	+	0.652934i	$0.773538\pi$
$80$	−5.00000	−0.559017
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−15.4641	−1.69741	−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641	−1.69741	−0.848703	+	0.528870i	$0.822616\pi$
$84$	0	0
$85$	0	0
$86$	15.4641	1.66754
$87$	−3.46410	−0.371391
$88$	1.73205	0.184637
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	1.73205	0.182574
$91$	0	0
$92$	−6.92820	−0.722315
$93$	2.92820	0.303641
$94$	−12.0000	−1.23771
$95$	1.46410	0.150214
$96$	5.19615	0.530330
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−2.53590	−0.248665
$105$	0	0
$106$	−22.3923	−2.17493
$107$	15.4641	1.49497	0.747486	−	0.664278i	$-0.231261\pi$
$107$	15.4641	1.49497	0.747486	+	0.664278i	$0.231261\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−1.73205	−0.165145
$111$	−8.92820	−0.847428
$112$	0	0
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	−2.53590	−0.237509
$115$	−6.92820	−0.646058
$116$	3.46410	0.321634
$117$	1.46410	0.135356
$118$	−12.0000	−1.10469
$119$	0	0
$120$	1.73205	0.158114
$121$	1.00000	0.0909091
$122$	−3.46410	−0.313625
$123$	−3.46410	−0.312348
$124$	−2.92820	−0.262960
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.92820	−0.437307	−0.218654	−	0.975803i	$-0.570166\pi$
$127$	−4.92820	−0.437307	−0.218654	+	0.975803i	$0.570166\pi$
$128$	12.1244	1.07165
$129$	−8.92820	−0.786084
$130$	2.53590	0.222413
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	13.8564	1.19701
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	12.0000	1.02151
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	6.92820	0.583460
$142$	−24.0000	−2.01404
$143$	−1.46410	−0.122434
$144$	−5.00000	−0.416667
$145$	3.46410	0.287678
$146$	−21.4641	−1.77638
$147$	0	0
$148$	8.92820	0.733894
$149$	−8.53590	−0.699288	−0.349644	−	0.936883i	$-0.613697\pi$
$149$	−8.53590	−0.699288	−0.349644	+	0.936883i	$0.613697\pi$
$150$	−1.73205	−0.141421
$151$	0.392305	0.0319253	0.0159627	−	0.999873i	$-0.494919\pi$
$151$	0.392305	0.0319253	0.0159627	+	0.999873i	$0.494919\pi$
$152$	−2.53590	−0.205689
$153$	0	0
$154$	0	0
$155$	−2.92820	−0.235199
$156$	−1.46410	−0.117222
$157$	16.9282	1.35102	0.675509	−	0.737352i	$-0.263924\pi$
$157$	16.9282	1.35102	0.675509	+	0.737352i	$0.263924\pi$
$158$	−23.3205	−1.85528
$159$	12.9282	1.02527
$160$	−5.19615	−0.410792
$161$	0	0
$162$	1.73205	0.136083
$163$	−17.8564	−1.39862	−0.699311	−	0.714818i	$-0.746510\pi$
$163$	−17.8564	−1.39862	−0.699311	+	0.714818i	$0.746510\pi$
$164$	3.46410	0.270501
$165$	1.00000	0.0778499
$166$	−26.7846	−2.07889
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	1.46410	0.111963
$172$	8.92820	0.680769
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	5.00000	0.376889
$177$	6.92820	0.520756
$178$	22.3923	1.67837
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	1.00000	0.0745356
$181$	11.8564	0.881280	0.440640	−	0.897684i	$-0.354752\pi$
$181$	11.8564	0.881280	0.440640	+	0.897684i	$0.354752\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	12.0000	0.884652
$185$	8.92820	0.656415
$186$	5.07180	0.371882
$187$	0	0
$188$	−6.92820	−0.505291
$189$	0	0
$190$	2.53590	0.183973
$191$	−5.07180	−0.366982	−0.183491	−	0.983021i	$-0.558740\pi$
$191$	−5.07180	−0.366982	−0.183491	+	0.983021i	$0.558740\pi$
$192$	−1.00000	−0.0721688
$193$	3.60770	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$193$	3.60770	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$194$	17.3205	1.24354
$195$	−1.46410	−0.104846
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	−1.73205	−0.123091
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	−1.73205	−0.122474
$201$	−8.00000	−0.564276
$202$	−18.0000	−1.26648
$203$	0	0
$204$	0	0
$205$	3.46410	0.241943
$206$	−13.8564	−0.965422
$207$	−6.92820	−0.481543
$208$	−7.32051	−0.507586
$209$	−1.46410	−0.101274
$210$	0	0
$211$	12.3923	0.853121	0.426561	−	0.904459i	$-0.359725\pi$
$211$	12.3923	0.853121	0.426561	+	0.904459i	$0.359725\pi$
$212$	−12.9282	−0.887913
$213$	13.8564	0.949425
$214$	26.7846	1.83096
$215$	8.92820	0.608898
$216$	1.73205	0.117851
$217$	0	0
$218$	−17.3205	−1.17309
$219$	12.3923	0.837394
$220$	−1.00000	−0.0674200
$221$	0	0
$222$	−15.4641	−1.03788
$223$	17.8564	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	1.60770	0.106942
$227$	−8.53590	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$227$	−8.53590	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$228$	−1.46410	−0.0969625
$229$	−3.85641	−0.254839	−0.127419	−	0.991849i	$-0.540669\pi$
$229$	−3.85641	−0.254839	−0.127419	+	0.991849i	$0.540669\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	−6.00000	−0.393919
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	2.53590	0.165777
$235$	−6.92820	−0.451946
$236$	−6.92820	−0.450988
$237$	13.4641	0.874587
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	5.00000	0.322749
$241$	−27.8564	−1.79439	−0.897194	−	0.441636i	$-0.854398\pi$
$241$	−27.8564	−1.79439	−0.897194	+	0.441636i	$0.854398\pi$
$242$	1.73205	0.111340
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	2.14359	0.136394
$248$	5.07180	0.322059
$249$	15.4641	0.979998
$250$	1.73205	0.109545
$251$	−25.8564	−1.63204	−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−25.8564	−1.63204	−0.816021	+	0.578022i	$0.803825\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	−8.53590	−0.535590
$255$	0	0
$256$	19.0000	1.18750
$257$	−7.85641	−0.490069	−0.245035	−	0.969514i	$-0.578799\pi$
$257$	−7.85641	−0.490069	−0.245035	+	0.969514i	$0.578799\pi$
$258$	−15.4641	−0.962753
$259$	0	0
$260$	1.46410	0.0907997
$261$	3.46410	0.214423
$262$	−8.78461	−0.542715
$263$	27.4641	1.69351	0.846755	−	0.531984i	$-0.178553\pi$
$263$	27.4641	1.69351	0.846755	+	0.531984i	$0.178553\pi$
$264$	−1.73205	−0.106600
$265$	−12.9282	−0.794173
$266$	0	0
$267$	−12.9282	−0.791193
$268$	8.00000	0.488678
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	−1.73205	−0.105409
$271$	32.3923	1.96769	0.983846	−	0.179016i	$-0.0572913\pi$
$271$	32.3923	1.96769	0.983846	+	0.179016i	$0.0572913\pi$
$272$	0	0
$273$	0	0
$274$	−31.1769	−1.88347
$275$	−1.00000	−0.0603023
$276$	6.92820	0.417029
$277$	22.5359	1.35405	0.677025	−	0.735960i	$-0.263269\pi$
$277$	22.5359	1.35405	0.677025	+	0.735960i	$0.263269\pi$
$278$	14.5359	0.871805
$279$	−2.92820	−0.175307
$280$	0	0
$281$	−3.46410	−0.206651	−0.103325	−	0.994648i	$-0.532948\pi$
$281$	−3.46410	−0.206651	−0.103325	+	0.994648i	$0.532948\pi$
$282$	12.0000	0.714590
$283$	−8.92820	−0.530727	−0.265363	−	0.964148i	$-0.585492\pi$
$283$	−8.92820	−0.530727	−0.265363	+	0.964148i	$0.585492\pi$
$284$	−13.8564	−0.822226
$285$	−1.46410	−0.0867259
$286$	−2.53590	−0.149951
$287$	0	0
$288$	−5.19615	−0.306186
$289$	−17.0000	−1.00000
$290$	6.00000	0.352332
$291$	−10.0000	−0.586210
$292$	−12.3923	−0.725205
$293$	−13.8564	−0.809500	−0.404750	−	0.914427i	$-0.632641\pi$
$293$	−13.8564	−0.809500	−0.404750	+	0.914427i	$0.632641\pi$
$294$	0	0
$295$	−6.92820	−0.403376
$296$	−15.4641	−0.898833
$297$	1.00000	0.0580259
$298$	−14.7846	−0.856449
$299$	−10.1436	−0.586619
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0.679492	0.0391004
$303$	10.3923	0.597022
$304$	−7.32051	−0.419860
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−5.07180	−0.288059
$311$	−18.9282	−1.07332	−0.536660	−	0.843799i	$-0.680314\pi$
$311$	−18.9282	−1.07332	−0.536660	+	0.843799i	$0.680314\pi$
$312$	2.53590	0.143567
$313$	−7.07180	−0.399722	−0.199861	−	0.979824i	$-0.564049\pi$
$313$	−7.07180	−0.399722	−0.199861	+	0.979824i	$0.564049\pi$
$314$	29.3205	1.65465
$315$	0	0
$316$	−13.4641	−0.757415
$317$	−11.0718	−0.621854	−0.310927	−	0.950434i	$-0.600639\pi$
$317$	−11.0718	−0.621854	−0.310927	+	0.950434i	$0.600639\pi$
$318$	22.3923	1.25570
$319$	−3.46410	−0.193952
$320$	1.00000	0.0559017
$321$	−15.4641	−0.863122
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.46410	0.0812137
$326$	−30.9282	−1.71295
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	0	0
$330$	1.73205	0.0953463
$331$	−17.8564	−0.981477	−0.490738	−	0.871307i	$-0.663273\pi$
$331$	−17.8564	−0.981477	−0.490738	+	0.871307i	$0.663273\pi$
$332$	−15.4641	−0.848703
$333$	8.92820	0.489263
$334$	−18.0000	−0.984916
$335$	8.00000	0.437087
$336$	0	0
$337$	−29.1769	−1.58937	−0.794684	−	0.607023i	$-0.792363\pi$
$337$	−29.1769	−1.58937	−0.794684	+	0.607023i	$0.792363\pi$
$338$	−18.8038	−1.02279
$339$	−0.928203	−0.0504131
$340$	0	0
$341$	2.92820	0.158571
$342$	2.53590	0.137126
$343$	0	0
$344$	−15.4641	−0.833768
$345$	6.92820	0.373002
$346$	20.7846	1.11739
$347$	1.60770	0.0863056	0.0431528	−	0.999068i	$-0.486260\pi$
$347$	1.60770	0.0863056	0.0431528	+	0.999068i	$0.486260\pi$
$348$	−3.46410	−0.185695
$349$	35.8564	1.91935	0.959675	−	0.281113i	$-0.0907035\pi$
$349$	35.8564	1.91935	0.959675	+	0.281113i	$0.0907035\pi$
$350$	0	0
$351$	−1.46410	−0.0781480
$352$	5.19615	0.276956
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	12.0000	0.637793
$355$	−13.8564	−0.735422
$356$	12.9282	0.685193
$357$	0	0
$358$	−12.0000	−0.634220
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	−1.73205	−0.0912871
$361$	−16.8564	−0.887179
$362$	20.5359	1.07934
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−12.3923	−0.648643
$366$	3.46410	0.181071
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	34.6410	1.80579
$369$	3.46410	0.180334
$370$	15.4641	0.803940
$371$	0	0
$372$	2.92820	0.151820
$373$	0.392305	0.0203128	0.0101564	−	0.999948i	$-0.496767\pi$
$373$	0.392305	0.0203128	0.0101564	+	0.999948i	$0.496767\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	12.0000	0.618853
$377$	5.07180	0.261211
$378$	0	0
$379$	9.85641	0.506290	0.253145	−	0.967428i	$-0.418535\pi$
$379$	9.85641	0.506290	0.253145	+	0.967428i	$0.418535\pi$
$380$	1.46410	0.0751068
$381$	4.92820	0.252479
$382$	−8.78461	−0.449460
$383$	−13.8564	−0.708029	−0.354015	−	0.935240i	$-0.615184\pi$
$383$	−13.8564	−0.708029	−0.354015	+	0.935240i	$0.615184\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	6.24871	0.318051
$387$	8.92820	0.453846
$388$	10.0000	0.507673
$389$	24.9282	1.26391	0.631955	−	0.775005i	$-0.282253\pi$
$389$	24.9282	1.26391	0.631955	+	0.775005i	$0.282253\pi$
$390$	−2.53590	−0.128410
$391$	0	0
$392$	0	0
$393$	5.07180	0.255838
$394$	−20.7846	−1.04711
$395$	−13.4641	−0.677452
$396$	−1.00000	−0.0502519
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−29.0718	−1.45724
$399$	0	0
$400$	−5.00000	−0.250000
$401$	19.8564	0.991582	0.495791	−	0.868442i	$-0.334878\pi$
$401$	19.8564	0.991582	0.495791	+	0.868442i	$0.334878\pi$
$402$	−13.8564	−0.691095
$403$	−4.28719	−0.213560
$404$	−10.3923	−0.517036
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.92820	−0.442555
$408$	0	0
$409$	−34.7846	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$409$	−34.7846	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$410$	6.00000	0.296319
$411$	18.0000	0.887875
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−12.0000	−0.589768
$415$	−15.4641	−0.759103
$416$	−7.60770	−0.372998
$417$	−8.39230	−0.410973
$418$	−2.53590	−0.124035
$419$	−17.0718	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$419$	−17.0718	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	21.4641	1.04486
$423$	−6.92820	−0.336861
$424$	22.3923	1.08747
$425$	0	0
$426$	24.0000	1.16280
$427$	0	0
$428$	15.4641	0.747486
$429$	1.46410	0.0706875
$430$	15.4641	0.745745
$431$	−32.7846	−1.57918	−0.789590	−	0.613635i	$-0.789706\pi$
$431$	−32.7846	−1.57918	−0.789590	+	0.613635i	$0.789706\pi$
$432$	5.00000	0.240563
$433$	−27.8564	−1.33869	−0.669347	−	0.742950i	$-0.733426\pi$
$433$	−27.8564	−1.33869	−0.669347	+	0.742950i	$0.733426\pi$
$434$	0	0
$435$	−3.46410	−0.166091
$436$	−10.0000	−0.478913
$437$	−10.1436	−0.485234
$438$	21.4641	1.02559
$439$	29.1769	1.39254	0.696269	−	0.717781i	$-0.254842\pi$
$439$	29.1769	1.39254	0.696269	+	0.717781i	$0.254842\pi$
$440$	1.73205	0.0825723
$441$	0	0
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−8.92820	−0.423714
$445$	12.9282	0.612856
$446$	30.9282	1.46449
$447$	8.53590	0.403734
$448$	0	0
$449$	14.7846	0.697729	0.348864	−	0.937173i	$-0.386567\pi$
$449$	14.7846	0.697729	0.348864	+	0.937173i	$0.386567\pi$
$450$	1.73205	0.0816497
$451$	−3.46410	−0.163118
$452$	0.928203	0.0436590
$453$	−0.392305	−0.0184321
$454$	−14.7846	−0.693876
$455$	0	0
$456$	2.53590	0.118754
$457$	−8.39230	−0.392575	−0.196288	−	0.980546i	$-0.562889\pi$
$457$	−8.39230	−0.392575	−0.196288	+	0.980546i	$0.562889\pi$
$458$	−6.67949	−0.312112
$459$	0	0
$460$	−6.92820	−0.323029
$461$	12.2487	0.570479	0.285240	−	0.958456i	$-0.407927\pi$
$461$	12.2487	0.570479	0.285240	+	0.958456i	$0.407927\pi$
$462$	0	0
$463$	−28.0000	−1.30127	−0.650635	−	0.759390i	$-0.725497\pi$
$463$	−28.0000	−1.30127	−0.650635	+	0.759390i	$0.725497\pi$
$464$	−17.3205	−0.804084
$465$	2.92820	0.135792
$466$	20.7846	0.962828
$467$	−18.9282	−0.875893	−0.437946	−	0.899001i	$-0.644294\pi$
$467$	−18.9282	−0.875893	−0.437946	+	0.899001i	$0.644294\pi$
$468$	1.46410	0.0676781
$469$	0	0
$470$	−12.0000	−0.553519
$471$	−16.9282	−0.780010
$472$	12.0000	0.552345
$473$	−8.92820	−0.410519
$474$	23.3205	1.07115
$475$	1.46410	0.0671776
$476$	0	0
$477$	−12.9282	−0.591942
$478$	−20.7846	−0.950666
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	5.19615	0.237171
$481$	13.0718	0.596023
$482$	−48.2487	−2.19767
$483$	0	0
$484$	1.00000	0.0454545
$485$	10.0000	0.454077
$486$	−1.73205	−0.0785674
$487$	23.7128	1.07453	0.537265	−	0.843413i	$-0.319457\pi$
$487$	23.7128	1.07453	0.537265	+	0.843413i	$0.319457\pi$
$488$	3.46410	0.156813
$489$	17.8564	0.807495
$490$	0	0
$491$	−17.0718	−0.770439	−0.385220	−	0.922825i	$-0.625874\pi$
$491$	−17.0718	−0.770439	−0.385220	+	0.922825i	$0.625874\pi$
$492$	−3.46410	−0.156174
$493$	0	0
$494$	3.71281	0.167047
$495$	−1.00000	−0.0449467
$496$	14.6410	0.657401
$497$	0	0
$498$	26.7846	1.20025
$499$	−12.7846	−0.572318	−0.286159	−	0.958182i	$-0.592379\pi$
$499$	−12.7846	−0.572318	−0.286159	+	0.958182i	$0.592379\pi$
$500$	1.00000	0.0447214
$501$	10.3923	0.464294
$502$	−44.7846	−1.99883
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	12.0000	0.533465
$507$	10.8564	0.482150
$508$	−4.92820	−0.218654
$509$	7.85641	0.348229	0.174115	−	0.984725i	$-0.444294\pi$
$509$	7.85641	0.348229	0.174115	+	0.984725i	$0.444294\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	−1.46410	−0.0646417
$514$	−13.6077	−0.600210
$515$	−8.00000	−0.352522
$516$	−8.92820	−0.393042
$517$	6.92820	0.304702
$518$	0	0
$519$	−12.0000	−0.526742
$520$	−2.53590	−0.111207
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−5.07180	−0.221562
$525$	0	0
$526$	47.5692	2.07412
$527$	0	0
$528$	−5.00000	−0.217597
$529$	25.0000	1.08696
$530$	−22.3923	−0.972660
$531$	−6.92820	−0.300658
$532$	0	0
$533$	5.07180	0.219684
$534$	−22.3923	−0.969010
$535$	15.4641	0.668571
$536$	−13.8564	−0.598506
$537$	6.92820	0.298974
$538$	13.6077	0.586669
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	56.1051	2.40992
$543$	−11.8564	−0.508807
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−18.0000	−0.768922
$549$	−2.00000	−0.0853579
$550$	−1.73205	−0.0738549
$551$	5.07180	0.216066
$552$	−12.0000	−0.510754
$553$	0	0
$554$	39.0333	1.65837
$555$	−8.92820	−0.378981
$556$	8.39230	0.355913
$557$	44.7846	1.89758	0.948792	−	0.315900i	$-0.102306\pi$
$557$	44.7846	1.89758	0.948792	+	0.315900i	$0.102306\pi$
$558$	−5.07180	−0.214706
$559$	13.0718	0.552878
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	10.3923	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$564$	6.92820	0.291730
$565$	0.928203	0.0390498
$566$	−15.4641	−0.650005
$567$	0	0
$568$	24.0000	1.00702
$569$	−29.3205	−1.22918	−0.614590	−	0.788847i	$-0.710678\pi$
$569$	−29.3205	−1.22918	−0.614590	+	0.788847i	$0.710678\pi$
$570$	−2.53590	−0.106217
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	−1.46410	−0.0612172
$573$	5.07180	0.211877
$574$	0	0
$575$	−6.92820	−0.288926
$576$	1.00000	0.0416667
$577$	−22.7846	−0.948536	−0.474268	−	0.880381i	$-0.657287\pi$
$577$	−22.7846	−0.948536	−0.474268	+	0.880381i	$0.657287\pi$
$578$	−29.4449	−1.22474
$579$	−3.60770	−0.149931
$580$	3.46410	0.143839
$581$	0	0
$582$	−17.3205	−0.717958
$583$	12.9282	0.535431
$584$	21.4641	0.888191
$585$	1.46410	0.0605332
$586$	−24.0000	−0.991431
$587$	5.07180	0.209335	0.104668	−	0.994507i	$-0.466622\pi$
$587$	5.07180	0.209335	0.104668	+	0.994507i	$0.466622\pi$
$588$	0	0
$589$	−4.28719	−0.176650
$590$	−12.0000	−0.494032
$591$	12.0000	0.493614
$592$	−44.6410	−1.83473
$593$	32.7846	1.34630	0.673151	−	0.739505i	$-0.264940\pi$
$593$	32.7846	1.34630	0.673151	+	0.739505i	$0.264940\pi$
$594$	1.73205	0.0710669
$595$	0	0
$596$	−8.53590	−0.349644
$597$	16.7846	0.686948
$598$	−17.5692	−0.718459
$599$	−10.1436	−0.414456	−0.207228	−	0.978293i	$-0.566444\pi$
$599$	−10.1436	−0.414456	−0.207228	+	0.978293i	$0.566444\pi$
$600$	1.73205	0.0707107
$601$	−36.6410	−1.49462	−0.747309	−	0.664477i	$-0.768655\pi$
$601$	−36.6410	−1.49462	−0.747309	+	0.664477i	$0.768655\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0.392305	0.0159627
$605$	1.00000	0.0406558
$606$	18.0000	0.731200
$607$	−22.7846	−0.924799	−0.462399	−	0.886672i	$-0.653011\pi$
$607$	−22.7846	−0.924799	−0.462399	+	0.886672i	$0.653011\pi$
$608$	−7.60770	−0.308533
$609$	0	0
$610$	−3.46410	−0.140257
$611$	−10.1436	−0.410366
$612$	0	0
$613$	0.392305	0.0158450	0.00792252	−	0.999969i	$-0.497478\pi$
$613$	0.392305	0.0158450	0.00792252	+	0.999969i	$0.497478\pi$
$614$	−24.2487	−0.978598
$615$	−3.46410	−0.139686
$616$	0	0
$617$	−23.0718	−0.928836	−0.464418	−	0.885616i	$-0.653736\pi$
$617$	−23.0718	−0.928836	−0.464418	+	0.885616i	$0.653736\pi$
$618$	13.8564	0.557386
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	−2.92820	−0.117599
$621$	6.92820	0.278019
$622$	−32.7846	−1.31454
$623$	0	0
$624$	7.32051	0.293055
$625$	1.00000	0.0400000
$626$	−12.2487	−0.489557
$627$	1.46410	0.0584706
$628$	16.9282	0.675509
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	23.3205	0.927640
$633$	−12.3923	−0.492550
$634$	−19.1769	−0.761613
$635$	−4.92820	−0.195570
$636$	12.9282	0.512637
$637$	0	0
$638$	−6.00000	−0.237542
$639$	−13.8564	−0.548151
$640$	12.1244	0.479257
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	−26.7846	−1.05710
$643$	45.5692	1.79707	0.898537	−	0.438897i	$-0.144631\pi$
$643$	45.5692	1.79707	0.898537	+	0.438897i	$0.144631\pi$
$644$	0	0
$645$	−8.92820	−0.351548
$646$	0	0
$647$	27.7128	1.08950	0.544752	−	0.838597i	$-0.316624\pi$
$647$	27.7128	1.08950	0.544752	+	0.838597i	$0.316624\pi$
$648$	−1.73205	−0.0680414
$649$	6.92820	0.271956
$650$	2.53590	0.0994661
$651$	0	0
$652$	−17.8564	−0.699311
$653$	−7.85641	−0.307445	−0.153722	−	0.988114i	$-0.549126\pi$
$653$	−7.85641	−0.307445	−0.153722	+	0.988114i	$0.549126\pi$
$654$	17.3205	0.677285
$655$	−5.07180	−0.198171
$656$	−17.3205	−0.676252
$657$	−12.3923	−0.483470
$658$	0	0
$659$	39.7128	1.54699	0.773496	−	0.633801i	$-0.218506\pi$
$659$	39.7128	1.54699	0.773496	+	0.633801i	$0.218506\pi$
$660$	1.00000	0.0389249
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−30.9282	−1.20206
$663$	0	0
$664$	26.7846	1.03944
$665$	0	0
$666$	15.4641	0.599222
$667$	−24.0000	−0.929284
$668$	−10.3923	−0.402090
$669$	−17.8564	−0.690369
$670$	13.8564	0.535320
$671$	2.00000	0.0772091
$672$	0	0
$673$	31.3205	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$673$	31.3205	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$674$	−50.5359	−1.94657
$675$	−1.00000	−0.0384900
$676$	−10.8564	−0.417554
$677$	32.7846	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$677$	32.7846	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$678$	−1.60770	−0.0617432
$679$	0	0
$680$	0	0
$681$	8.53590	0.327096
$682$	5.07180	0.194209
$683$	8.78461	0.336134	0.168067	−	0.985776i	$-0.446248\pi$
$683$	8.78461	0.336134	0.168067	+	0.985776i	$0.446248\pi$
$684$	1.46410	0.0559813
$685$	−18.0000	−0.687745
$686$	0	0
$687$	3.85641	0.147131
$688$	−44.6410	−1.70192
$689$	−18.9282	−0.721107
$690$	12.0000	0.456832
$691$	7.71281	0.293409	0.146705	−	0.989180i	$-0.453133\pi$
$691$	7.71281	0.293409	0.146705	+	0.989180i	$0.453133\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	2.78461	0.105702
$695$	8.39230	0.318338
$696$	6.00000	0.227429
$697$	0	0
$698$	62.1051	2.35071
$699$	−12.0000	−0.453882
$700$	0	0
$701$	32.5359	1.22886	0.614432	−	0.788970i	$-0.289385\pi$
$701$	32.5359	1.22886	0.614432	+	0.788970i	$0.289385\pi$
$702$	−2.53590	−0.0957113
$703$	13.0718	0.493012
$704$	−1.00000	−0.0376889
$705$	6.92820	0.260931
$706$	1.60770	0.0605064
$707$	0	0
$708$	6.92820	0.260378
$709$	15.8564	0.595500	0.297750	−	0.954644i	$-0.403764\pi$
$709$	15.8564	0.595500	0.297750	+	0.954644i	$0.403764\pi$
$710$	−24.0000	−0.900704
$711$	−13.4641	−0.504943
$712$	−22.3923	−0.839187
$713$	20.2872	0.759761
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	−6.92820	−0.258919
$717$	12.0000	0.448148
$718$	36.0000	1.34351
$719$	18.9282	0.705903	0.352951	−	0.935642i	$-0.385178\pi$
$719$	18.9282	0.705903	0.352951	+	0.935642i	$0.385178\pi$
$720$	−5.00000	−0.186339
$721$	0	0
$722$	−29.1962	−1.08657
$723$	27.8564	1.03599
$724$	11.8564	0.440640
$725$	3.46410	0.128654
$726$	−1.73205	−0.0642824
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−21.4641	−0.794422
$731$	0	0
$732$	2.00000	0.0739221
$733$	49.9615	1.84537	0.922686	−	0.385553i	$-0.125989\pi$
$733$	49.9615	1.84537	0.922686	+	0.385553i	$0.125989\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	−8.00000	−0.294684
$738$	6.00000	0.220863
$739$	10.5359	0.387569	0.193785	−	0.981044i	$-0.437924\pi$
$739$	10.5359	0.387569	0.193785	+	0.981044i	$0.437924\pi$
$740$	8.92820	0.328207
$741$	−2.14359	−0.0787469
$742$	0	0
$743$	46.3923	1.70197	0.850984	−	0.525191i	$-0.176006\pi$
$743$	46.3923	1.70197	0.850984	+	0.525191i	$0.176006\pi$
$744$	−5.07180	−0.185941
$745$	−8.53590	−0.312731
$746$	0.679492	0.0248780
$747$	−15.4641	−0.565802
$748$	0	0
$749$	0	0
$750$	−1.73205	−0.0632456
$751$	13.0718	0.476997	0.238498	−	0.971143i	$-0.423345\pi$
$751$	13.0718	0.476997	0.238498	+	0.971143i	$0.423345\pi$
$752$	34.6410	1.26323
$753$	25.8564	0.942260
$754$	8.78461	0.319917
$755$	0.392305	0.0142774
$756$	0	0
$757$	−6.78461	−0.246591	−0.123295	−	0.992370i	$-0.539346\pi$
$757$	−6.78461	−0.246591	−0.123295	+	0.992370i	$0.539346\pi$
$758$	17.0718	0.620076
$759$	−6.92820	−0.251478
$760$	−2.53590	−0.0919867
$761$	39.4641	1.43057	0.715286	−	0.698832i	$-0.246296\pi$
$761$	39.4641	1.43057	0.715286	+	0.698832i	$0.246296\pi$
$762$	8.53590	0.309223
$763$	0	0
$764$	−5.07180	−0.183491
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−10.1436	−0.366264
$768$	−19.0000	−0.685603
$769$	46.4974	1.67674	0.838370	−	0.545102i	$-0.183509\pi$
$769$	46.4974	1.67674	0.838370	+	0.545102i	$0.183509\pi$
$770$	0	0
$771$	7.85641	0.282942
$772$	3.60770	0.129844
$773$	31.8564	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$773$	31.8564	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$774$	15.4641	0.555846
$775$	−2.92820	−0.105184
$776$	−17.3205	−0.621770
$777$	0	0
$778$	43.1769	1.54797
$779$	5.07180	0.181716
$780$	−1.46410	−0.0524232
$781$	13.8564	0.495821
$782$	0	0
$783$	−3.46410	−0.123797
$784$	0	0
$785$	16.9282	0.604193
$786$	8.78461	0.313337
$787$	18.7846	0.669599	0.334800	−	0.942289i	$-0.391331\pi$
$787$	18.7846	0.669599	0.334800	+	0.942289i	$0.391331\pi$
$788$	−12.0000	−0.427482
$789$	−27.4641	−0.977748
$790$	−23.3205	−0.829706
$791$	0	0
$792$	1.73205	0.0615457
$793$	−2.92820	−0.103984
$794$	−3.46410	−0.122936
$795$	12.9282	0.458516
$796$	−16.7846	−0.594915
$797$	−16.6410	−0.589455	−0.294728	−	0.955581i	$-0.595229\pi$
$797$	−16.6410	−0.589455	−0.294728	+	0.955581i	$0.595229\pi$
$798$	0	0
$799$	0	0
$800$	−5.19615	−0.183712
$801$	12.9282	0.456796
$802$	34.3923	1.21443
$803$	12.3923	0.437315
$804$	−8.00000	−0.282138
$805$	0	0
$806$	−7.42563	−0.261557
$807$	−7.85641	−0.276559
$808$	18.0000	0.633238
$809$	−8.53590	−0.300106	−0.150053	−	0.988678i	$-0.547944\pi$
$809$	−8.53590	−0.300106	−0.150053	+	0.988678i	$0.547944\pi$
$810$	1.73205	0.0608581
$811$	8.39230	0.294694	0.147347	−	0.989085i	$-0.452927\pi$
$811$	8.39230	0.294694	0.147347	+	0.989085i	$0.452927\pi$
$812$	0	0
$813$	−32.3923	−1.13605
$814$	−15.4641	−0.542016
$815$	−17.8564	−0.625483
$816$	0	0
$817$	13.0718	0.457324
$818$	−60.2487	−2.10655
$819$	0	0
$820$	3.46410	0.120972
$821$	27.4641	0.958504	0.479252	−	0.877677i	$-0.340908\pi$
$821$	27.4641	0.958504	0.479252	+	0.877677i	$0.340908\pi$
$822$	31.1769	1.08742
$823$	49.5692	1.72787	0.863937	−	0.503600i	$-0.167991\pi$
$823$	49.5692	1.72787	0.863937	+	0.503600i	$0.167991\pi$
$824$	13.8564	0.482711
$825$	1.00000	0.0348155
$826$	0	0
$827$	1.60770	0.0559050	0.0279525	−	0.999609i	$-0.491101\pi$
$827$	1.60770	0.0559050	0.0279525	+	0.999609i	$0.491101\pi$
$828$	−6.92820	−0.240772
$829$	25.7128	0.893043	0.446521	−	0.894773i	$-0.352663\pi$
$829$	25.7128	0.893043	0.446521	+	0.894773i	$0.352663\pi$
$830$	−26.7846	−0.929707
$831$	−22.5359	−0.781762
$832$	1.46410	0.0507586
$833$	0	0
$834$	−14.5359	−0.503337
$835$	−10.3923	−0.359641
$836$	−1.46410	−0.0506370
$837$	2.92820	0.101214
$838$	−29.5692	−1.02145
$839$	−15.2154	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$839$	−15.2154	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	3.46410	0.119381
$843$	3.46410	0.119310
$844$	12.3923	0.426561
$845$	−10.8564	−0.373472
$846$	−12.0000	−0.412568
$847$	0	0
$848$	64.6410	2.21978
$849$	8.92820	0.306415
$850$	0	0
$851$	−61.8564	−2.12041
$852$	13.8564	0.474713
$853$	−24.3923	−0.835177	−0.417588	−	0.908636i	$-0.637125\pi$
$853$	−24.3923	−0.835177	−0.417588	+	0.908636i	$0.637125\pi$
$854$	0	0
$855$	1.46410	0.0500712
$856$	−26.7846	−0.915479
$857$	10.1436	0.346499	0.173249	−	0.984878i	$-0.444573\pi$
$857$	10.1436	0.346499	0.173249	+	0.984878i	$0.444573\pi$
$858$	2.53590	0.0865741
$859$	−47.7128	−1.62794	−0.813970	−	0.580907i	$-0.802698\pi$
$859$	−47.7128	−1.62794	−0.813970	+	0.580907i	$0.802698\pi$
$860$	8.92820	0.304449
$861$	0	0
$862$	−56.7846	−1.93409
$863$	−10.1436	−0.345292	−0.172646	−	0.984984i	$-0.555232\pi$
$863$	−10.1436	−0.345292	−0.172646	+	0.984984i	$0.555232\pi$
$864$	5.19615	0.176777
$865$	12.0000	0.408012
$866$	−48.2487	−1.63956
$867$	17.0000	0.577350
$868$	0	0
$869$	13.4641	0.456738
$870$	−6.00000	−0.203419
$871$	11.7128	0.396874
$872$	17.3205	0.586546
$873$	10.0000	0.338449
$874$	−17.5692	−0.594288
$875$	0	0
$876$	12.3923	0.418697
$877$	14.2487	0.481145	0.240572	−	0.970631i	$-0.422665\pi$
$877$	14.2487	0.481145	0.240572	+	0.970631i	$0.422665\pi$
$878$	50.5359	1.70550
$879$	13.8564	0.467365
$880$	5.00000	0.168550
$881$	−12.9282	−0.435562	−0.217781	−	0.975998i	$-0.569882\pi$
$881$	−12.9282	−0.435562	−0.217781	+	0.975998i	$0.569882\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	6.92820	0.232889
$886$	20.7846	0.698273
$887$	36.2487	1.21711	0.608556	−	0.793511i	$-0.291749\pi$
$887$	36.2487	1.21711	0.608556	+	0.793511i	$0.291749\pi$
$888$	15.4641	0.518941
$889$	0	0
$890$	22.3923	0.750592
$891$	−1.00000	−0.0335013
$892$	17.8564	0.597877
$893$	−10.1436	−0.339442
$894$	14.7846	0.494471
$895$	−6.92820	−0.231584
$896$	0	0
$897$	10.1436	0.338685
$898$	25.6077	0.854540
$899$	−10.1436	−0.338308
$900$	1.00000	0.0333333
$901$	0	0
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−1.60770	−0.0534711
$905$	11.8564	0.394120
$906$	−0.679492	−0.0225746
$907$	45.8564	1.52264	0.761318	−	0.648378i	$-0.224552\pi$
$907$	45.8564	1.52264	0.761318	+	0.648378i	$0.224552\pi$
$908$	−8.53590	−0.283274
$909$	−10.3923	−0.344691
$910$	0	0
$911$	5.07180	0.168036	0.0840181	−	0.996464i	$-0.473225\pi$
$911$	5.07180	0.168036	0.0840181	+	0.996464i	$0.473225\pi$
$912$	7.32051	0.242406
$913$	15.4641	0.511787
$914$	−14.5359	−0.480805
$915$	2.00000	0.0661180
$916$	−3.85641	−0.127419
$917$	0	0
$918$	0	0
$919$	−11.6077	−0.382903	−0.191451	−	0.981502i	$-0.561319\pi$
$919$	−11.6077	−0.382903	−0.191451	+	0.981502i	$0.561319\pi$
$920$	12.0000	0.395628
$921$	14.0000	0.461316
$922$	21.2154	0.698692
$923$	−20.2872	−0.667761
$924$	0	0
$925$	8.92820	0.293558
$926$	−48.4974	−1.59372
$927$	−8.00000	−0.262754
$928$	−18.0000	−0.590879
$929$	38.7846	1.27248	0.636241	−	0.771490i	$-0.280488\pi$
$929$	38.7846	1.27248	0.636241	+	0.771490i	$0.280488\pi$
$930$	5.07180	0.166311
$931$	0	0
$932$	12.0000	0.393073
$933$	18.9282	0.619682
$934$	−32.7846	−1.07275
$935$	0	0
$936$	−2.53590	−0.0828884
$937$	−0.392305	−0.0128160	−0.00640802	−	0.999979i	$-0.502040\pi$
$937$	−0.392305	−0.0128160	−0.00640802	+	0.999979i	$0.502040\pi$
$938$	0	0
$939$	7.07180	0.230779
$940$	−6.92820	−0.225973
$941$	−20.5359	−0.669451	−0.334726	−	0.942316i	$-0.608644\pi$
$941$	−20.5359	−0.669451	−0.334726	+	0.942316i	$0.608644\pi$
$942$	−29.3205	−0.955314
$943$	−24.0000	−0.781548
$944$	34.6410	1.12747
$945$	0	0
$946$	−15.4641	−0.502781
$947$	−5.07180	−0.164811	−0.0824056	−	0.996599i	$-0.526260\pi$
$947$	−5.07180	−0.164811	−0.0824056	+	0.996599i	$0.526260\pi$
$948$	13.4641	0.437294
$949$	−18.1436	−0.588966
$950$	2.53590	0.0822754
$951$	11.0718	0.359028
$952$	0	0
$953$	−44.7846	−1.45072	−0.725358	−	0.688372i	$-0.758326\pi$
$953$	−44.7846	−1.45072	−0.725358	+	0.688372i	$0.758326\pi$
$954$	−22.3923	−0.724978
$955$	−5.07180	−0.164119
$956$	−12.0000	−0.388108
$957$	3.46410	0.111979
$958$	−20.7846	−0.671520
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	−22.4256	−0.723407
$962$	22.6410	0.729976
$963$	15.4641	0.498324
$964$	−27.8564	−0.897194
$965$	3.60770	0.116136
$966$	0	0
$967$	−18.7846	−0.604072	−0.302036	−	0.953296i	$-0.597666\pi$
$967$	−18.7846	−0.604072	−0.302036	+	0.953296i	$0.597666\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	17.3205	0.556128
$971$	1.85641	0.0595749	0.0297875	−	0.999556i	$-0.490517\pi$
$971$	1.85641	0.0595749	0.0297875	+	0.999556i	$0.490517\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	41.0718	1.31603
$975$	−1.46410	−0.0468888
$976$	10.0000	0.320092
$977$	−35.5692	−1.13796	−0.568980	−	0.822351i	$-0.692662\pi$
$977$	−35.5692	−1.13796	−0.568980	+	0.822351i	$0.692662\pi$
$978$	30.9282	0.988975
$979$	−12.9282	−0.413187
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−29.5692	−0.943592
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	6.00000	0.191273
$985$	−12.0000	−0.382352
$986$	0	0
$987$	0	0
$988$	2.14359	0.0681968
$989$	−61.8564	−1.96692
$990$	−1.73205	−0.0550482
$991$	−48.7846	−1.54969	−0.774847	−	0.632149i	$-0.782173\pi$
$991$	−48.7846	−1.54969	−0.774847	+	0.632149i	$0.782173\pi$
$992$	15.2154	0.483089
$993$	17.8564	0.566656
$994$	0	0
$995$	−16.7846	−0.532108
$996$	15.4641	0.489999
$997$	−48.3923	−1.53260	−0.766300	−	0.642483i	$-0.777904\pi$
$997$	−48.3923	−1.53260	−0.766300	+	0.642483i	$0.777904\pi$
$998$	−22.1436	−0.700943
$999$	−8.92820	−0.282476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.bd.1.2		2
7.6	odd	2		165.2.a.b.1.2	✓	2
21.20	even	2		495.2.a.c.1.1		2
28.27	even	2		2640.2.a.x.1.1		2
35.13	even	4		825.2.c.c.199.2		4
35.27	even	4		825.2.c.c.199.3		4
35.34	odd	2		825.2.a.e.1.1		2
77.76	even	2		1815.2.a.i.1.1		2
84.83	odd	2		7920.2.a.bz.1.1		2
105.62	odd	4		2475.2.c.n.199.2		4
105.83	odd	4		2475.2.c.n.199.3		4
105.104	even	2		2475.2.a.r.1.2		2
231.230	odd	2		5445.2.a.s.1.2		2
385.384	even	2		9075.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.2	✓	2	7.6	odd	2
495.2.a.c.1.1		2	21.20	even	2
825.2.a.e.1.1		2	35.34	odd	2
825.2.c.c.199.2		4	35.13	even	4
825.2.c.c.199.3		4	35.27	even	4
1815.2.a.i.1.1		2	77.76	even	2
2475.2.a.r.1.2		2	105.104	even	2
2475.2.c.n.199.2		4	105.62	odd	4
2475.2.c.n.199.3		4	105.83	odd	4
2640.2.a.x.1.1		2	28.27	even	2
5445.2.a.s.1.2		2	231.230	odd	2
7920.2.a.bz.1.1		2	84.83	odd	2
8085.2.a.bd.1.2		2	1.1	even	1	trivial
9075.2.a.bh.1.2		2	385.384	even	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:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{6} -1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} -1.00000 q^{11} -1.00000 q^{12} +1.46410 q^{13} -1.00000 q^{15} -5.00000 q^{16} +1.73205 q^{18} +1.46410 q^{19} +1.00000 q^{20} -1.73205 q^{22} -6.92820 q^{23} +1.73205 q^{24} +1.00000 q^{25} +2.53590 q^{26} -1.00000 q^{27} +3.46410 q^{29} -1.73205 q^{30} -2.92820 q^{31} -5.19615 q^{32} +1.00000 q^{33} +1.00000 q^{36} +8.92820 q^{37} +2.53590 q^{38} -1.46410 q^{39} -1.73205 q^{40} +3.46410 q^{41} +8.92820 q^{43} -1.00000 q^{44} +1.00000 q^{45} -12.0000 q^{46} -6.92820 q^{47} +5.00000 q^{48} +1.73205 q^{50} +1.46410 q^{52} -12.9282 q^{53} -1.73205 q^{54} -1.00000 q^{55} -1.46410 q^{57} +6.00000 q^{58} -6.92820 q^{59} -1.00000 q^{60} -2.00000 q^{61} -5.07180 q^{62} +1.00000 q^{64} +1.46410 q^{65} +1.73205 q^{66} +8.00000 q^{67} +6.92820 q^{69} -13.8564 q^{71} -1.73205 q^{72} -12.3923 q^{73} +15.4641 q^{74} -1.00000 q^{75} +1.46410 q^{76} -2.53590 q^{78} -13.4641 q^{79} -5.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -15.4641 q^{83} +15.4641 q^{86} -3.46410 q^{87} +1.73205 q^{88} +12.9282 q^{89} +1.73205 q^{90} -6.92820 q^{92} +2.92820 q^{93} -12.0000 q^{94} +1.46410 q^{95} +5.19615 q^{96} +10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{15} - 10 q^{16} - 4 q^{19} + 2 q^{20} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 8 q^{31} + 2 q^{33} + 2 q^{36} + 4 q^{37} + 12 q^{38} + 4 q^{39} + 4 q^{43} - 2 q^{44} + 2 q^{45} - 24 q^{46} + 10 q^{48} - 4 q^{52} - 12 q^{53} - 2 q^{55} + 4 q^{57} + 12 q^{58} - 2 q^{60} - 4 q^{61} - 24 q^{62} + 2 q^{64} - 4 q^{65} + 16 q^{67} - 4 q^{73} + 24 q^{74} - 2 q^{75} - 4 q^{76} - 12 q^{78} - 20 q^{79} - 10 q^{80} + 2 q^{81} + 12 q^{82} - 24 q^{83} + 24 q^{86} + 12 q^{89} - 8 q^{93} - 24 q^{94} - 4 q^{95} + 20 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^12 - 4 * q^13 - 2 * q^15 - 10 * q^16 - 4 * q^19 + 2 * q^20 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 8 * q^31 + 2 * q^33 + 2 * q^36 + 4 * q^37 + 12 * q^38 + 4 * q^39 + 4 * q^43 - 2 * q^44 + 2 * q^45 - 24 * q^46 + 10 * q^48 - 4 * q^52 - 12 * q^53 - 2 * q^55 + 4 * q^57 + 12 * q^58 - 2 * q^60 - 4 * q^61 - 24 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 - 4 * q^73 + 24 * q^74 - 2 * q^75 - 4 * q^76 - 12 * q^78 - 20 * q^79 - 10 * q^80 + 2 * q^81 + 12 * q^82 - 24 * q^83 + 24 * q^86 + 12 * q^89 - 8 * q^93 - 24 * q^94 - 4 * q^95 + 20 * q^97 - 2 * q^99

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