LMFDB - Embedded newform 8085.2.a.bw.1.6

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:	$6$
Coefficient field:	6.6.2803712.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 8x^{2} - 2 $ x^6 - 6x^4 + 8x^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			1.6
Root			$2.05288$ 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)

$f(q)$	$=$	$q+2.05288 q^{2} -1.00000 q^{3} +2.21432 q^{4} +1.00000 q^{5} -2.05288 q^{6} +0.439973 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.05288 q^{2} -1.00000 q^{3} +2.21432 q^{4} +1.00000 q^{5} -2.05288 q^{6} +0.439973 q^{8} +1.00000 q^{9} +2.05288 q^{10} +1.00000 q^{11} -2.21432 q^{12} -1.18175 q^{13} -1.00000 q^{15} -3.52543 q^{16} -0.774347 q^{17} +2.05288 q^{18} +7.17427 q^{19} +2.21432 q^{20} +2.05288 q^{22} -9.19488 q^{23} -0.439973 q^{24} +1.00000 q^{25} -2.42598 q^{26} -1.00000 q^{27} -6.98451 q^{29} -2.05288 q^{30} -3.18610 q^{31} -8.11723 q^{32} -1.00000 q^{33} -1.58964 q^{34} +2.21432 q^{36} +1.71013 q^{37} +14.7279 q^{38} +1.18175 q^{39} +0.439973 q^{40} -9.62703 q^{41} -12.2246 q^{43} +2.21432 q^{44} +1.00000 q^{45} -18.8760 q^{46} +9.07877 q^{47} +3.52543 q^{48} +2.05288 q^{50} +0.774347 q^{51} -2.61676 q^{52} -8.85867 q^{53} -2.05288 q^{54} +1.00000 q^{55} -7.17427 q^{57} -14.3384 q^{58} -5.10474 q^{59} -2.21432 q^{60} +0.496006 q^{61} -6.54068 q^{62} -9.61285 q^{64} -1.18175 q^{65} -2.05288 q^{66} +10.4943 q^{67} -1.71465 q^{68} +9.19488 q^{69} +11.6583 q^{71} +0.439973 q^{72} +8.49614 q^{73} +3.51070 q^{74} -1.00000 q^{75} +15.8861 q^{76} +2.42598 q^{78} -6.83524 q^{79} -3.52543 q^{80} +1.00000 q^{81} -19.7631 q^{82} +2.73247 q^{83} -0.774347 q^{85} -25.0956 q^{86} +6.98451 q^{87} +0.439973 q^{88} -13.2266 q^{89} +2.05288 q^{90} -20.3604 q^{92} +3.18610 q^{93} +18.6376 q^{94} +7.17427 q^{95} +8.11723 q^{96} +15.9929 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} + 6 q^{11} + 8 q^{13} - 6 q^{15} - 8 q^{16} + 6 q^{17} - 6 q^{19} - 14 q^{23} + 6 q^{25} - 4 q^{26} - 6 q^{27} - 6 q^{29} - 12 q^{31} - 6 q^{33} - 8 q^{34} - 12 q^{37} + 20 q^{38} - 8 q^{39} - 8 q^{41} - 30 q^{43} + 6 q^{45} - 16 q^{46} + 8 q^{48} - 6 q^{51} + 4 q^{52} - 2 q^{53} + 6 q^{55} + 6 q^{57} - 20 q^{58} - 2 q^{59} - 10 q^{61} - 16 q^{62} - 4 q^{64} + 8 q^{65} - 24 q^{67} - 16 q^{68} + 14 q^{69} - 4 q^{71} + 32 q^{73} + 4 q^{74} - 6 q^{75} + 16 q^{76} + 4 q^{78} - 20 q^{79} - 8 q^{80} + 6 q^{81} - 36 q^{82} - 6 q^{83} + 6 q^{85} - 8 q^{86} + 6 q^{87} - 6 q^{89} - 24 q^{92} + 12 q^{93} + 4 q^{94} - 6 q^{95} + 34 q^{97} + 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 + 6 * q^11 + 8 * q^13 - 6 * q^15 - 8 * q^16 + 6 * q^17 - 6 * q^19 - 14 * q^23 + 6 * q^25 - 4 * q^26 - 6 * q^27 - 6 * q^29 - 12 * q^31 - 6 * q^33 - 8 * q^34 - 12 * q^37 + 20 * q^38 - 8 * q^39 - 8 * q^41 - 30 * q^43 + 6 * q^45 - 16 * q^46 + 8 * q^48 - 6 * q^51 + 4 * q^52 - 2 * q^53 + 6 * q^55 + 6 * q^57 - 20 * q^58 - 2 * q^59 - 10 * q^61 - 16 * q^62 - 4 * q^64 + 8 * q^65 - 24 * q^67 - 16 * q^68 + 14 * q^69 - 4 * q^71 + 32 * q^73 + 4 * q^74 - 6 * q^75 + 16 * q^76 + 4 * q^78 - 20 * q^79 - 8 * q^80 + 6 * q^81 - 36 * q^82 - 6 * q^83 + 6 * q^85 - 8 * q^86 + 6 * q^87 - 6 * q^89 - 24 * q^92 + 12 * q^93 + 4 * q^94 - 6 * q^95 + 34 * q^97 + 6 * 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 $

$2$	2.05288	1.45161	0.725803	−	0.687903i	$-0.241468\pi$
$2$	2.05288	1.45161	0.725803	+	0.687903i	$0.241468\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.21432	1.10716
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.05288	−0.838085
$7$	0	0
$7$	0	0
$8$	0.439973	0.155554
$9$	1.00000	0.333333
$10$	2.05288	0.649178
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−2.21432	−0.639219
$13$	−1.18175	−0.327757	−0.163879	−	0.986481i	$-0.552401\pi$
$13$	−1.18175	−0.327757	−0.163879	+	0.986481i	$0.552401\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	−3.52543	−0.881357
$17$	−0.774347	−0.187807	−0.0939033	−	0.995581i	$-0.529934\pi$
$17$	−0.774347	−0.187807	−0.0939033	+	0.995581i	$0.529934\pi$
$18$	2.05288	0.483869
$19$	7.17427	1.64589	0.822945	−	0.568121i	$-0.192329\pi$
$19$	7.17427	1.64589	0.822945	+	0.568121i	$0.192329\pi$
$20$	2.21432	0.495137
$21$	0	0
$22$	2.05288	0.437676
$23$	−9.19488	−1.91726	−0.958632	−	0.284647i	$-0.908124\pi$
$23$	−9.19488	−1.91726	−0.958632	+	0.284647i	$0.908124\pi$
$24$	−0.439973	−0.0898091
$25$	1.00000	0.200000
$26$	−2.42598	−0.475775
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.98451	−1.29699	−0.648495	−	0.761219i	$-0.724601\pi$
$29$	−6.98451	−1.29699	−0.648495	+	0.761219i	$0.724601\pi$
$30$	−2.05288	−0.374803
$31$	−3.18610	−0.572240	−0.286120	−	0.958194i	$-0.592366\pi$
$31$	−3.18610	−0.572240	−0.286120	+	0.958194i	$0.592366\pi$
$32$	−8.11723	−1.43494
$33$	−1.00000	−0.174078
$34$	−1.58964	−0.272621
$35$	0	0
$36$	2.21432	0.369053
$37$	1.71013	0.281144	0.140572	−	0.990070i	$-0.455106\pi$
$37$	1.71013	0.281144	0.140572	+	0.990070i	$0.455106\pi$
$38$	14.7279	2.38918
$39$	1.18175	0.189231
$40$	0.439973	0.0695658
$41$	−9.62703	−1.50349	−0.751745	−	0.659454i	$-0.770787\pi$
$41$	−9.62703	−1.50349	−0.751745	+	0.659454i	$0.770787\pi$
$42$	0	0
$43$	−12.2246	−1.86423	−0.932116	−	0.362160i	$-0.882039\pi$
$43$	−12.2246	−1.86423	−0.932116	+	0.362160i	$0.882039\pi$
$44$	2.21432	0.333821
$45$	1.00000	0.149071
$46$	−18.8760	−2.78311
$47$	9.07877	1.32428	0.662138	−	0.749382i	$-0.269649\pi$
$47$	9.07877	1.32428	0.662138	+	0.749382i	$0.269649\pi$
$48$	3.52543	0.508852
$49$	0	0
$50$	2.05288	0.290321
$51$	0.774347	0.108430
$52$	−2.61676	−0.362880
$53$	−8.85867	−1.21683	−0.608416	−	0.793618i	$-0.708195\pi$
$53$	−8.85867	−1.21683	−0.608416	+	0.793618i	$0.708195\pi$
$54$	−2.05288	−0.279362
$55$	1.00000	0.134840
$56$	0	0
$57$	−7.17427	−0.950255
$58$	−14.3384	−1.88272
$59$	−5.10474	−0.664581	−0.332290	−	0.943177i	$-0.607821\pi$
$59$	−5.10474	−0.664581	−0.332290	+	0.943177i	$0.607821\pi$
$60$	−2.21432	−0.285867
$61$	0.496006	0.0635071	0.0317536	−	0.999496i	$-0.489891\pi$
$61$	0.496006	0.0635071	0.0317536	+	0.999496i	$0.489891\pi$
$62$	−6.54068	−0.830667
$63$	0	0
$64$	−9.61285	−1.20161
$65$	−1.18175	−0.146578
$66$	−2.05288	−0.252692
$67$	10.4943	1.28209	0.641044	−	0.767504i	$-0.278501\pi$
$67$	10.4943	1.28209	0.641044	+	0.767504i	$0.278501\pi$
$68$	−1.71465	−0.207932
$69$	9.19488	1.10693
$70$	0	0
$71$	11.6583	1.38358	0.691792	−	0.722097i	$-0.256822\pi$
$71$	11.6583	1.38358	0.691792	+	0.722097i	$0.256822\pi$
$72$	0.439973	0.0518513
$73$	8.49614	0.994398	0.497199	−	0.867636i	$-0.334362\pi$
$73$	8.49614	0.994398	0.497199	+	0.867636i	$0.334362\pi$
$74$	3.51070	0.408110
$75$	−1.00000	−0.115470
$76$	15.8861	1.82226
$77$	0	0
$78$	2.42598	0.274689
$79$	−6.83524	−0.769025	−0.384512	−	0.923120i	$-0.625630\pi$
$79$	−6.83524	−0.769025	−0.384512	+	0.923120i	$0.625630\pi$
$80$	−3.52543	−0.394155
$81$	1.00000	0.111111
$82$	−19.7631	−2.18247
$83$	2.73247	0.299927	0.149964	−	0.988692i	$-0.452084\pi$
$83$	2.73247	0.299927	0.149964	+	0.988692i	$0.452084\pi$
$84$	0	0
$85$	−0.774347	−0.0839897
$86$	−25.0956	−2.70613
$87$	6.98451	0.748818
$88$	0.439973	0.0469013
$89$	−13.2266	−1.40201	−0.701006	−	0.713155i	$-0.747265\pi$
$89$	−13.2266	−1.40201	−0.701006	+	0.713155i	$0.747265\pi$
$90$	2.05288	0.216393
$91$	0	0
$92$	−20.3604	−2.12272
$93$	3.18610	0.330383
$94$	18.6376	1.92233
$95$	7.17427	0.736064
$96$	8.11723	0.828461
$97$	15.9929	1.62383	0.811915	−	0.583776i	$-0.198425\pi$
$97$	15.9929	1.62383	0.811915	+	0.583776i	$0.198425\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	2.21432	0.221432
$101$	−6.33721	−0.630576	−0.315288	−	0.948996i	$-0.602101\pi$
$101$	−6.33721	−0.630576	−0.315288	+	0.948996i	$0.602101\pi$
$102$	1.58964	0.157398
$103$	4.78329	0.471311	0.235656	−	0.971837i	$-0.424276\pi$
$103$	4.78329	0.471311	0.235656	+	0.971837i	$0.424276\pi$
$104$	−0.519936	−0.0509839
$105$	0	0
$106$	−18.1858	−1.76636
$107$	0.194133	0.0187676	0.00938379	−	0.999956i	$-0.497013\pi$
$107$	0.194133	0.0187676	0.00938379	+	0.999956i	$0.497013\pi$
$108$	−2.21432	−0.213073
$109$	15.3776	1.47291	0.736453	−	0.676489i	$-0.236499\pi$
$109$	15.3776	1.47291	0.736453	+	0.676489i	$0.236499\pi$
$110$	2.05288	0.195735
$111$	−1.71013	−0.162319
$112$	0	0
$113$	−5.67603	−0.533956	−0.266978	−	0.963703i	$-0.586025\pi$
$113$	−5.67603	−0.533956	−0.266978	+	0.963703i	$0.586025\pi$
$114$	−14.7279	−1.37940
$115$	−9.19488	−0.857427
$116$	−15.4659	−1.43598
$117$	−1.18175	−0.109252
$118$	−10.4794	−0.964709
$119$	0	0
$120$	−0.439973	−0.0401638
$121$	1.00000	0.0909091
$122$	1.01824	0.0921873
$123$	9.62703	0.868040
$124$	−7.05504	−0.633561
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.94847	−0.172899	−0.0864493	−	0.996256i	$-0.527552\pi$
$127$	−1.94847	−0.172899	−0.0864493	+	0.996256i	$0.527552\pi$
$128$	−3.49957	−0.309322
$129$	12.2246	1.07631
$130$	−2.42598	−0.212773
$131$	−18.5876	−1.62401	−0.812004	−	0.583652i	$-0.801623\pi$
$131$	−18.5876	−1.62401	−0.812004	+	0.583652i	$0.801623\pi$
$132$	−2.21432	−0.192732
$133$	0	0
$134$	21.5436	1.86109
$135$	−1.00000	−0.0860663
$136$	−0.340692	−0.0292141
$137$	−23.0152	−1.96633	−0.983163	−	0.182733i	$-0.941506\pi$
$137$	−23.0152	−1.96633	−0.983163	+	0.182733i	$0.941506\pi$
$138$	18.8760	1.60683
$139$	−2.89018	−0.245142	−0.122571	−	0.992460i	$-0.539114\pi$
$139$	−2.89018	−0.245142	−0.122571	+	0.992460i	$0.539114\pi$
$140$	0	0
$141$	−9.07877	−0.764571
$142$	23.9331	2.00842
$143$	−1.18175	−0.0988226
$144$	−3.52543	−0.293786
$145$	−6.98451	−0.580032
$146$	17.4416	1.44347
$147$	0	0
$148$	3.78678	0.311271
$149$	−15.1209	−1.23875	−0.619376	−	0.785095i	$-0.712614\pi$
$149$	−15.1209	−1.23875	−0.619376	+	0.785095i	$0.712614\pi$
$150$	−2.05288	−0.167617
$151$	−9.45909	−0.769770	−0.384885	−	0.922965i	$-0.625759\pi$
$151$	−9.45909	−0.769770	−0.384885	+	0.922965i	$0.625759\pi$
$152$	3.15648	0.256025
$153$	−0.774347	−0.0626022
$154$	0	0
$155$	−3.18610	−0.255914
$156$	2.61676	0.209509
$157$	9.20973	0.735017	0.367508	−	0.930020i	$-0.380211\pi$
$157$	9.20973	0.735017	0.367508	+	0.930020i	$0.380211\pi$
$158$	−14.0319	−1.11632
$159$	8.85867	0.702538
$160$	−8.11723	−0.641723
$161$	0	0
$162$	2.05288	0.161290
$163$	−16.2166	−1.27018	−0.635090	−	0.772438i	$-0.719037\pi$
$163$	−16.2166	−1.27018	−0.635090	+	0.772438i	$0.719037\pi$
$164$	−21.3173	−1.66460
$165$	−1.00000	−0.0778499
$166$	5.60943	0.435376
$167$	5.57559	0.431452	0.215726	−	0.976454i	$-0.430788\pi$
$167$	5.57559	0.431452	0.215726	+	0.976454i	$0.430788\pi$
$168$	0	0
$169$	−11.6035	−0.892575
$170$	−1.58964	−0.121920
$171$	7.17427	0.548630
$172$	−27.0691	−2.06400
$173$	−10.1133	−0.768900	−0.384450	−	0.923146i	$-0.625609\pi$
$173$	−10.1133	−0.768900	−0.384450	+	0.923146i	$0.625609\pi$
$174$	14.3384	1.08699
$175$	0	0
$176$	−3.52543	−0.265739
$177$	5.10474	0.383696
$178$	−27.1525	−2.03517
$179$	−10.4437	−0.780600	−0.390300	−	0.920688i	$-0.627629\pi$
$179$	−10.4437	−0.780600	−0.390300	+	0.920688i	$0.627629\pi$
$180$	2.21432	0.165046
$181$	−21.5313	−1.60041	−0.800206	−	0.599725i	$-0.795276\pi$
$181$	−21.5313	−1.60041	−0.800206	+	0.599725i	$0.795276\pi$
$182$	0	0
$183$	−0.496006	−0.0366659
$184$	−4.04550	−0.298238
$185$	1.71013	0.125731
$186$	6.54068	0.479586
$187$	−0.774347	−0.0566258
$188$	20.1033	1.46618
$189$	0	0
$190$	14.7279	1.06848
$191$	−25.4208	−1.83939	−0.919693	−	0.392638i	$-0.871563\pi$
$191$	−25.4208	−1.83939	−0.919693	+	0.392638i	$0.871563\pi$
$192$	9.61285	0.693748
$193$	3.34149	0.240526	0.120263	−	0.992742i	$-0.461626\pi$
$193$	3.34149	0.240526	0.120263	+	0.992742i	$0.461626\pi$
$194$	32.8314	2.35716
$195$	1.18175	0.0846266
$196$	0	0
$197$	27.7272	1.97548	0.987740	−	0.156106i	$-0.0498940\pi$
$197$	27.7272	1.97548	0.987740	+	0.156106i	$0.0498940\pi$
$198$	2.05288	0.145892
$199$	4.67504	0.331405	0.165702	−	0.986176i	$-0.447011\pi$
$199$	4.67504	0.331405	0.165702	+	0.986176i	$0.447011\pi$
$200$	0.439973	0.0311108
$201$	−10.4943	−0.740214
$202$	−13.0095	−0.915348
$203$	0	0
$204$	1.71465	0.120050
$205$	−9.62703	−0.672381
$206$	9.81952	0.684158
$207$	−9.19488	−0.639088
$208$	4.16616	0.288871
$209$	7.17427	0.496254
$210$	0	0
$211$	−7.89959	−0.543830	−0.271915	−	0.962321i	$-0.587657\pi$
$211$	−7.89959	−0.543830	−0.271915	+	0.962321i	$0.587657\pi$
$212$	−19.6159	−1.34723
$213$	−11.6583	−0.798812
$214$	0.398533	0.0272431
$215$	−12.2246	−0.833710
$216$	−0.439973	−0.0299364
$217$	0	0
$218$	31.5683	2.13808
$219$	−8.49614	−0.574116
$220$	2.21432	0.149289
$221$	0.915081	0.0615550
$222$	−3.51070	−0.235623
$223$	−15.0340	−1.00675	−0.503374	−	0.864069i	$-0.667908\pi$
$223$	−15.0340	−1.00675	−0.503374	+	0.864069i	$0.667908\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−11.6522	−0.775094
$227$	26.6872	1.77129	0.885644	−	0.464364i	$-0.153717\pi$
$227$	26.6872	1.77129	0.885644	+	0.464364i	$0.153717\pi$
$228$	−15.8861	−1.05208
$229$	−10.6312	−0.702528	−0.351264	−	0.936276i	$-0.614248\pi$
$229$	−10.6312	−0.702528	−0.351264	+	0.936276i	$0.614248\pi$
$230$	−18.8760	−1.24465
$231$	0	0
$232$	−3.07299	−0.201752
$233$	29.8617	1.95630	0.978152	−	0.207892i	$-0.0666604\pi$
$233$	29.8617	1.95630	0.978152	+	0.207892i	$0.0666604\pi$
$234$	−2.42598	−0.158592
$235$	9.07877	0.592234
$236$	−11.3035	−0.735797
$237$	6.83524	0.443997
$238$	0	0
$239$	−7.13508	−0.461530	−0.230765	−	0.973010i	$-0.574123\pi$
$239$	−7.13508	−0.461530	−0.230765	+	0.973010i	$0.574123\pi$
$240$	3.52543	0.227565
$241$	1.30406	0.0840018	0.0420009	−	0.999118i	$-0.486627\pi$
$241$	1.30406	0.0840018	0.0420009	+	0.999118i	$0.486627\pi$
$242$	2.05288	0.131964
$243$	−1.00000	−0.0641500
$244$	1.09832	0.0703125
$245$	0	0
$246$	19.7631	1.26005
$247$	−8.47816	−0.539453
$248$	−1.40180	−0.0890142
$249$	−2.73247	−0.173163
$250$	2.05288	0.129836
$251$	29.1143	1.83768	0.918840	−	0.394629i	$-0.129127\pi$
$251$	29.1143	1.83768	0.918840	+	0.394629i	$0.129127\pi$
$252$	0	0
$253$	−9.19488	−0.578077
$254$	−3.99997	−0.250981
$255$	0.774347	0.0484915
$256$	12.0415	0.752593
$257$	−28.3278	−1.76704	−0.883521	−	0.468391i	$-0.844834\pi$
$257$	−28.3278	−1.76704	−0.883521	+	0.468391i	$0.844834\pi$
$258$	25.0956	1.56239
$259$	0	0
$260$	−2.61676	−0.162285
$261$	−6.98451	−0.432330
$262$	−38.1582	−2.35742
$263$	−0.361001	−0.0222603	−0.0111301	−	0.999938i	$-0.503543\pi$
$263$	−0.361001	−0.0222603	−0.0111301	+	0.999938i	$0.503543\pi$
$264$	−0.439973	−0.0270785
$265$	−8.85867	−0.544184
$266$	0	0
$267$	13.2266	0.809452
$268$	23.2378	1.41948
$269$	−28.5430	−1.74030	−0.870149	−	0.492788i	$-0.835978\pi$
$269$	−28.5430	−1.74030	−0.870149	+	0.492788i	$0.835978\pi$
$270$	−2.05288	−0.124934
$271$	−9.88169	−0.600270	−0.300135	−	0.953897i	$-0.597032\pi$
$271$	−9.88169	−0.600270	−0.300135	+	0.953897i	$0.597032\pi$
$272$	2.72990	0.165525
$273$	0	0
$274$	−47.2476	−2.85433
$275$	1.00000	0.0603023
$276$	20.3604	1.22555
$277$	21.3096	1.28037	0.640186	−	0.768220i	$-0.278857\pi$
$277$	21.3096	1.28037	0.640186	+	0.768220i	$0.278857\pi$
$278$	−5.93319	−0.355849
$279$	−3.18610	−0.190747
$280$	0	0
$281$	−6.61109	−0.394385	−0.197192	−	0.980365i	$-0.563182\pi$
$281$	−6.61109	−0.394385	−0.197192	+	0.980365i	$0.563182\pi$
$282$	−18.6376	−1.10986
$283$	−21.3981	−1.27198	−0.635991	−	0.771696i	$-0.719409\pi$
$283$	−21.3981	−1.27198	−0.635991	+	0.771696i	$0.719409\pi$
$284$	25.8152	1.53185
$285$	−7.17427	−0.424967
$286$	−2.42598	−0.143451
$287$	0	0
$288$	−8.11723	−0.478312
$289$	−16.4004	−0.964729
$290$	−14.3384	−0.841978
$291$	−15.9929	−0.937518
$292$	18.8132	1.10096
$293$	−3.24225	−0.189414	−0.0947072	−	0.995505i	$-0.530191\pi$
$293$	−3.24225	−0.189414	−0.0947072	+	0.995505i	$0.530191\pi$
$294$	0	0
$295$	−5.10474	−0.297210
$296$	0.752412	0.0437331
$297$	−1.00000	−0.0580259
$298$	−31.0414	−1.79818
$299$	10.8660	0.628398
$300$	−2.21432	−0.127844
$301$	0	0
$302$	−19.4184	−1.11740
$303$	6.33721	0.364063
$304$	−25.2924	−1.45062
$305$	0.496006	0.0284012
$306$	−1.58964	−0.0908738
$307$	29.1505	1.66371	0.831855	−	0.554994i	$-0.187279\pi$
$307$	29.1505	1.66371	0.831855	+	0.554994i	$0.187279\pi$
$308$	0	0
$309$	−4.78329	−0.272112
$310$	−6.54068	−0.371486
$311$	−9.34779	−0.530065	−0.265032	−	0.964240i	$-0.585383\pi$
$311$	−9.34779	−0.530065	−0.265032	+	0.964240i	$0.585383\pi$
$312$	0.519936	0.0294356
$313$	20.6192	1.16546	0.582732	−	0.812665i	$-0.301984\pi$
$313$	20.6192	1.16546	0.582732	+	0.812665i	$0.301984\pi$
$314$	18.9065	1.06695
$315$	0	0
$316$	−15.1354	−0.851433
$317$	30.3250	1.70322	0.851610	−	0.524176i	$-0.175627\pi$
$317$	30.3250	1.70322	0.851610	+	0.524176i	$0.175627\pi$
$318$	18.1858	1.01981
$319$	−6.98451	−0.391057
$320$	−9.61285	−0.537375
$321$	−0.194133	−0.0108355
$322$	0	0
$323$	−5.55537	−0.309109
$324$	2.21432	0.123018
$325$	−1.18175	−0.0655515
$326$	−33.2907	−1.84380
$327$	−15.3776	−0.850382
$328$	−4.23563	−0.233874
$329$	0	0
$330$	−2.05288	−0.113007
$331$	−4.92222	−0.270550	−0.135275	−	0.990808i	$-0.543192\pi$
$331$	−4.92222	−0.270550	−0.135275	+	0.990808i	$0.543192\pi$
$332$	6.05055	0.332067
$333$	1.71013	0.0937147
$334$	11.4460	0.626299
$335$	10.4943	0.573368
$336$	0	0
$337$	−12.3988	−0.675405	−0.337703	−	0.941253i	$-0.609650\pi$
$337$	−12.3988	−0.675405	−0.337703	+	0.941253i	$0.609650\pi$
$338$	−23.8206	−1.29567
$339$	5.67603	0.308280
$340$	−1.71465	−0.0929900
$341$	−3.18610	−0.172537
$342$	14.7279	0.796395
$343$	0	0
$344$	−5.37849	−0.289989
$345$	9.19488	0.495036
$346$	−20.7614	−1.11614
$347$	−11.1149	−0.596678	−0.298339	−	0.954460i	$-0.596433\pi$
$347$	−11.1149	−0.596678	−0.298339	+	0.954460i	$0.596433\pi$
$348$	15.4659	0.829061
$349$	28.3720	1.51872	0.759360	−	0.650670i	$-0.225512\pi$
$349$	28.3720	1.51872	0.759360	+	0.650670i	$0.225512\pi$
$350$	0	0
$351$	1.18175	0.0630769
$352$	−8.11723	−0.432650
$353$	18.9239	1.00722	0.503609	−	0.863932i	$-0.332005\pi$
$353$	18.9239	1.00722	0.503609	+	0.863932i	$0.332005\pi$
$354$	10.4794	0.556975
$355$	11.6583	0.618757
$356$	−29.2878	−1.55225
$357$	0	0
$358$	−21.4397	−1.13312
$359$	−0.715359	−0.0377552	−0.0188776	−	0.999822i	$-0.506009\pi$
$359$	−0.715359	−0.0377552	−0.0188776	+	0.999822i	$0.506009\pi$
$360$	0.439973	0.0231886
$361$	32.4701	1.70895
$362$	−44.2013	−2.32317
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	8.49614	0.444708
$366$	−1.01824	−0.0532244
$367$	−11.5861	−0.604787	−0.302394	−	0.953183i	$-0.597786\pi$
$367$	−11.5861	−0.604787	−0.302394	+	0.953183i	$0.597786\pi$
$368$	32.4159	1.68979
$369$	−9.62703	−0.501163
$370$	3.51070	0.182513
$371$	0	0
$372$	7.05504	0.365787
$373$	−7.53669	−0.390235	−0.195118	−	0.980780i	$-0.562509\pi$
$373$	−7.53669	−0.390235	−0.195118	+	0.980780i	$0.562509\pi$
$374$	−1.58964	−0.0821984
$375$	−1.00000	−0.0516398
$376$	3.99441	0.205996
$377$	8.25391	0.425098
$378$	0	0
$379$	−22.9367	−1.17818	−0.589090	−	0.808067i	$-0.700514\pi$
$379$	−22.9367	−1.17818	−0.589090	+	0.808067i	$0.700514\pi$
$380$	15.8861	0.814941
$381$	1.94847	0.0998231
$382$	−52.1859	−2.67006
$383$	16.1650	0.825994	0.412997	−	0.910732i	$-0.364482\pi$
$383$	16.1650	0.825994	0.412997	+	0.910732i	$0.364482\pi$
$384$	3.49957	0.178587
$385$	0	0
$386$	6.85968	0.349148
$387$	−12.2246	−0.621411
$388$	35.4133	1.79784
$389$	19.0853	0.967664	0.483832	−	0.875161i	$-0.339245\pi$
$389$	19.0853	0.967664	0.483832	+	0.875161i	$0.339245\pi$
$390$	2.42598	0.122844
$391$	7.12002	0.360075
$392$	0	0
$393$	18.5876	0.937622
$394$	56.9206	2.86762
$395$	−6.83524	−0.343918
$396$	2.21432	0.111274
$397$	9.03584	0.453496	0.226748	−	0.973953i	$-0.427191\pi$
$397$	9.03584	0.453496	0.226748	+	0.973953i	$0.427191\pi$
$398$	9.59729	0.481069
$399$	0	0
$400$	−3.52543	−0.176271
$401$	−28.7160	−1.43401	−0.717005	−	0.697069i	$-0.754487\pi$
$401$	−28.7160	−1.43401	−0.717005	+	0.697069i	$0.754487\pi$
$402$	−21.5436	−1.07450
$403$	3.76516	0.187556
$404$	−14.0326	−0.698148
$405$	1.00000	0.0496904
$406$	0	0
$407$	1.71013	0.0847681
$408$	0.340692	0.0168667
$409$	22.0098	1.08831	0.544157	−	0.838984i	$-0.316850\pi$
$409$	22.0098	1.08831	0.544157	+	0.838984i	$0.316850\pi$
$410$	−19.7631	−0.976032
$411$	23.0152	1.13526
$412$	10.5917	0.521817
$413$	0	0
$414$	−18.8760	−0.927704
$415$	2.73247	0.134131
$416$	9.59250	0.470311
$417$	2.89018	0.141533
$418$	14.7279	0.720366
$419$	−12.8793	−0.629193	−0.314597	−	0.949225i	$-0.601869\pi$
$419$	−12.8793	−0.629193	−0.314597	+	0.949225i	$0.601869\pi$
$420$	0	0
$421$	17.7808	0.866585	0.433292	−	0.901253i	$-0.357352\pi$
$421$	17.7808	0.866585	0.433292	+	0.901253i	$0.357352\pi$
$422$	−16.2169	−0.789427
$423$	9.07877	0.441425
$424$	−3.89758	−0.189283
$425$	−0.774347	−0.0375613
$426$	−23.9331	−1.15956
$427$	0	0
$428$	0.429873	0.0207787
$429$	1.18175	0.0570552
$430$	−25.0956	−1.21022
$431$	30.5637	1.47220	0.736101	−	0.676871i	$-0.236665\pi$
$431$	30.5637	1.47220	0.736101	+	0.676871i	$0.236665\pi$
$432$	3.52543	0.169617
$433$	27.7407	1.33313	0.666567	−	0.745445i	$-0.267763\pi$
$433$	27.7407	1.33313	0.666567	+	0.745445i	$0.267763\pi$
$434$	0	0
$435$	6.98451	0.334882
$436$	34.0509	1.63074
$437$	−65.9665	−3.15561
$438$	−17.4416	−0.833390
$439$	−12.8245	−0.612078	−0.306039	−	0.952019i	$-0.599004\pi$
$439$	−12.8245	−0.612078	−0.306039	+	0.952019i	$0.599004\pi$
$440$	0.439973	0.0209749
$441$	0	0
$442$	1.87855	0.0893536
$443$	−9.55340	−0.453896	−0.226948	−	0.973907i	$-0.572875\pi$
$443$	−9.55340	−0.453896	−0.226948	+	0.973907i	$0.572875\pi$
$444$	−3.78678	−0.179713
$445$	−13.2266	−0.626999
$446$	−30.8629	−1.46140
$447$	15.1209	0.715194
$448$	0	0
$449$	−6.51474	−0.307449	−0.153725	−	0.988114i	$-0.549127\pi$
$449$	−6.51474	−0.307449	−0.153725	+	0.988114i	$0.549127\pi$
$450$	2.05288	0.0967737
$451$	−9.62703	−0.453319
$452$	−12.5685	−0.591175
$453$	9.45909	0.444427
$454$	54.7856	2.57121
$455$	0	0
$456$	−3.15648	−0.147816
$457$	16.4987	0.771777	0.385888	−	0.922545i	$-0.373895\pi$
$457$	16.4987	0.771777	0.385888	+	0.922545i	$0.373895\pi$
$458$	−21.8245	−1.01979
$459$	0.774347	0.0361434
$460$	−20.3604	−0.949309
$461$	−0.871211	−0.0405764	−0.0202882	−	0.999794i	$-0.506458\pi$
$461$	−0.871211	−0.0405764	−0.0202882	+	0.999794i	$0.506458\pi$
$462$	0	0
$463$	11.8549	0.550944	0.275472	−	0.961309i	$-0.411166\pi$
$463$	11.8549	0.550944	0.275472	+	0.961309i	$0.411166\pi$
$464$	24.6234	1.14311
$465$	3.18610	0.147752
$466$	61.3025	2.83978
$467$	22.4384	1.03832	0.519162	−	0.854676i	$-0.326244\pi$
$467$	22.4384	1.03832	0.519162	+	0.854676i	$0.326244\pi$
$468$	−2.61676	−0.120960
$469$	0	0
$470$	18.6376	0.859690
$471$	−9.20973	−0.424362
$472$	−2.24595	−0.103378
$473$	−12.2246	−0.562087
$474$	14.0319	0.644508
$475$	7.17427	0.329178
$476$	0	0
$477$	−8.85867	−0.405611
$478$	−14.6475	−0.669960
$479$	−3.08962	−0.141168	−0.0705842	−	0.997506i	$-0.522486\pi$
$479$	−3.08962	−0.141168	−0.0705842	+	0.997506i	$0.522486\pi$
$480$	8.11723	0.370499
$481$	−2.02094	−0.0921470
$482$	2.67708	0.121937
$483$	0	0
$484$	2.21432	0.100651
$485$	15.9929	0.726199
$486$	−2.05288	−0.0931206
$487$	−6.27941	−0.284547	−0.142274	−	0.989827i	$-0.545441\pi$
$487$	−6.27941	−0.284547	−0.142274	+	0.989827i	$0.545441\pi$
$488$	0.218229	0.00987878
$489$	16.2166	0.733339
$490$	0	0
$491$	−0.724085	−0.0326775	−0.0163387	−	0.999867i	$-0.505201\pi$
$491$	−0.724085	−0.0326775	−0.0163387	+	0.999867i	$0.505201\pi$
$492$	21.3173	0.961059
$493$	5.40843	0.243584
$494$	−17.4047	−0.783072
$495$	1.00000	0.0449467
$496$	11.2324	0.504348
$497$	0	0
$498$	−5.60943	−0.251364
$499$	−14.4673	−0.647646	−0.323823	−	0.946118i	$-0.604968\pi$
$499$	−14.4673	−0.647646	−0.323823	+	0.946118i	$0.604968\pi$
$500$	2.21432	0.0990274
$501$	−5.57559	−0.249099
$502$	59.7683	2.66759
$503$	11.1625	0.497709	0.248855	−	0.968541i	$-0.419946\pi$
$503$	11.1625	0.497709	0.248855	+	0.968541i	$0.419946\pi$
$504$	0	0
$505$	−6.33721	−0.282002
$506$	−18.8760	−0.839140
$507$	11.6035	0.515328
$508$	−4.31453	−0.191426
$509$	13.4299	0.595271	0.297635	−	0.954680i	$-0.403802\pi$
$509$	13.4299	0.595271	0.297635	+	0.954680i	$0.403802\pi$
$510$	1.58964	0.0703905
$511$	0	0
$512$	31.7189	1.40179
$513$	−7.17427	−0.316752
$514$	−58.1537	−2.56505
$515$	4.78329	0.210777
$516$	27.0691	1.19165
$517$	9.07877	0.399284
$518$	0	0
$519$	10.1133	0.443925
$520$	−0.519936	−0.0228007
$521$	8.66905	0.379798	0.189899	−	0.981804i	$-0.439184\pi$
$521$	8.66905	0.379798	0.189899	+	0.981804i	$0.439184\pi$
$522$	−14.3384	−0.627573
$523$	−15.4472	−0.675459	−0.337729	−	0.941243i	$-0.609659\pi$
$523$	−15.4472	−0.675459	−0.337729	+	0.941243i	$0.609659\pi$
$524$	−41.1589	−1.79804
$525$	0	0
$526$	−0.741092	−0.0323132
$527$	2.46715	0.107471
$528$	3.52543	0.153425
$529$	61.5458	2.67590
$530$	−18.1858	−0.789941
$531$	−5.10474	−0.221527
$532$	0	0
$533$	11.3767	0.492780
$534$	27.1525	1.17501
$535$	0.194133	0.00839312
$536$	4.61723	0.199434
$537$	10.4437	0.450680
$538$	−58.5954	−2.52623
$539$	0	0
$540$	−2.21432	−0.0952891
$541$	−15.3340	−0.659261	−0.329631	−	0.944110i	$-0.606924\pi$
$541$	−15.3340	−0.659261	−0.329631	+	0.944110i	$0.606924\pi$
$542$	−20.2859	−0.871355
$543$	21.5313	0.923998
$544$	6.28555	0.269491
$545$	15.3776	0.658703
$546$	0	0
$547$	−27.4663	−1.17437	−0.587187	−	0.809451i	$-0.699765\pi$
$547$	−27.4663	−1.17437	−0.587187	+	0.809451i	$0.699765\pi$
$548$	−50.9631	−2.17704
$549$	0.496006	0.0211690
$550$	2.05288	0.0875351
$551$	−50.1087	−2.13470
$552$	4.04550	0.172188
$553$	0	0
$554$	43.7462	1.85860
$555$	−1.71013	−0.0725911
$556$	−6.39978	−0.271411
$557$	−38.7998	−1.64400	−0.822000	−	0.569487i	$-0.807142\pi$
$557$	−38.7998	−1.64400	−0.822000	+	0.569487i	$0.807142\pi$
$558$	−6.54068	−0.276889
$559$	14.4464	0.611016
$560$	0	0
$561$	0.774347	0.0326929
$562$	−13.5718	−0.572491
$563$	−32.6562	−1.37630	−0.688148	−	0.725570i	$-0.741576\pi$
$563$	−32.6562	−1.37630	−0.688148	+	0.725570i	$0.741576\pi$
$564$	−20.1033	−0.846502
$565$	−5.67603	−0.238792
$566$	−43.9277	−1.84642
$567$	0	0
$568$	5.12933	0.215222
$569$	−17.2073	−0.721368	−0.360684	−	0.932688i	$-0.617457\pi$
$569$	−17.2073	−0.721368	−0.360684	+	0.932688i	$0.617457\pi$
$570$	−14.7279	−0.616885
$571$	−8.26678	−0.345954	−0.172977	−	0.984926i	$-0.555339\pi$
$571$	−8.26678	−0.345954	−0.172977	+	0.984926i	$0.555339\pi$
$572$	−2.61676	−0.109412
$573$	25.4208	1.06197
$574$	0	0
$575$	−9.19488	−0.383453
$576$	−9.61285	−0.400535
$577$	−16.2244	−0.675429	−0.337714	−	0.941249i	$-0.609654\pi$
$577$	−16.2244	−0.675429	−0.337714	+	0.941249i	$0.609654\pi$
$578$	−33.6680	−1.40041
$579$	−3.34149	−0.138868
$580$	−15.4659	−0.642188
$581$	0	0
$582$	−32.8314	−1.36091
$583$	−8.85867	−0.366889
$584$	3.73807	0.154683
$585$	−1.18175	−0.0488592
$586$	−6.65596	−0.274955
$587$	−17.6303	−0.727682	−0.363841	−	0.931461i	$-0.618535\pi$
$587$	−17.6303	−0.727682	−0.363841	+	0.931461i	$0.618535\pi$
$588$	0	0
$589$	−22.8579	−0.941844
$590$	−10.4794	−0.431431
$591$	−27.7272	−1.14054
$592$	−6.02895	−0.247788
$593$	−8.51365	−0.349614	−0.174807	−	0.984603i	$-0.555930\pi$
$593$	−8.51365	−0.349614	−0.174807	+	0.984603i	$0.555930\pi$
$594$	−2.05288	−0.0842307
$595$	0	0
$596$	−33.4825	−1.37150
$597$	−4.67504	−0.191336
$598$	22.3066	0.912186
$599$	39.0409	1.59517	0.797583	−	0.603209i	$-0.206112\pi$
$599$	39.0409	1.59517	0.797583	+	0.603209i	$0.206112\pi$
$600$	−0.439973	−0.0179618
$601$	−27.3324	−1.11491	−0.557456	−	0.830206i	$-0.688223\pi$
$601$	−27.3324	−1.11491	−0.557456	+	0.830206i	$0.688223\pi$
$602$	0	0
$603$	10.4943	0.427363
$604$	−20.9455	−0.852259
$605$	1.00000	0.0406558
$606$	13.0095	0.528476
$607$	−43.9488	−1.78383	−0.891914	−	0.452205i	$-0.850638\pi$
$607$	−43.9488	−1.78383	−0.891914	+	0.452205i	$0.850638\pi$
$608$	−58.2352	−2.36175
$609$	0	0
$610$	1.01824	0.0412274
$611$	−10.7288	−0.434041
$612$	−1.71465	−0.0693107
$613$	13.3196	0.537973	0.268986	−	0.963144i	$-0.413311\pi$
$613$	13.3196	0.537973	0.268986	+	0.963144i	$0.413311\pi$
$614$	59.8426	2.41505
$615$	9.62703	0.388199
$616$	0	0
$617$	24.6797	0.993566	0.496783	−	0.867875i	$-0.334515\pi$
$617$	24.6797	0.993566	0.496783	+	0.867875i	$0.334515\pi$
$618$	−9.81952	−0.394999
$619$	−10.3078	−0.414307	−0.207153	−	0.978309i	$-0.566420\pi$
$619$	−10.3078	−0.414307	−0.207153	+	0.978309i	$0.566420\pi$
$620$	−7.05504	−0.283337
$621$	9.19488	0.368978
$622$	−19.1899	−0.769445
$623$	0	0
$624$	−4.16616	−0.166780
$625$	1.00000	0.0400000
$626$	42.3287	1.69179
$627$	−7.17427	−0.286513
$628$	20.3933	0.813781
$629$	−1.32424	−0.0528007
$630$	0	0
$631$	3.78540	0.150694	0.0753472	−	0.997157i	$-0.475993\pi$
$631$	3.78540	0.150694	0.0753472	+	0.997157i	$0.475993\pi$
$632$	−3.00732	−0.119625
$633$	7.89959	0.313981
$634$	62.2535	2.47240
$635$	−1.94847	−0.0773226
$636$	19.6159	0.777822
$637$	0	0
$638$	−14.3384	−0.567661
$639$	11.6583	0.461194
$640$	−3.49957	−0.138333
$641$	31.9358	1.26139	0.630695	−	0.776031i	$-0.282770\pi$
$641$	31.9358	1.26139	0.630695	+	0.776031i	$0.282770\pi$
$642$	−0.398533	−0.0157288
$643$	−14.8549	−0.585821	−0.292910	−	0.956140i	$-0.594624\pi$
$643$	−14.8549	−0.585821	−0.292910	+	0.956140i	$0.594624\pi$
$644$	0	0
$645$	12.2246	0.481343
$646$	−11.4045	−0.448705
$647$	−9.20733	−0.361978	−0.180989	−	0.983485i	$-0.557930\pi$
$647$	−9.20733	−0.361978	−0.180989	+	0.983485i	$0.557930\pi$
$648$	0.439973	0.0172838
$649$	−5.10474	−0.200379
$650$	−2.42598	−0.0951549
$651$	0	0
$652$	−35.9087	−1.40629
$653$	35.5407	1.39082	0.695408	−	0.718615i	$-0.255224\pi$
$653$	35.5407	1.39082	0.695408	+	0.718615i	$0.255224\pi$
$654$	−31.5683	−1.23442
$655$	−18.5876	−0.726279
$656$	33.9394	1.32511
$657$	8.49614	0.331466
$658$	0	0
$659$	18.0578	0.703432	0.351716	−	0.936107i	$-0.385598\pi$
$659$	18.0578	0.703432	0.351716	+	0.936107i	$0.385598\pi$
$660$	−2.21432	−0.0861923
$661$	−22.1577	−0.861835	−0.430917	−	0.902391i	$-0.641810\pi$
$661$	−22.1577	−0.861835	−0.430917	+	0.902391i	$0.641810\pi$
$662$	−10.1047	−0.392731
$663$	−0.915081	−0.0355388
$664$	1.20221	0.0466548
$665$	0	0
$666$	3.51070	0.136037
$667$	64.2217	2.48667
$668$	12.3461	0.477687
$669$	15.0340	0.581246
$670$	21.5436	0.832304
$671$	0.496006	0.0191481
$672$	0	0
$673$	8.63603	0.332895	0.166447	−	0.986050i	$-0.446770\pi$
$673$	8.63603	0.332895	0.166447	+	0.986050i	$0.446770\pi$
$674$	−25.4532	−0.980422
$675$	−1.00000	−0.0384900
$676$	−25.6938	−0.988223
$677$	22.5078	0.865044	0.432522	−	0.901623i	$-0.357624\pi$
$677$	22.5078	0.865044	0.432522	+	0.901623i	$0.357624\pi$
$678$	11.6522	0.447501
$679$	0	0
$680$	−0.340692	−0.0130649
$681$	−26.6872	−1.02265
$682$	−6.54068	−0.250456
$683$	18.4936	0.707638	0.353819	−	0.935314i	$-0.384883\pi$
$683$	18.4936	0.707638	0.353819	+	0.935314i	$0.384883\pi$
$684$	15.8861	0.607421
$685$	−23.0152	−0.879367
$686$	0	0
$687$	10.6312	0.405605
$688$	43.0969	1.64305
$689$	10.4687	0.398826
$690$	18.8760	0.718597
$691$	17.4492	0.663798	0.331899	−	0.943315i	$-0.392311\pi$
$691$	17.4492	0.663798	0.331899	+	0.943315i	$0.392311\pi$
$692$	−22.3941	−0.851295
$693$	0	0
$694$	−22.8175	−0.866141
$695$	−2.89018	−0.109631
$696$	3.07299	0.116482
$697$	7.45466	0.282365
$698$	58.2444	2.20458
$699$	−29.8617	−1.12947
$700$	0	0
$701$	36.2220	1.36809	0.684044	−	0.729441i	$-0.260220\pi$
$701$	36.2220	1.36809	0.684044	+	0.729441i	$0.260220\pi$
$702$	2.42598	0.0915628
$703$	12.2689	0.462732
$704$	−9.61285	−0.362298
$705$	−9.07877	−0.341926
$706$	38.8485	1.46208
$707$	0	0
$708$	11.3035	0.424813
$709$	−0.933784	−0.0350690	−0.0175345	−	0.999846i	$-0.505582\pi$
$709$	−0.933784	−0.0350690	−0.0175345	+	0.999846i	$0.505582\pi$
$710$	23.9331	0.898192
$711$	−6.83524	−0.256342
$712$	−5.81932	−0.218088
$713$	29.2958	1.09714
$714$	0	0
$715$	−1.18175	−0.0441948
$716$	−23.1257	−0.864249
$717$	7.13508	0.266464
$718$	−1.46855	−0.0548057
$719$	−7.17767	−0.267682	−0.133841	−	0.991003i	$-0.542731\pi$
$719$	−7.17767	−0.267682	−0.133841	+	0.991003i	$0.542731\pi$
$720$	−3.52543	−0.131385
$721$	0	0
$722$	66.6573	2.48073
$723$	−1.30406	−0.0484984
$724$	−47.6773	−1.77191
$725$	−6.98451	−0.259398
$726$	−2.05288	−0.0761896
$727$	5.16019	0.191381	0.0956904	−	0.995411i	$-0.469494\pi$
$727$	5.16019	0.191381	0.0956904	+	0.995411i	$0.469494\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	17.4416	0.645541
$731$	9.46607	0.350115
$732$	−1.09832	−0.0405950
$733$	23.9879	0.886014	0.443007	−	0.896518i	$-0.353912\pi$
$733$	23.9879	0.886014	0.443007	+	0.896518i	$0.353912\pi$
$734$	−23.7848	−0.877913
$735$	0	0
$736$	74.6369	2.75115
$737$	10.4943	0.386564
$738$	−19.7631	−0.727491
$739$	34.8237	1.28101	0.640504	−	0.767955i	$-0.278725\pi$
$739$	34.8237	1.28101	0.640504	+	0.767955i	$0.278725\pi$
$740$	3.78678	0.139205
$741$	8.47816	0.311453
$742$	0	0
$743$	20.9385	0.768159	0.384079	−	0.923300i	$-0.374519\pi$
$743$	20.9385	0.768159	0.384079	+	0.923300i	$0.374519\pi$
$744$	1.40180	0.0513924
$745$	−15.1209	−0.553987
$746$	−15.4719	−0.566468
$747$	2.73247	0.0999757
$748$	−1.71465	−0.0626939
$749$	0	0
$750$	−2.05288	−0.0749606
$751$	25.4386	0.928268	0.464134	−	0.885765i	$-0.346366\pi$
$751$	25.4386	0.928268	0.464134	+	0.885765i	$0.346366\pi$
$752$	−32.0066	−1.16716
$753$	−29.1143	−1.06099
$754$	16.9443	0.617075
$755$	−9.45909	−0.344252
$756$	0	0
$757$	−24.0730	−0.874947	−0.437473	−	0.899231i	$-0.644127\pi$
$757$	−24.0730	−0.874947	−0.437473	+	0.899231i	$0.644127\pi$
$758$	−47.0864	−1.71025
$759$	9.19488	0.333753
$760$	3.15648	0.114498
$761$	−6.13610	−0.222433	−0.111217	−	0.993796i	$-0.535475\pi$
$761$	−6.13610	−0.222433	−0.111217	+	0.993796i	$0.535475\pi$
$762$	3.99997	0.144904
$763$	0	0
$764$	−56.2898	−2.03649
$765$	−0.774347	−0.0279966
$766$	33.1849	1.19902
$767$	6.03251	0.217821
$768$	−12.0415	−0.434510
$769$	−31.5094	−1.13626	−0.568130	−	0.822939i	$-0.692333\pi$
$769$	−31.5094	−1.13626	−0.568130	+	0.822939i	$0.692333\pi$
$770$	0	0
$771$	28.3278	1.02020
$772$	7.39912	0.266300
$773$	30.3594	1.09195	0.545976	−	0.837801i	$-0.316159\pi$
$773$	30.3594	1.09195	0.545976	+	0.837801i	$0.316159\pi$
$774$	−25.0956	−0.902043
$775$	−3.18610	−0.114448
$776$	7.03643	0.252593
$777$	0	0
$778$	39.1799	1.40467
$779$	−69.0669	−2.47458
$780$	2.61676	0.0936952
$781$	11.6583	0.417166
$782$	14.6166	0.522687
$783$	6.98451	0.249606
$784$	0	0
$785$	9.20973	0.328709
$786$	38.1582	1.36106
$787$	−23.0006	−0.819882	−0.409941	−	0.912112i	$-0.634451\pi$
$787$	−23.0006	−0.819882	−0.409941	+	0.912112i	$0.634451\pi$
$788$	61.3969	2.18717
$789$	0.361001	0.0128520
$790$	−14.0319	−0.499234
$791$	0	0
$792$	0.439973	0.0156338
$793$	−0.586154	−0.0208149
$794$	18.5495	0.658297
$795$	8.85867	0.314185
$796$	10.3520	0.366918
$797$	10.7217	0.379783	0.189892	−	0.981805i	$-0.439186\pi$
$797$	10.7217	0.379783	0.189892	+	0.981805i	$0.439186\pi$
$798$	0	0
$799$	−7.03012	−0.248708
$800$	−8.11723	−0.286987
$801$	−13.2266	−0.467337
$802$	−58.9505	−2.08162
$803$	8.49614	0.299822
$804$	−23.2378	−0.819536
$805$	0	0
$806$	7.72942	0.272257
$807$	28.5430	1.00476
$808$	−2.78820	−0.0980885
$809$	10.3644	0.364392	0.182196	−	0.983262i	$-0.441680\pi$
$809$	10.3644	0.364392	0.182196	+	0.983262i	$0.441680\pi$
$810$	2.05288	0.0721309
$811$	44.9382	1.57800	0.788998	−	0.614396i	$-0.210600\pi$
$811$	44.9382	1.57800	0.788998	+	0.614396i	$0.210600\pi$
$812$	0	0
$813$	9.88169	0.346566
$814$	3.51070	0.123050
$815$	−16.2166	−0.568042
$816$	−2.72990	−0.0955657
$817$	−87.7025	−3.06832
$818$	45.1835	1.57980
$819$	0	0
$820$	−21.3173	−0.744433
$821$	−33.7511	−1.17792	−0.588961	−	0.808161i	$-0.700463\pi$
$821$	−33.7511	−1.17792	−0.588961	+	0.808161i	$0.700463\pi$
$822$	47.2476	1.64795
$823$	−21.9955	−0.766715	−0.383358	−	0.923600i	$-0.625232\pi$
$823$	−21.9955	−0.766715	−0.383358	+	0.923600i	$0.625232\pi$
$824$	2.10452	0.0733143
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−13.1948	−0.458827	−0.229413	−	0.973329i	$-0.573681\pi$
$827$	−13.1948	−0.458827	−0.229413	+	0.973329i	$0.573681\pi$
$828$	−20.3604	−0.707573
$829$	2.16010	0.0750235	0.0375117	−	0.999296i	$-0.488057\pi$
$829$	2.16010	0.0750235	0.0375117	+	0.999296i	$0.488057\pi$
$830$	5.60943	0.194706
$831$	−21.3096	−0.739223
$832$	11.3599	0.393835
$833$	0	0
$834$	5.93319	0.205450
$835$	5.57559	0.192951
$836$	15.8861	0.549433
$837$	3.18610	0.110128
$838$	−26.4396	−0.913340
$839$	8.58101	0.296249	0.148125	−	0.988969i	$-0.452676\pi$
$839$	8.58101	0.296249	0.148125	+	0.988969i	$0.452676\pi$
$840$	0	0
$841$	19.7834	0.682185
$842$	36.5019	1.25794
$843$	6.61109	0.227698
$844$	−17.4922	−0.602107
$845$	−11.6035	−0.399172
$846$	18.6376	0.640775
$847$	0	0
$848$	31.2306	1.07246
$849$	21.3981	0.734379
$850$	−1.58964	−0.0545243
$851$	−15.7245	−0.539028
$852$	−25.8152	−0.884413
$853$	48.2642	1.65253	0.826267	−	0.563279i	$-0.190460\pi$
$853$	48.2642	1.65253	0.826267	+	0.563279i	$0.190460\pi$
$854$	0	0
$855$	7.17427	0.245355
$856$	0.0854134	0.00291937
$857$	17.8631	0.610191	0.305095	−	0.952322i	$-0.401312\pi$
$857$	17.8631	0.610191	0.305095	+	0.952322i	$0.401312\pi$
$858$	2.42598	0.0828217
$859$	46.9709	1.60263	0.801313	−	0.598246i	$-0.204135\pi$
$859$	46.9709	1.60263	0.801313	+	0.598246i	$0.204135\pi$
$860$	−27.0691	−0.923050
$861$	0	0
$862$	62.7437	2.13706
$863$	21.0431	0.716316	0.358158	−	0.933661i	$-0.383405\pi$
$863$	21.0431	0.716316	0.358158	+	0.933661i	$0.383405\pi$
$864$	8.11723	0.276154
$865$	−10.1133	−0.343863
$866$	56.9484	1.93518
$867$	16.4004	0.556986
$868$	0	0
$869$	−6.83524	−0.231870
$870$	14.3384	0.486116
$871$	−12.4017	−0.420214
$872$	6.76572	0.229116
$873$	15.9929	0.541276
$874$	−135.421	−4.58070
$875$	0	0
$876$	−18.8132	−0.635638
$877$	−23.8166	−0.804229	−0.402115	−	0.915589i	$-0.631725\pi$
$877$	−23.8166	−0.804229	−0.402115	+	0.915589i	$0.631725\pi$
$878$	−26.3271	−0.888496
$879$	3.24225	0.109358
$880$	−3.52543	−0.118842
$881$	−27.2269	−0.917299	−0.458649	−	0.888617i	$-0.651667\pi$
$881$	−27.2269	−0.917299	−0.458649	+	0.888617i	$0.651667\pi$
$882$	0	0
$883$	−43.1402	−1.45178	−0.725892	−	0.687808i	$-0.758573\pi$
$883$	−43.1402	−1.45178	−0.725892	+	0.687808i	$0.758573\pi$
$884$	2.02628	0.0681513
$885$	5.10474	0.171594
$886$	−19.6120	−0.658878
$887$	−37.0515	−1.24407	−0.622034	−	0.782990i	$-0.713693\pi$
$887$	−37.0515	−1.24407	−0.622034	+	0.782990i	$0.713693\pi$
$888$	−0.752412	−0.0252493
$889$	0	0
$890$	−27.1525	−0.910155
$891$	1.00000	0.0335013
$892$	−33.2900	−1.11463
$893$	65.1336	2.17961
$894$	31.0414	1.03818
$895$	−10.4437	−0.349095
$896$	0	0
$897$	−10.8660	−0.362806
$898$	−13.3740	−0.446295
$899$	22.2533	0.742190
$900$	2.21432	0.0738107
$901$	6.85968	0.228529
$902$	−19.7631	−0.658041
$903$	0	0
$904$	−2.49730	−0.0830589
$905$	−21.5313	−0.715726
$906$	19.4184	0.645133
$907$	−6.14258	−0.203961	−0.101981	−	0.994786i	$-0.532518\pi$
$907$	−6.14258	−0.203961	−0.101981	+	0.994786i	$0.532518\pi$
$908$	59.0939	1.96110
$909$	−6.33721	−0.210192
$910$	0	0
$911$	−30.5029	−1.01061	−0.505303	−	0.862942i	$-0.668619\pi$
$911$	−30.5029	−1.01061	−0.505303	+	0.862942i	$0.668619\pi$
$912$	25.2924	0.837514
$913$	2.73247	0.0904314
$914$	33.8699	1.12032
$915$	−0.496006	−0.0163975
$916$	−23.5408	−0.777811
$917$	0	0
$918$	1.58964	0.0524660
$919$	53.2656	1.75707	0.878536	−	0.477677i	$-0.158521\pi$
$919$	53.2656	1.75707	0.878536	+	0.477677i	$0.158521\pi$
$920$	−4.04550	−0.133376
$921$	−29.1505	−0.960543
$922$	−1.78849	−0.0589009
$923$	−13.7771	−0.453480
$924$	0	0
$925$	1.71013	0.0562288
$926$	24.3367	0.799753
$927$	4.78329	0.157104
$928$	56.6948	1.86110
$929$	−33.4444	−1.09728	−0.548638	−	0.836060i	$-0.684853\pi$
$929$	−33.4444	−1.09728	−0.548638	+	0.836060i	$0.684853\pi$
$930$	6.54068	0.214477
$931$	0	0
$932$	66.1233	2.16594
$933$	9.34779	0.306033
$934$	46.0633	1.50724
$935$	−0.774347	−0.0253238
$936$	−0.519936	−0.0169946
$937$	−10.4294	−0.340713	−0.170357	−	0.985382i	$-0.554492\pi$
$937$	−10.4294	−0.340713	−0.170357	+	0.985382i	$0.554492\pi$
$938$	0	0
$939$	−20.6192	−0.672881
$940$	20.1033	0.655698
$941$	38.0284	1.23969	0.619845	−	0.784724i	$-0.287195\pi$
$941$	38.0284	1.23969	0.619845	+	0.784724i	$0.287195\pi$
$942$	−18.9065	−0.616007
$943$	88.5194	2.88259
$944$	17.9964	0.585733
$945$	0	0
$946$	−25.0956	−0.815929
$947$	44.9923	1.46205	0.731027	−	0.682348i	$-0.239041\pi$
$947$	44.9923	1.46205	0.731027	+	0.682348i	$0.239041\pi$
$948$	15.1354	0.491575
$949$	−10.0403	−0.325921
$950$	14.7279	0.477837
$951$	−30.3250	−0.983354
$952$	0	0
$953$	−3.80303	−0.123192	−0.0615961	−	0.998101i	$-0.519619\pi$
$953$	−3.80303	−0.123192	−0.0615961	+	0.998101i	$0.519619\pi$
$954$	−18.1858	−0.588787
$955$	−25.4208	−0.822599
$956$	−15.7994	−0.510988
$957$	6.98451	0.225777
$958$	−6.34262	−0.204921
$959$	0	0
$960$	9.61285	0.310253
$961$	−20.8488	−0.672541
$962$	−4.14875	−0.133761
$963$	0.194133	0.00625586
$964$	2.88760	0.0930034
$965$	3.34149	0.107566
$966$	0	0
$967$	−10.5436	−0.339060	−0.169530	−	0.985525i	$-0.554225\pi$
$967$	−10.5436	−0.339060	−0.169530	+	0.985525i	$0.554225\pi$
$968$	0.439973	0.0141413
$969$	5.55537	0.178464
$970$	32.8314	1.05415
$971$	3.44816	0.110657	0.0553283	−	0.998468i	$-0.482379\pi$
$971$	3.44816	0.110657	0.0553283	+	0.998468i	$0.482379\pi$
$972$	−2.21432	−0.0710243
$973$	0	0
$974$	−12.8909	−0.413051
$975$	1.18175	0.0378462
$976$	−1.74863	−0.0559724
$977$	48.5845	1.55436	0.777178	−	0.629281i	$-0.216651\pi$
$977$	48.5845	1.55436	0.777178	+	0.629281i	$0.216651\pi$
$978$	33.2907	1.06452
$979$	−13.2266	−0.422722
$980$	0	0
$981$	15.3776	0.490968
$982$	−1.48646	−0.0474348
$983$	18.8903	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$983$	18.8903	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$984$	4.23563	0.135027
$985$	27.7272	0.883462
$986$	11.1029	0.353587
$987$	0	0
$988$	−18.7734	−0.597260
$989$	112.404	3.57423
$990$	2.05288	0.0652448
$991$	22.0494	0.700424	0.350212	−	0.936671i	$-0.386110\pi$
$991$	22.0494	0.700424	0.350212	+	0.936671i	$0.386110\pi$
$992$	25.8623	0.821128
$993$	4.92222	0.156202
$994$	0	0
$995$	4.67504	0.148209
$996$	−6.05055	−0.191719
$997$	−11.4649	−0.363097	−0.181549	−	0.983382i	$-0.558111\pi$
$997$	−11.4649	−0.363097	−0.181549	+	0.983382i	$0.558111\pi$
$998$	−29.6997	−0.940126
$999$	−1.71013	−0.0541062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8085.2.a.bw.1.6	✓	6
7.6	odd	2		8085.2.a.by.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8085.2.a.bw.1.6	✓	6	1.1	even	1	trivial
8085.2.a.by.1.6	yes	6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8085.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05288 q^{2} -1.00000 q^{3} +2.21432 q^{4} +1.00000 q^{5} -2.05288 q^{6} +0.439973 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.05288 q^{2} -1.00000 q^{3} +2.21432 q^{4} +1.00000 q^{5} -2.05288 q^{6} +0.439973 q^{8} +1.00000 q^{9} +2.05288 q^{10} +1.00000 q^{11} -2.21432 q^{12} -1.18175 q^{13} -1.00000 q^{15} -3.52543 q^{16} -0.774347 q^{17} +2.05288 q^{18} +7.17427 q^{19} +2.21432 q^{20} +2.05288 q^{22} -9.19488 q^{23} -0.439973 q^{24} +1.00000 q^{25} -2.42598 q^{26} -1.00000 q^{27} -6.98451 q^{29} -2.05288 q^{30} -3.18610 q^{31} -8.11723 q^{32} -1.00000 q^{33} -1.58964 q^{34} +2.21432 q^{36} +1.71013 q^{37} +14.7279 q^{38} +1.18175 q^{39} +0.439973 q^{40} -9.62703 q^{41} -12.2246 q^{43} +2.21432 q^{44} +1.00000 q^{45} -18.8760 q^{46} +9.07877 q^{47} +3.52543 q^{48} +2.05288 q^{50} +0.774347 q^{51} -2.61676 q^{52} -8.85867 q^{53} -2.05288 q^{54} +1.00000 q^{55} -7.17427 q^{57} -14.3384 q^{58} -5.10474 q^{59} -2.21432 q^{60} +0.496006 q^{61} -6.54068 q^{62} -9.61285 q^{64} -1.18175 q^{65} -2.05288 q^{66} +10.4943 q^{67} -1.71465 q^{68} +9.19488 q^{69} +11.6583 q^{71} +0.439973 q^{72} +8.49614 q^{73} +3.51070 q^{74} -1.00000 q^{75} +15.8861 q^{76} +2.42598 q^{78} -6.83524 q^{79} -3.52543 q^{80} +1.00000 q^{81} -19.7631 q^{82} +2.73247 q^{83} -0.774347 q^{85} -25.0956 q^{86} +6.98451 q^{87} +0.439973 q^{88} -13.2266 q^{89} +2.05288 q^{90} -20.3604 q^{92} +3.18610 q^{93} +18.6376 q^{94} +7.17427 q^{95} +8.11723 q^{96} +15.9929 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} + 6 q^{11} + 8 q^{13} - 6 q^{15} - 8 q^{16} + 6 q^{17} - 6 q^{19} - 14 q^{23} + 6 q^{25} - 4 q^{26} - 6 q^{27} - 6 q^{29} - 12 q^{31} - 6 q^{33} - 8 q^{34} - 12 q^{37} + 20 q^{38} - 8 q^{39} - 8 q^{41} - 30 q^{43} + 6 q^{45} - 16 q^{46} + 8 q^{48} - 6 q^{51} + 4 q^{52} - 2 q^{53} + 6 q^{55} + 6 q^{57} - 20 q^{58} - 2 q^{59} - 10 q^{61} - 16 q^{62} - 4 q^{64} + 8 q^{65} - 24 q^{67} - 16 q^{68} + 14 q^{69} - 4 q^{71} + 32 q^{73} + 4 q^{74} - 6 q^{75} + 16 q^{76} + 4 q^{78} - 20 q^{79} - 8 q^{80} + 6 q^{81} - 36 q^{82} - 6 q^{83} + 6 q^{85} - 8 q^{86} + 6 q^{87} - 6 q^{89} - 24 q^{92} + 12 q^{93} + 4 q^{94} - 6 q^{95} + 34 q^{97} + 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 + 6 * q^11 + 8 * q^13 - 6 * q^15 - 8 * q^16 + 6 * q^17 - 6 * q^19 - 14 * q^23 + 6 * q^25 - 4 * q^26 - 6 * q^27 - 6 * q^29 - 12 * q^31 - 6 * q^33 - 8 * q^34 - 12 * q^37 + 20 * q^38 - 8 * q^39 - 8 * q^41 - 30 * q^43 + 6 * q^45 - 16 * q^46 + 8 * q^48 - 6 * q^51 + 4 * q^52 - 2 * q^53 + 6 * q^55 + 6 * q^57 - 20 * q^58 - 2 * q^59 - 10 * q^61 - 16 * q^62 - 4 * q^64 + 8 * q^65 - 24 * q^67 - 16 * q^68 + 14 * q^69 - 4 * q^71 + 32 * q^73 + 4 * q^74 - 6 * q^75 + 16 * q^76 + 4 * q^78 - 20 * q^79 - 8 * q^80 + 6 * q^81 - 36 * q^82 - 6 * q^83 + 6 * q^85 - 8 * q^86 + 6 * q^87 - 6 * q^89 - 24 * q^92 + 12 * q^93 + 4 * q^94 - 6 * q^95 + 34 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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