LMFDB - Embedded newform 810.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(1,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("810.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.46788256372$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	810.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.267949 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.46410 q^{11} +0.267949 q^{13} -0.267949 q^{14} +1.00000 q^{16} +3.46410 q^{17} +5.92820 q^{19} -1.00000 q^{20} +3.46410 q^{22} +6.46410 q^{23} +1.00000 q^{25} -0.267949 q^{26} +0.267949 q^{28} -6.92820 q^{29} -1.46410 q^{31} -1.00000 q^{32} -3.46410 q^{34} -0.267949 q^{35} +8.00000 q^{37} -5.92820 q^{38} +1.00000 q^{40} +5.19615 q^{41} +5.46410 q^{43} -3.46410 q^{44} -6.46410 q^{46} -0.464102 q^{47} -6.92820 q^{49} -1.00000 q^{50} +0.267949 q^{52} +5.53590 q^{53} +3.46410 q^{55} -0.267949 q^{56} +6.92820 q^{58} +8.66025 q^{59} +12.3923 q^{61} +1.46410 q^{62} +1.00000 q^{64} -0.267949 q^{65} +8.00000 q^{67} +3.46410 q^{68} +0.267949 q^{70} +6.00000 q^{71} -14.3923 q^{73} -8.00000 q^{74} +5.92820 q^{76} -0.928203 q^{77} -14.3923 q^{79} -1.00000 q^{80} -5.19615 q^{82} +15.4641 q^{83} -3.46410 q^{85} -5.46410 q^{86} +3.46410 q^{88} -12.0000 q^{89} +0.0717968 q^{91} +6.46410 q^{92} +0.464102 q^{94} -5.92820 q^{95} +14.9282 q^{97} +6.92820 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 4 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{35} + 16 q^{37}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 4 * q^13 - 4 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 + 6 * q^23 + 2 * q^25 - 4 * q^26 + 4 * q^28 + 4 * q^31 - 2 * q^32 - 4 * q^35 + 16 * q^37 + 2 * q^38 + 2 * q^40 + 4 * q^43 - 6 * q^46 + 6 * q^47 - 2 * q^50 + 4 * q^52 + 18 * q^53 - 4 * q^56 + 4 * q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 + 4 * q^70 + 12 * q^71 - 8 * q^73 - 16 * q^74 - 2 * q^76 + 12 * q^77 - 8 * q^79 - 2 * q^80 + 24 * q^83 - 4 * q^86 - 24 * q^89 + 14 * q^91 + 6 * q^92 - 6 * q^94 + 2 * q^95 + 16 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.267949	0.101275	0.0506376	−	0.998717i	$-0.483875\pi$
$7$	0.267949	0.101275	0.0506376	+	0.998717i	$0.483875\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0.267949	0.0743157	0.0371579	−	0.999309i	$-0.488170\pi$
$13$	0.267949	0.0743157	0.0371579	+	0.999309i	$0.488170\pi$
$14$	−0.267949	−0.0716124
$15$	0	0
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	5.92820	1.36002	0.680012	−	0.733201i	$-0.261975\pi$
$19$	5.92820	1.36002	0.680012	+	0.733201i	$0.261975\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	3.46410	0.738549
$23$	6.46410	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$23$	6.46410	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.267949	−0.0525492
$27$	0	0
$28$	0.267949	0.0506376
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	−0.267949	−0.0452917
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−5.92820	−0.961682
$39$	0	0
$40$	1.00000	0.158114
$41$	5.19615	0.811503	0.405751	−	0.913984i	$-0.367010\pi$
$41$	5.19615	0.811503	0.405751	+	0.913984i	$0.367010\pi$
$42$	0	0
$43$	5.46410	0.833268	0.416634	−	0.909074i	$-0.363210\pi$
$43$	5.46410	0.833268	0.416634	+	0.909074i	$0.363210\pi$
$44$	−3.46410	−0.522233
$45$	0	0
$46$	−6.46410	−0.953080
$47$	−0.464102	−0.0676962	−0.0338481	−	0.999427i	$-0.510776\pi$
$47$	−0.464102	−0.0676962	−0.0338481	+	0.999427i	$0.510776\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0.267949	0.0371579
$53$	5.53590	0.760414	0.380207	−	0.924901i	$-0.375853\pi$
$53$	5.53590	0.760414	0.380207	+	0.924901i	$0.375853\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	−0.267949	−0.0358062
$57$	0	0
$58$	6.92820	0.909718
$59$	8.66025	1.12747	0.563735	−	0.825956i	$-0.309364\pi$
$59$	8.66025	1.12747	0.563735	+	0.825956i	$0.309364\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	1.46410	0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.267949	−0.0332350
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	3.46410	0.420084
$69$	0	0
$70$	0.267949	0.0320261
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−14.3923	−1.68449	−0.842246	−	0.539093i	$-0.818767\pi$
$73$	−14.3923	−1.68449	−0.842246	+	0.539093i	$0.818767\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	5.92820	0.680012
$77$	−0.928203	−0.105779
$78$	0	0
$79$	−14.3923	−1.61926	−0.809630	−	0.586940i	$-0.800332\pi$
$79$	−14.3923	−1.61926	−0.809630	+	0.586940i	$0.800332\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−5.19615	−0.573819
$83$	15.4641	1.69741	0.848703	−	0.528870i	$-0.177384\pi$
$83$	15.4641	1.69741	0.848703	+	0.528870i	$0.177384\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	−5.46410	−0.589209
$87$	0	0
$88$	3.46410	0.369274
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	0.0717968	0.00752635
$92$	6.46410	0.673929
$93$	0	0
$94$	0.464102	0.0478684
$95$	−5.92820	−0.608221
$96$	0	0
$97$	14.9282	1.51573	0.757865	−	0.652412i	$-0.226243\pi$
$97$	14.9282	1.51573	0.757865	+	0.652412i	$0.226243\pi$
$98$	6.92820	0.699854
$99$	0	0
$100$	1.00000	0.100000
$101$	2.53590	0.252331	0.126166	−	0.992009i	$-0.459733\pi$
$101$	2.53590	0.252331	0.126166	+	0.992009i	$0.459733\pi$
$102$	0	0
$103$	−4.80385	−0.473337	−0.236669	−	0.971590i	$-0.576056\pi$
$103$	−4.80385	−0.473337	−0.236669	+	0.971590i	$0.576056\pi$
$104$	−0.267949	−0.0262746
$105$	0	0
$106$	−5.53590	−0.537694
$107$	15.4641	1.49497	0.747486	−	0.664278i	$-0.231261\pi$
$107$	15.4641	1.49497	0.747486	+	0.664278i	$0.231261\pi$
$108$	0	0
$109$	9.85641	0.944073	0.472036	−	0.881579i	$-0.343519\pi$
$109$	9.85641	0.944073	0.472036	+	0.881579i	$0.343519\pi$
$110$	−3.46410	−0.330289
$111$	0	0
$112$	0.267949	0.0253188
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	−6.46410	−0.602781
$116$	−6.92820	−0.643268
$117$	0	0
$118$	−8.66025	−0.797241
$119$	0.928203	0.0850883
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.3923	−1.12195
$123$	0	0
$124$	−1.46410	−0.131480
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.80385	−0.426273	−0.213136	−	0.977022i	$-0.568368\pi$
$127$	−4.80385	−0.426273	−0.213136	+	0.977022i	$0.568368\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.267949	0.0235007
$131$	10.2679	0.897115	0.448557	−	0.893754i	$-0.351938\pi$
$131$	10.2679	0.897115	0.448557	+	0.893754i	$0.351938\pi$
$132$	0	0
$133$	1.58846	0.137737
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−3.46410	−0.297044
$137$	10.3923	0.887875	0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923	0.887875	0.443937	+	0.896058i	$0.353581\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−0.267949	−0.0226458
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−0.928203	−0.0776203
$144$	0	0
$145$	6.92820	0.575356
$146$	14.3923	1.19112
$147$	0	0
$148$	8.00000	0.657596
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−2.39230	−0.194683	−0.0973415	−	0.995251i	$-0.531034\pi$
$151$	−2.39230	−0.194683	−0.0973415	+	0.995251i	$0.531034\pi$
$152$	−5.92820	−0.480841
$153$	0	0
$154$	0.928203	0.0747967
$155$	1.46410	0.117599
$156$	0	0
$157$	−10.1244	−0.808012	−0.404006	−	0.914756i	$-0.632382\pi$
$157$	−10.1244	−0.808012	−0.404006	+	0.914756i	$0.632382\pi$
$158$	14.3923	1.14499
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.73205	0.136505
$162$	0	0
$163$	−17.8564	−1.39862	−0.699311	−	0.714818i	$-0.746510\pi$
$163$	−17.8564	−1.39862	−0.699311	+	0.714818i	$0.746510\pi$
$164$	5.19615	0.405751
$165$	0	0
$166$	−15.4641	−1.20025
$167$	−6.92820	−0.536120	−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820	−0.536120	−0.268060	+	0.963402i	$0.586383\pi$
$168$	0	0
$169$	−12.9282	−0.994477
$170$	3.46410	0.265684
$171$	0	0
$172$	5.46410	0.416634
$173$	14.3205	1.08877	0.544384	−	0.838836i	$-0.316763\pi$
$173$	14.3205	1.08877	0.544384	+	0.838836i	$0.316763\pi$
$174$	0	0
$175$	0.267949	0.0202551
$176$	−3.46410	−0.261116
$177$	0	0
$178$	12.0000	0.899438
$179$	−12.1244	−0.906217	−0.453108	−	0.891455i	$-0.649685\pi$
$179$	−12.1244	−0.906217	−0.453108	+	0.891455i	$0.649685\pi$
$180$	0	0
$181$	−19.4641	−1.44676	−0.723378	−	0.690453i	$-0.757411\pi$
$181$	−19.4641	−1.44676	−0.723378	+	0.690453i	$0.757411\pi$
$182$	−0.0717968	−0.00532193
$183$	0	0
$184$	−6.46410	−0.476540
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−0.464102	−0.0338481
$189$	0	0
$190$	5.92820	0.430077
$191$	9.46410	0.684798	0.342399	−	0.939555i	$-0.388760\pi$
$191$	9.46410	0.684798	0.342399	+	0.939555i	$0.388760\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	−14.9282	−1.07178
$195$	0	0
$196$	−6.92820	−0.494872
$197$	−18.4641	−1.31551	−0.657756	−	0.753231i	$-0.728494\pi$
$197$	−18.4641	−1.31551	−0.657756	+	0.753231i	$0.728494\pi$
$198$	0	0
$199$	26.2487	1.86072	0.930361	−	0.366645i	$-0.119494\pi$
$199$	26.2487	1.86072	0.930361	+	0.366645i	$0.119494\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−2.53590	−0.178425
$203$	−1.85641	−0.130294
$204$	0	0
$205$	−5.19615	−0.362915
$206$	4.80385	0.334700
$207$	0	0
$208$	0.267949	0.0185789
$209$	−20.5359	−1.42050
$210$	0	0
$211$	18.8564	1.29813	0.649064	−	0.760734i	$-0.275161\pi$
$211$	18.8564	1.29813	0.649064	+	0.760734i	$0.275161\pi$
$212$	5.53590	0.380207
$213$	0	0
$214$	−15.4641	−1.05710
$215$	−5.46410	−0.372649
$216$	0	0
$217$	−0.392305	−0.0266314
$218$	−9.85641	−0.667560
$219$	0	0
$220$	3.46410	0.233550
$221$	0.928203	0.0624377
$222$	0	0
$223$	−19.4641	−1.30341	−0.651706	−	0.758471i	$-0.725947\pi$
$223$	−19.4641	−1.30341	−0.651706	+	0.758471i	$0.725947\pi$
$224$	−0.267949	−0.0179031
$225$	0	0
$226$	0.928203	0.0617432
$227$	−4.39230	−0.291528	−0.145764	−	0.989319i	$-0.546564\pi$
$227$	−4.39230	−0.291528	−0.145764	+	0.989319i	$0.546564\pi$
$228$	0	0
$229$	−5.85641	−0.387002	−0.193501	−	0.981100i	$-0.561984\pi$
$229$	−5.85641	−0.387002	−0.193501	+	0.981100i	$0.561984\pi$
$230$	6.46410	0.426230
$231$	0	0
$232$	6.92820	0.454859
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	0.464102	0.0302747
$236$	8.66025	0.563735
$237$	0	0
$238$	−0.928203	−0.0601665
$239$	−15.4641	−1.00029	−0.500145	−	0.865942i	$-0.666720\pi$
$239$	−15.4641	−1.00029	−0.500145	+	0.865942i	$0.666720\pi$
$240$	0	0
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	12.3923	0.793336
$245$	6.92820	0.442627
$246$	0	0
$247$	1.58846	0.101071
$248$	1.46410	0.0929705
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.1244	0.765283	0.382641	−	0.923897i	$-0.375015\pi$
$251$	12.1244	0.765283	0.382641	+	0.923897i	$0.375015\pi$
$252$	0	0
$253$	−22.3923	−1.40779
$254$	4.80385	0.301420
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.14359	−0.258470	−0.129235	−	0.991614i	$-0.541252\pi$
$257$	−4.14359	−0.258470	−0.129235	+	0.991614i	$0.541252\pi$
$258$	0	0
$259$	2.14359	0.133196
$260$	−0.267949	−0.0166175
$261$	0	0
$262$	−10.2679	−0.634356
$263$	26.3205	1.62299	0.811496	−	0.584358i	$-0.198654\pi$
$263$	26.3205	1.62299	0.811496	+	0.584358i	$0.198654\pi$
$264$	0	0
$265$	−5.53590	−0.340068
$266$	−1.58846	−0.0973946
$267$	0	0
$268$	8.00000	0.488678
$269$	0.928203	0.0565935	0.0282968	−	0.999600i	$-0.490992\pi$
$269$	0.928203	0.0565935	0.0282968	+	0.999600i	$0.490992\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	−10.3923	−0.627822
$275$	−3.46410	−0.208893
$276$	0	0
$277$	15.7321	0.945247	0.472624	−	0.881264i	$-0.343307\pi$
$277$	15.7321	0.945247	0.472624	+	0.881264i	$0.343307\pi$
$278$	13.0000	0.779688
$279$	0	0
$280$	0.267949	0.0160130
$281$	−31.0526	−1.85244	−0.926220	−	0.376983i	$-0.876962\pi$
$281$	−31.0526	−1.85244	−0.926220	+	0.376983i	$0.876962\pi$
$282$	0	0
$283$	31.3205	1.86181	0.930905	−	0.365260i	$-0.119020\pi$
$283$	31.3205	1.86181	0.930905	+	0.365260i	$0.119020\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0.928203	0.0548858
$287$	1.39230	0.0821852
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−6.92820	−0.406838
$291$	0	0
$292$	−14.3923	−0.842246
$293$	−13.3923	−0.782387	−0.391193	−	0.920308i	$-0.627938\pi$
$293$	−13.3923	−0.782387	−0.391193	+	0.920308i	$0.627938\pi$
$294$	0	0
$295$	−8.66025	−0.504219
$296$	−8.00000	−0.464991
$297$	0	0
$298$	18.0000	1.04271
$299$	1.73205	0.100167
$300$	0	0
$301$	1.46410	0.0843894
$302$	2.39230	0.137662
$303$	0	0
$304$	5.92820	0.340006
$305$	−12.3923	−0.709581
$306$	0	0
$307$	−8.39230	−0.478974	−0.239487	−	0.970900i	$-0.576979\pi$
$307$	−8.39230	−0.478974	−0.239487	+	0.970900i	$0.576979\pi$
$308$	−0.928203	−0.0528893
$309$	0	0
$310$	−1.46410	−0.0831554
$311$	15.4641	0.876889	0.438444	−	0.898758i	$-0.355530\pi$
$311$	15.4641	0.876889	0.438444	+	0.898758i	$0.355530\pi$
$312$	0	0
$313$	31.3205	1.77034	0.885170	−	0.465268i	$-0.154042\pi$
$313$	31.3205	1.77034	0.885170	+	0.465268i	$0.154042\pi$
$314$	10.1244	0.571350
$315$	0	0
$316$	−14.3923	−0.809630
$317$	−24.4641	−1.37404	−0.687020	−	0.726638i	$-0.741082\pi$
$317$	−24.4641	−1.37404	−0.687020	+	0.726638i	$0.741082\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−1.73205	−0.0965234
$323$	20.5359	1.14265
$324$	0	0
$325$	0.267949	0.0148631
$326$	17.8564	0.988975
$327$	0	0
$328$	−5.19615	−0.286910
$329$	−0.124356	−0.00685595
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	15.4641	0.848703
$333$	0	0
$334$	6.92820	0.379094
$335$	−8.00000	−0.437087
$336$	0	0
$337$	20.9282	1.14003	0.570016	−	0.821634i	$-0.306937\pi$
$337$	20.9282	1.14003	0.570016	+	0.821634i	$0.306937\pi$
$338$	12.9282	0.703202
$339$	0	0
$340$	−3.46410	−0.187867
$341$	5.07180	0.274653
$342$	0	0
$343$	−3.73205	−0.201512
$344$	−5.46410	−0.294605
$345$	0	0
$346$	−14.3205	−0.769875
$347$	26.7846	1.43787	0.718937	−	0.695076i	$-0.244629\pi$
$347$	26.7846	1.43787	0.718937	+	0.695076i	$0.244629\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−0.267949	−0.0143225
$351$	0	0
$352$	3.46410	0.184637
$353$	1.60770	0.0855690	0.0427845	−	0.999084i	$-0.486377\pi$
$353$	1.60770	0.0855690	0.0427845	+	0.999084i	$0.486377\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	−12.0000	−0.635999
$357$	0	0
$358$	12.1244	0.640792
$359$	−15.4641	−0.816164	−0.408082	−	0.912945i	$-0.633802\pi$
$359$	−15.4641	−0.816164	−0.408082	+	0.912945i	$0.633802\pi$
$360$	0	0
$361$	16.1436	0.849663
$362$	19.4641	1.02301
$363$	0	0
$364$	0.0717968	0.00376317
$365$	14.3923	0.753328
$366$	0	0
$367$	−14.3923	−0.751272	−0.375636	−	0.926767i	$-0.622576\pi$
$367$	−14.3923	−0.751272	−0.375636	+	0.926767i	$0.622576\pi$
$368$	6.46410	0.336965
$369$	0	0
$370$	8.00000	0.415900
$371$	1.48334	0.0770111
$372$	0	0
$373$	2.92820	0.151617	0.0758083	−	0.997122i	$-0.475846\pi$
$373$	2.92820	0.151617	0.0758083	+	0.997122i	$0.475846\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	0.464102	0.0239342
$377$	−1.85641	−0.0956098
$378$	0	0
$379$	9.14359	0.469675	0.234837	−	0.972035i	$-0.424544\pi$
$379$	9.14359	0.469675	0.234837	+	0.972035i	$0.424544\pi$
$380$	−5.92820	−0.304110
$381$	0	0
$382$	−9.46410	−0.484226
$383$	0.464102	0.0237145	0.0118572	−	0.999930i	$-0.496226\pi$
$383$	0.464102	0.0237145	0.0118572	+	0.999930i	$0.496226\pi$
$384$	0	0
$385$	0.928203	0.0473056
$386$	16.0000	0.814379
$387$	0	0
$388$	14.9282	0.757865
$389$	−11.3205	−0.573973	−0.286986	−	0.957935i	$-0.592653\pi$
$389$	−11.3205	−0.573973	−0.286986	+	0.957935i	$0.592653\pi$
$390$	0	0
$391$	22.3923	1.13243
$392$	6.92820	0.349927
$393$	0	0
$394$	18.4641	0.930208
$395$	14.3923	0.724155
$396$	0	0
$397$	−17.8564	−0.896187	−0.448094	−	0.893987i	$-0.647897\pi$
$397$	−17.8564	−0.896187	−0.448094	+	0.893987i	$0.647897\pi$
$398$	−26.2487	−1.31573
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.41154	−0.420052	−0.210026	−	0.977696i	$-0.567355\pi$
$401$	−8.41154	−0.420052	−0.210026	+	0.977696i	$0.567355\pi$
$402$	0	0
$403$	−0.392305	−0.0195421
$404$	2.53590	0.126166
$405$	0	0
$406$	1.85641	0.0921319
$407$	−27.7128	−1.37367
$408$	0	0
$409$	−0.0717968	−0.00355012	−0.00177506	−	0.999998i	$-0.500565\pi$
$409$	−0.0717968	−0.00355012	−0.00177506	+	0.999998i	$0.500565\pi$
$410$	5.19615	0.256620
$411$	0	0
$412$	−4.80385	−0.236669
$413$	2.32051	0.114185
$414$	0	0
$415$	−15.4641	−0.759103
$416$	−0.267949	−0.0131373
$417$	0	0
$418$	20.5359	1.00444
$419$	3.46410	0.169232	0.0846162	−	0.996414i	$-0.473034\pi$
$419$	3.46410	0.169232	0.0846162	+	0.996414i	$0.473034\pi$
$420$	0	0
$421$	30.3923	1.48123	0.740615	−	0.671929i	$-0.234534\pi$
$421$	30.3923	1.48123	0.740615	+	0.671929i	$0.234534\pi$
$422$	−18.8564	−0.917916
$423$	0	0
$424$	−5.53590	−0.268847
$425$	3.46410	0.168034
$426$	0	0
$427$	3.32051	0.160691
$428$	15.4641	0.747486
$429$	0	0
$430$	5.46410	0.263502
$431$	−7.60770	−0.366450	−0.183225	−	0.983071i	$-0.558654\pi$
$431$	−7.60770	−0.366450	−0.183225	+	0.983071i	$0.558654\pi$
$432$	0	0
$433$	−10.9282	−0.525176	−0.262588	−	0.964908i	$-0.584576\pi$
$433$	−10.9282	−0.525176	−0.262588	+	0.964908i	$0.584576\pi$
$434$	0.392305	0.0188312
$435$	0	0
$436$	9.85641	0.472036
$437$	38.3205	1.83312
$438$	0	0
$439$	−2.39230	−0.114178	−0.0570892	−	0.998369i	$-0.518182\pi$
$439$	−2.39230	−0.114178	−0.0570892	+	0.998369i	$0.518182\pi$
$440$	−3.46410	−0.165145
$441$	0	0
$442$	−0.928203	−0.0441501
$443$	−4.14359	−0.196868	−0.0984340	−	0.995144i	$-0.531383\pi$
$443$	−4.14359	−0.196868	−0.0984340	+	0.995144i	$0.531383\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	19.4641	0.921652
$447$	0	0
$448$	0.267949	0.0126594
$449$	32.9090	1.55307	0.776535	−	0.630074i	$-0.216975\pi$
$449$	32.9090	1.55307	0.776535	+	0.630074i	$0.216975\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	−0.928203	−0.0436590
$453$	0	0
$454$	4.39230	0.206141
$455$	−0.0717968	−0.00336588
$456$	0	0
$457$	0.392305	0.0183512	0.00917562	−	0.999958i	$-0.497079\pi$
$457$	0.392305	0.0183512	0.00917562	+	0.999958i	$0.497079\pi$
$458$	5.85641	0.273652
$459$	0	0
$460$	−6.46410	−0.301390
$461$	−32.7846	−1.52693	−0.763466	−	0.645848i	$-0.776504\pi$
$461$	−32.7846	−1.52693	−0.763466	+	0.645848i	$0.776504\pi$
$462$	0	0
$463$	−15.1962	−0.706225	−0.353113	−	0.935581i	$-0.614877\pi$
$463$	−15.1962	−0.706225	−0.353113	+	0.935581i	$0.614877\pi$
$464$	−6.92820	−0.321634
$465$	0	0
$466$	12.0000	0.555889
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	2.14359	0.0989820
$470$	−0.464102	−0.0214074
$471$	0	0
$472$	−8.66025	−0.398621
$473$	−18.9282	−0.870320
$474$	0	0
$475$	5.92820	0.272005
$476$	0.928203	0.0425441
$477$	0	0
$478$	15.4641	0.707312
$479$	−0.928203	−0.0424107	−0.0212053	−	0.999775i	$-0.506750\pi$
$479$	−0.928203	−0.0424107	−0.0212053	+	0.999775i	$0.506750\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	3.07180	0.139917
$483$	0	0
$484$	1.00000	0.0454545
$485$	−14.9282	−0.677855
$486$	0	0
$487$	31.4449	1.42490	0.712451	−	0.701721i	$-0.247585\pi$
$487$	31.4449	1.42490	0.712451	+	0.701721i	$0.247585\pi$
$488$	−12.3923	−0.560973
$489$	0	0
$490$	−6.92820	−0.312984
$491$	12.1244	0.547165	0.273582	−	0.961849i	$-0.411791\pi$
$491$	12.1244	0.547165	0.273582	+	0.961849i	$0.411791\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	−1.58846	−0.0714681
$495$	0	0
$496$	−1.46410	−0.0657401
$497$	1.60770	0.0721150
$498$	0	0
$499$	37.7846	1.69147	0.845736	−	0.533602i	$-0.179162\pi$
$499$	37.7846	1.69147	0.845736	+	0.533602i	$0.179162\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.1244	−0.541136
$503$	−18.9282	−0.843967	−0.421983	−	0.906604i	$-0.638666\pi$
$503$	−18.9282	−0.843967	−0.421983	+	0.906604i	$0.638666\pi$
$504$	0	0
$505$	−2.53590	−0.112846
$506$	22.3923	0.995459
$507$	0	0
$508$	−4.80385	−0.213136
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−3.85641	−0.170597
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.14359	0.182766
$515$	4.80385	0.211683
$516$	0	0
$517$	1.60770	0.0707064
$518$	−2.14359	−0.0941841
$519$	0	0
$520$	0.267949	0.0117503
$521$	−6.80385	−0.298082	−0.149041	−	0.988831i	$-0.547619\pi$
$521$	−6.80385	−0.298082	−0.149041	+	0.988831i	$0.547619\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	10.2679	0.448557
$525$	0	0
$526$	−26.3205	−1.14763
$527$	−5.07180	−0.220931
$528$	0	0
$529$	18.7846	0.816722
$530$	5.53590	0.240464
$531$	0	0
$532$	1.58846	0.0688684
$533$	1.39230	0.0603074
$534$	0	0
$535$	−15.4641	−0.668571
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−0.928203	−0.0400177
$539$	24.0000	1.03375
$540$	0	0
$541$	13.0718	0.562000	0.281000	−	0.959708i	$-0.409334\pi$
$541$	13.0718	0.562000	0.281000	+	0.959708i	$0.409334\pi$
$542$	−14.0000	−0.601351
$543$	0	0
$544$	−3.46410	−0.148522
$545$	−9.85641	−0.422202
$546$	0	0
$547$	−8.39230	−0.358829	−0.179415	−	0.983774i	$-0.557420\pi$
$547$	−8.39230	−0.358829	−0.179415	+	0.983774i	$0.557420\pi$
$548$	10.3923	0.443937
$549$	0	0
$550$	3.46410	0.147710
$551$	−41.0718	−1.74972
$552$	0	0
$553$	−3.85641	−0.163991
$554$	−15.7321	−0.668391
$555$	0	0
$556$	−13.0000	−0.551323
$557$	−40.1769	−1.70235	−0.851175	−	0.524882i	$-0.824110\pi$
$557$	−40.1769	−1.70235	−0.851175	+	0.524882i	$0.824110\pi$
$558$	0	0
$559$	1.46410	0.0619249
$560$	−0.267949	−0.0113229
$561$	0	0
$562$	31.0526	1.30987
$563$	−13.8564	−0.583978	−0.291989	−	0.956422i	$-0.594317\pi$
$563$	−13.8564	−0.583978	−0.291989	+	0.956422i	$0.594317\pi$
$564$	0	0
$565$	0.928203	0.0390498
$566$	−31.3205	−1.31650
$567$	0	0
$568$	−6.00000	−0.251754
$569$	12.1244	0.508279	0.254140	−	0.967168i	$-0.418208\pi$
$569$	12.1244	0.508279	0.254140	+	0.967168i	$0.418208\pi$
$570$	0	0
$571$	1.07180	0.0448533	0.0224266	−	0.999748i	$-0.492861\pi$
$571$	1.07180	0.0448533	0.0224266	+	0.999748i	$0.492861\pi$
$572$	−0.928203	−0.0388101
$573$	0	0
$574$	−1.39230	−0.0581137
$575$	6.46410	0.269572
$576$	0	0
$577$	16.7846	0.698752	0.349376	−	0.936983i	$-0.386394\pi$
$577$	16.7846	0.698752	0.349376	+	0.936983i	$0.386394\pi$
$578$	5.00000	0.207973
$579$	0	0
$580$	6.92820	0.287678
$581$	4.14359	0.171905
$582$	0	0
$583$	−19.1769	−0.794227
$584$	14.3923	0.595558
$585$	0	0
$586$	13.3923	0.553231
$587$	1.60770	0.0663567	0.0331783	−	0.999449i	$-0.489437\pi$
$587$	1.60770	0.0663567	0.0331783	+	0.999449i	$0.489437\pi$
$588$	0	0
$589$	−8.67949	−0.357632
$590$	8.66025	0.356537
$591$	0	0
$592$	8.00000	0.328798
$593$	−3.46410	−0.142254	−0.0711268	−	0.997467i	$-0.522659\pi$
$593$	−3.46410	−0.142254	−0.0711268	+	0.997467i	$0.522659\pi$
$594$	0	0
$595$	−0.928203	−0.0380526
$596$	−18.0000	−0.737309
$597$	0	0
$598$	−1.73205	−0.0708288
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	−7.92820	−0.323398	−0.161699	−	0.986840i	$-0.551697\pi$
$601$	−7.92820	−0.323398	−0.161699	+	0.986840i	$0.551697\pi$
$602$	−1.46410	−0.0596723
$603$	0	0
$604$	−2.39230	−0.0973415
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−35.1769	−1.42779	−0.713893	−	0.700254i	$-0.753070\pi$
$607$	−35.1769	−1.42779	−0.713893	+	0.700254i	$0.753070\pi$
$608$	−5.92820	−0.240420
$609$	0	0
$610$	12.3923	0.501750
$611$	−0.124356	−0.00503089
$612$	0	0
$613$	39.9808	1.61481	0.807404	−	0.589999i	$-0.200872\pi$
$613$	39.9808	1.61481	0.807404	+	0.589999i	$0.200872\pi$
$614$	8.39230	0.338686
$615$	0	0
$616$	0.928203	0.0373984
$617$	−1.60770	−0.0647234	−0.0323617	−	0.999476i	$-0.510303\pi$
$617$	−1.60770	−0.0647234	−0.0323617	+	0.999476i	$0.510303\pi$
$618$	0	0
$619$	25.7846	1.03637	0.518185	−	0.855268i	$-0.326608\pi$
$619$	25.7846	1.03637	0.518185	+	0.855268i	$0.326608\pi$
$620$	1.46410	0.0587997
$621$	0	0
$622$	−15.4641	−0.620054
$623$	−3.21539	−0.128822
$624$	0	0
$625$	1.00000	0.0400000
$626$	−31.3205	−1.25182
$627$	0	0
$628$	−10.1244	−0.404006
$629$	27.7128	1.10498
$630$	0	0
$631$	35.7128	1.42170	0.710852	−	0.703341i	$-0.248309\pi$
$631$	35.7128	1.42170	0.710852	+	0.703341i	$0.248309\pi$
$632$	14.3923	0.572495
$633$	0	0
$634$	24.4641	0.971594
$635$	4.80385	0.190635
$636$	0	0
$637$	−1.85641	−0.0735535
$638$	−24.0000	−0.950169
$639$	0	0
$640$	1.00000	0.0395285
$641$	44.7846	1.76889	0.884443	−	0.466648i	$-0.154539\pi$
$641$	44.7846	1.76889	0.884443	+	0.466648i	$0.154539\pi$
$642$	0	0
$643$	−20.3923	−0.804194	−0.402097	−	0.915597i	$-0.631719\pi$
$643$	−20.3923	−0.804194	−0.402097	+	0.915597i	$0.631719\pi$
$644$	1.73205	0.0682524
$645$	0	0
$646$	−20.5359	−0.807974
$647$	46.6410	1.83365	0.916824	−	0.399292i	$-0.130744\pi$
$647$	46.6410	1.83365	0.916824	+	0.399292i	$0.130744\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	−0.267949	−0.0105098
$651$	0	0
$652$	−17.8564	−0.699311
$653$	−0.928203	−0.0363234	−0.0181617	−	0.999835i	$-0.505781\pi$
$653$	−0.928203	−0.0363234	−0.0181617	+	0.999835i	$0.505781\pi$
$654$	0	0
$655$	−10.2679	−0.401202
$656$	5.19615	0.202876
$657$	0	0
$658$	0.124356	0.00484789
$659$	43.0526	1.67709	0.838545	−	0.544833i	$-0.183407\pi$
$659$	43.0526	1.67709	0.838545	+	0.544833i	$0.183407\pi$
$660$	0	0
$661$	−32.3923	−1.25991	−0.629957	−	0.776630i	$-0.716928\pi$
$661$	−32.3923	−1.25991	−0.629957	+	0.776630i	$0.716928\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	−15.4641	−0.600124
$665$	−1.58846	−0.0615977
$666$	0	0
$667$	−44.7846	−1.73407
$668$	−6.92820	−0.268060
$669$	0	0
$670$	8.00000	0.309067
$671$	−42.9282	−1.65722
$672$	0	0
$673$	−23.1769	−0.893404	−0.446702	−	0.894683i	$-0.647402\pi$
$673$	−23.1769	−0.893404	−0.446702	+	0.894683i	$0.647402\pi$
$674$	−20.9282	−0.806124
$675$	0	0
$676$	−12.9282	−0.497239
$677$	33.2487	1.27785	0.638926	−	0.769268i	$-0.279379\pi$
$677$	33.2487	1.27785	0.638926	+	0.769268i	$0.279379\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	3.46410	0.132842
$681$	0	0
$682$	−5.07180	−0.194209
$683$	−33.7128	−1.28998	−0.644992	−	0.764189i	$-0.723140\pi$
$683$	−33.7128	−1.28998	−0.644992	+	0.764189i	$0.723140\pi$
$684$	0	0
$685$	−10.3923	−0.397070
$686$	3.73205	0.142490
$687$	0	0
$688$	5.46410	0.208317
$689$	1.48334	0.0565107
$690$	0	0
$691$	1.78461	0.0678898	0.0339449	−	0.999424i	$-0.489193\pi$
$691$	1.78461	0.0678898	0.0339449	+	0.999424i	$0.489193\pi$
$692$	14.3205	0.544384
$693$	0	0
$694$	−26.7846	−1.01673
$695$	13.0000	0.493118
$696$	0	0
$697$	18.0000	0.681799
$698$	−2.00000	−0.0757011
$699$	0	0
$700$	0.267949	0.0101275
$701$	−6.67949	−0.252281	−0.126140	−	0.992012i	$-0.540259\pi$
$701$	−6.67949	−0.252281	−0.126140	+	0.992012i	$0.540259\pi$
$702$	0	0
$703$	47.4256	1.78869
$704$	−3.46410	−0.130558
$705$	0	0
$706$	−1.60770	−0.0605064
$707$	0.679492	0.0255549
$708$	0	0
$709$	13.3205	0.500262	0.250131	−	0.968212i	$-0.419526\pi$
$709$	13.3205	0.500262	0.250131	+	0.968212i	$0.419526\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	12.0000	0.449719
$713$	−9.46410	−0.354433
$714$	0	0
$715$	0.928203	0.0347128
$716$	−12.1244	−0.453108
$717$	0	0
$718$	15.4641	0.577115
$719$	−47.5692	−1.77403	−0.887016	−	0.461738i	$-0.847226\pi$
$719$	−47.5692	−1.77403	−0.887016	+	0.461738i	$0.847226\pi$
$720$	0	0
$721$	−1.28719	−0.0479374
$722$	−16.1436	−0.600802
$723$	0	0
$724$	−19.4641	−0.723378
$725$	−6.92820	−0.257307
$726$	0	0
$727$	−39.1962	−1.45370	−0.726852	−	0.686794i	$-0.759018\pi$
$727$	−39.1962	−1.45370	−0.726852	+	0.686794i	$0.759018\pi$
$728$	−0.0717968	−0.00266097
$729$	0	0
$730$	−14.3923	−0.532683
$731$	18.9282	0.700085
$732$	0	0
$733$	−16.0000	−0.590973	−0.295487	−	0.955347i	$-0.595482\pi$
$733$	−16.0000	−0.590973	−0.295487	+	0.955347i	$0.595482\pi$
$734$	14.3923	0.531230
$735$	0	0
$736$	−6.46410	−0.238270
$737$	−27.7128	−1.02081
$738$	0	0
$739$	−45.5692	−1.67629	−0.838145	−	0.545447i	$-0.816360\pi$
$739$	−45.5692	−1.67629	−0.838145	+	0.545447i	$0.816360\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	−1.48334	−0.0544551
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	−2.92820	−0.107209
$747$	0	0
$748$	−12.0000	−0.438763
$749$	4.14359	0.151404
$750$	0	0
$751$	−18.7846	−0.685460	−0.342730	−	0.939434i	$-0.611352\pi$
$751$	−18.7846	−0.685460	−0.342730	+	0.939434i	$0.611352\pi$
$752$	−0.464102	−0.0169240
$753$	0	0
$754$	1.85641	0.0676063
$755$	2.39230	0.0870649
$756$	0	0
$757$	19.1962	0.697696	0.348848	−	0.937179i	$-0.386573\pi$
$757$	19.1962	0.697696	0.348848	+	0.937179i	$0.386573\pi$
$758$	−9.14359	−0.332110
$759$	0	0
$760$	5.92820	0.215039
$761$	10.2679	0.372213	0.186106	−	0.982530i	$-0.440413\pi$
$761$	10.2679	0.372213	0.186106	+	0.982530i	$0.440413\pi$
$762$	0	0
$763$	2.64102	0.0956112
$764$	9.46410	0.342399
$765$	0	0
$766$	−0.464102	−0.0167687
$767$	2.32051	0.0837887
$768$	0	0
$769$	−40.9282	−1.47591	−0.737954	−	0.674851i	$-0.764208\pi$
$769$	−40.9282	−1.47591	−0.737954	+	0.674851i	$0.764208\pi$
$770$	−0.928203	−0.0334501
$771$	0	0
$772$	−16.0000	−0.575853
$773$	11.0718	0.398225	0.199112	−	0.979977i	$-0.436194\pi$
$773$	11.0718	0.398225	0.199112	+	0.979977i	$0.436194\pi$
$774$	0	0
$775$	−1.46410	−0.0525921
$776$	−14.9282	−0.535891
$777$	0	0
$778$	11.3205	0.405860
$779$	30.8038	1.10366
$780$	0	0
$781$	−20.7846	−0.743732
$782$	−22.3923	−0.800747
$783$	0	0
$784$	−6.92820	−0.247436
$785$	10.1244	0.361354
$786$	0	0
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	−18.4641	−0.657756
$789$	0	0
$790$	−14.3923	−0.512055
$791$	−0.248711	−0.00884316
$792$	0	0
$793$	3.32051	0.117915
$794$	17.8564	0.633700
$795$	0	0
$796$	26.2487	0.930361
$797$	−43.8564	−1.55347	−0.776737	−	0.629825i	$-0.783126\pi$
$797$	−43.8564	−1.55347	−0.776737	+	0.629825i	$0.783126\pi$
$798$	0	0
$799$	−1.60770	−0.0568762
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	8.41154	0.297022
$803$	49.8564	1.75939
$804$	0	0
$805$	−1.73205	−0.0610468
$806$	0.392305	0.0138183
$807$	0	0
$808$	−2.53590	−0.0892126
$809$	20.4115	0.717632	0.358816	−	0.933408i	$-0.383181\pi$
$809$	20.4115	0.717632	0.358816	+	0.933408i	$0.383181\pi$
$810$	0	0
$811$	−29.8564	−1.04840	−0.524200	−	0.851595i	$-0.675636\pi$
$811$	−29.8564	−1.04840	−0.524200	+	0.851595i	$0.675636\pi$
$812$	−1.85641	−0.0651471
$813$	0	0
$814$	27.7128	0.971334
$815$	17.8564	0.625483
$816$	0	0
$817$	32.3923	1.13326
$818$	0.0717968	0.00251032
$819$	0	0
$820$	−5.19615	−0.181458
$821$	−6.67949	−0.233116	−0.116558	−	0.993184i	$-0.537186\pi$
$821$	−6.67949	−0.233116	−0.116558	+	0.993184i	$0.537186\pi$
$822$	0	0
$823$	−9.32051	−0.324892	−0.162446	−	0.986717i	$-0.551938\pi$
$823$	−9.32051	−0.324892	−0.162446	+	0.986717i	$0.551938\pi$
$824$	4.80385	0.167350
$825$	0	0
$826$	−2.32051	−0.0807408
$827$	20.1051	0.699123	0.349562	−	0.936913i	$-0.386330\pi$
$827$	20.1051	0.699123	0.349562	+	0.936913i	$0.386330\pi$
$828$	0	0
$829$	−45.5692	−1.58268	−0.791342	−	0.611373i	$-0.790617\pi$
$829$	−45.5692	−1.58268	−0.791342	+	0.611373i	$0.790617\pi$
$830$	15.4641	0.536767
$831$	0	0
$832$	0.267949	0.00928947
$833$	−24.0000	−0.831551
$834$	0	0
$835$	6.92820	0.239760
$836$	−20.5359	−0.710249
$837$	0	0
$838$	−3.46410	−0.119665
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	−30.3923	−1.04739
$843$	0	0
$844$	18.8564	0.649064
$845$	12.9282	0.444744
$846$	0	0
$847$	0.267949	0.00920684
$848$	5.53590	0.190104
$849$	0	0
$850$	−3.46410	−0.118818
$851$	51.7128	1.77269
$852$	0	0
$853$	−48.7846	−1.67035	−0.835177	−	0.549982i	$-0.814635\pi$
$853$	−48.7846	−1.67035	−0.835177	+	0.549982i	$0.814635\pi$
$854$	−3.32051	−0.113625
$855$	0	0
$856$	−15.4641	−0.528552
$857$	9.71281	0.331783	0.165892	−	0.986144i	$-0.446950\pi$
$857$	9.71281	0.331783	0.165892	+	0.986144i	$0.446950\pi$
$858$	0	0
$859$	11.2154	0.382664	0.191332	−	0.981525i	$-0.438719\pi$
$859$	11.2154	0.382664	0.191332	+	0.981525i	$0.438719\pi$
$860$	−5.46410	−0.186324
$861$	0	0
$862$	7.60770	0.259119
$863$	−44.3205	−1.50869	−0.754344	−	0.656480i	$-0.772045\pi$
$863$	−44.3205	−1.50869	−0.754344	+	0.656480i	$0.772045\pi$
$864$	0	0
$865$	−14.3205	−0.486912
$866$	10.9282	0.371355
$867$	0	0
$868$	−0.392305	−0.0133157
$869$	49.8564	1.69126
$870$	0	0
$871$	2.14359	0.0726329
$872$	−9.85641	−0.333780
$873$	0	0
$874$	−38.3205	−1.29621
$875$	−0.267949	−0.00905834
$876$	0	0
$877$	31.4449	1.06182	0.530909	−	0.847429i	$-0.321851\pi$
$877$	31.4449	1.06182	0.530909	+	0.847429i	$0.321851\pi$
$878$	2.39230	0.0807364
$879$	0	0
$880$	3.46410	0.116775
$881$	53.5692	1.80479	0.902396	−	0.430907i	$-0.141806\pi$
$881$	53.5692	1.80479	0.902396	+	0.430907i	$0.141806\pi$
$882$	0	0
$883$	16.5359	0.556477	0.278239	−	0.960512i	$-0.410249\pi$
$883$	16.5359	0.556477	0.278239	+	0.960512i	$0.410249\pi$
$884$	0.928203	0.0312189
$885$	0	0
$886$	4.14359	0.139207
$887$	−23.1051	−0.775794	−0.387897	−	0.921703i	$-0.626798\pi$
$887$	−23.1051	−0.775794	−0.387897	+	0.921703i	$0.626798\pi$
$888$	0	0
$889$	−1.28719	−0.0431709
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−19.4641	−0.651706
$893$	−2.75129	−0.0920684
$894$	0	0
$895$	12.1244	0.405273
$896$	−0.267949	−0.00895155
$897$	0	0
$898$	−32.9090	−1.09819
$899$	10.1436	0.338308
$900$	0	0
$901$	19.1769	0.638876
$902$	18.0000	0.599334
$903$	0	0
$904$	0.928203	0.0308716
$905$	19.4641	0.647009
$906$	0	0
$907$	55.5692	1.84515	0.922573	−	0.385823i	$-0.126082\pi$
$907$	55.5692	1.84515	0.922573	+	0.385823i	$0.126082\pi$
$908$	−4.39230	−0.145764
$909$	0	0
$910$	0.0717968	0.00238004
$911$	25.6077	0.848421	0.424210	−	0.905564i	$-0.360552\pi$
$911$	25.6077	0.848421	0.424210	+	0.905564i	$0.360552\pi$
$912$	0	0
$913$	−53.5692	−1.77288
$914$	−0.392305	−0.0129763
$915$	0	0
$916$	−5.85641	−0.193501
$917$	2.75129	0.0908556
$918$	0	0
$919$	−24.7846	−0.817569	−0.408784	−	0.912631i	$-0.634047\pi$
$919$	−24.7846	−0.817569	−0.408784	+	0.912631i	$0.634047\pi$
$920$	6.46410	0.213115
$921$	0	0
$922$	32.7846	1.07970
$923$	1.60770	0.0529179
$924$	0	0
$925$	8.00000	0.263038
$926$	15.1962	0.499377
$927$	0	0
$928$	6.92820	0.227429
$929$	17.0718	0.560107	0.280054	−	0.959984i	$-0.409648\pi$
$929$	17.0718	0.560107	0.280054	+	0.959984i	$0.409648\pi$
$930$	0	0
$931$	−41.0718	−1.34607
$932$	−12.0000	−0.393073
$933$	0	0
$934$	−6.00000	−0.196326
$935$	12.0000	0.392442
$936$	0	0
$937$	33.8564	1.10604	0.553020	−	0.833168i	$-0.313475\pi$
$937$	33.8564	1.10604	0.553020	+	0.833168i	$0.313475\pi$
$938$	−2.14359	−0.0699908
$939$	0	0
$940$	0.464102	0.0151373
$941$	−29.3205	−0.955821	−0.477911	−	0.878408i	$-0.658606\pi$
$941$	−29.3205	−0.955821	−0.477911	+	0.878408i	$0.658606\pi$
$942$	0	0
$943$	33.5885	1.09379
$944$	8.66025	0.281867
$945$	0	0
$946$	18.9282	0.615409
$947$	−21.4641	−0.697490	−0.348745	−	0.937218i	$-0.613392\pi$
$947$	−21.4641	−0.697490	−0.348745	+	0.937218i	$0.613392\pi$
$948$	0	0
$949$	−3.85641	−0.125184
$950$	−5.92820	−0.192336
$951$	0	0
$952$	−0.928203	−0.0300832
$953$	30.2487	0.979852	0.489926	−	0.871764i	$-0.337024\pi$
$953$	30.2487	0.979852	0.489926	+	0.871764i	$0.337024\pi$
$954$	0	0
$955$	−9.46410	−0.306251
$956$	−15.4641	−0.500145
$957$	0	0
$958$	0.928203	0.0299889
$959$	2.78461	0.0899197
$960$	0	0
$961$	−28.8564	−0.930852
$962$	−2.14359	−0.0691122
$963$	0	0
$964$	−3.07180	−0.0989359
$965$	16.0000	0.515058
$966$	0	0
$967$	−14.3923	−0.462825	−0.231413	−	0.972856i	$-0.574335\pi$
$967$	−14.3923	−0.462825	−0.231413	+	0.972856i	$0.574335\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	14.9282	0.479316
$971$	−43.3013	−1.38960	−0.694802	−	0.719201i	$-0.744508\pi$
$971$	−43.3013	−1.38960	−0.694802	+	0.719201i	$0.744508\pi$
$972$	0	0
$973$	−3.48334	−0.111671
$974$	−31.4449	−1.00756
$975$	0	0
$976$	12.3923	0.396668
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	41.5692	1.32856
$980$	6.92820	0.221313
$981$	0	0
$982$	−12.1244	−0.386904
$983$	−60.4974	−1.92957	−0.964784	−	0.263043i	$-0.915274\pi$
$983$	−60.4974	−1.92957	−0.964784	+	0.263043i	$0.915274\pi$
$984$	0	0
$985$	18.4641	0.588315
$986$	24.0000	0.764316
$987$	0	0
$988$	1.58846	0.0505356
$989$	35.3205	1.12313
$990$	0	0
$991$	52.7846	1.67676	0.838379	−	0.545087i	$-0.183504\pi$
$991$	52.7846	1.67676	0.838379	+	0.545087i	$0.183504\pi$
$992$	1.46410	0.0464853
$993$	0	0
$994$	−1.60770	−0.0509930
$995$	−26.2487	−0.832140
$996$	0	0
$997$	7.19615	0.227904	0.113952	−	0.993486i	$-0.463649\pi$
$997$	7.19615	0.227904	0.113952	+	0.993486i	$0.463649\pi$
$998$	−37.7846	−1.19605
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.a.j.1.1	✓	2
3.2	odd	2		810.2.a.l.1.1	yes	2
4.3	odd	2		6480.2.a.bb.1.2		2
5.2	odd	4		4050.2.c.z.649.1		4
5.3	odd	4		4050.2.c.z.649.4		4
5.4	even	2		4050.2.a.bt.1.2		2
9.2	odd	6		810.2.e.m.271.2		4
9.4	even	3		810.2.e.n.541.2		4
9.5	odd	6		810.2.e.m.541.2		4
9.7	even	3		810.2.e.n.271.2		4
12.11	even	2		6480.2.a.bj.1.2		2
15.2	even	4		4050.2.c.x.649.3		4
15.8	even	4		4050.2.c.x.649.2		4
15.14	odd	2		4050.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.a.j.1.1	✓	2	1.1	even	1	trivial
810.2.a.l.1.1	yes	2	3.2	odd	2
810.2.e.m.271.2		4	9.2	odd	6
810.2.e.m.541.2		4	9.5	odd	6
810.2.e.n.271.2		4	9.7	even	3
810.2.e.n.541.2		4	9.4	even	3
4050.2.a.bk.1.2		2	15.14	odd	2
4050.2.a.bt.1.2		2	5.4	even	2
4050.2.c.x.649.2		4	15.8	even	4
4050.2.c.x.649.3		4	15.2	even	4
4050.2.c.z.649.1		4	5.2	odd	4
4050.2.c.z.649.4		4	5.3	odd	4
6480.2.a.bb.1.2		2	4.3	odd	2
6480.2.a.bj.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.267949 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.46410 q^{11} +0.267949 q^{13} -0.267949 q^{14} +1.00000 q^{16} +3.46410 q^{17} +5.92820 q^{19} -1.00000 q^{20} +3.46410 q^{22} +6.46410 q^{23} +1.00000 q^{25} -0.267949 q^{26} +0.267949 q^{28} -6.92820 q^{29} -1.46410 q^{31} -1.00000 q^{32} -3.46410 q^{34} -0.267949 q^{35} +8.00000 q^{37} -5.92820 q^{38} +1.00000 q^{40} +5.19615 q^{41} +5.46410 q^{43} -3.46410 q^{44} -6.46410 q^{46} -0.464102 q^{47} -6.92820 q^{49} -1.00000 q^{50} +0.267949 q^{52} +5.53590 q^{53} +3.46410 q^{55} -0.267949 q^{56} +6.92820 q^{58} +8.66025 q^{59} +12.3923 q^{61} +1.46410 q^{62} +1.00000 q^{64} -0.267949 q^{65} +8.00000 q^{67} +3.46410 q^{68} +0.267949 q^{70} +6.00000 q^{71} -14.3923 q^{73} -8.00000 q^{74} +5.92820 q^{76} -0.928203 q^{77} -14.3923 q^{79} -1.00000 q^{80} -5.19615 q^{82} +15.4641 q^{83} -3.46410 q^{85} -5.46410 q^{86} +3.46410 q^{88} -12.0000 q^{89} +0.0717968 q^{91} +6.46410 q^{92} +0.464102 q^{94} -5.92820 q^{95} +14.9282 q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 4 q^{13} - 4 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} + 6 q^{23} + 2 q^{25} - 4 q^{26} + 4 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{35} + 16 q^{37}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 + 4 * q^13 - 4 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^20 + 6 * q^23 + 2 * q^25 - 4 * q^26 + 4 * q^28 + 4 * q^31 - 2 * q^32 - 4 * q^35 + 16 * q^37 + 2 * q^38 + 2 * q^40 + 4 * q^43 - 6 * q^46 + 6 * q^47 - 2 * q^50 + 4 * q^52 + 18 * q^53 - 4 * q^56 + 4 * q^61 - 4 * q^62 + 2 * q^64 - 4 * q^65 + 16 * q^67 + 4 * q^70 + 12 * q^71 - 8 * q^73 - 16 * q^74 - 2 * q^76 + 12 * q^77 - 8 * q^79 - 2 * q^80 + 24 * q^83 - 4 * q^86 - 24 * q^89 + 14 * q^91 + 6 * q^92 - 6 * q^94 + 2 * q^95 + 16 * q^97

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