LMFDB - Embedded newform 810.3.b.c.809.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.21
Character	$\chi$	$=$	810.809
Dual form			810.3.b.c.809.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(1.68174 - 4.70869i) q^{5} -9.41831i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(1.68174 - 4.70869i) q^{5} -9.41831i q^{7} +2.82843 q^{8} +(2.37833 - 6.65909i) q^{10} -2.35458i q^{11} +1.53833i q^{13} -13.3195i q^{14} +4.00000 q^{16} -11.0873 q^{17} +7.09220 q^{19} +(3.36347 - 9.41738i) q^{20} -3.32989i q^{22} +8.19590 q^{23} +(-19.3435 - 15.8376i) q^{25} +2.17553i q^{26} -18.8366i q^{28} -17.9495i q^{29} -58.7165 q^{31} +5.65685 q^{32} -15.6798 q^{34} +(-44.3479 - 15.8391i) q^{35} -20.7943i q^{37} +10.0299 q^{38} +(4.75667 - 13.3182i) q^{40} +48.9075i q^{41} -3.55509i q^{43} -4.70917i q^{44} +11.5908 q^{46} +69.7706 q^{47} -39.7046 q^{49} +(-27.3559 - 22.3977i) q^{50} +3.07666i q^{52} -69.0100 q^{53} +(-11.0870 - 3.95979i) q^{55} -26.6390i q^{56} -25.3844i q^{58} -39.3443i q^{59} +115.668 q^{61} -83.0377 q^{62} +8.00000 q^{64} +(7.24352 + 2.58707i) q^{65} -102.550i q^{67} -22.1745 q^{68} +(-62.7174 - 22.3999i) q^{70} +102.667i q^{71} -120.613i q^{73} -29.4075i q^{74} +14.1844 q^{76} -22.1762 q^{77} -18.2602 q^{79} +(6.72695 - 18.8348i) q^{80} +69.1657i q^{82} +161.941 q^{83} +(-18.6459 + 52.2065i) q^{85} -5.02765i q^{86} -6.65977i q^{88} +88.2337i q^{89} +14.4885 q^{91} +16.3918 q^{92} +98.6706 q^{94} +(11.9272 - 33.3950i) q^{95} -140.620i q^{97} -56.1508 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4}+O(q^{10}) $ 24 * q + 48 * q^4	$ 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	1.68174	−	4.70869i	0.336347	−	0.941738i
$5$	1.68174	−	4.70869i	0.336347	−	0.941738i
$6$		0			0
$7$		−	9.41831i		−	1.34547i	−0.739882	−	0.672737i	$-0.765119\pi$
$7$		−	9.41831i		−	1.34547i	0.739882	−	0.672737i	$-0.234881\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	2.37833	−	6.65909i	0.237833	−	0.665909i
$11$		−	2.35458i		−	0.214053i	−0.994256	−	0.107027i	$-0.965867\pi$
$11$		−	2.35458i		−	0.214053i	0.994256	−	0.107027i	$-0.0341330\pi$
$12$		0			0
$13$			1.53833i			0.118333i	0.998248	+	0.0591665i	$0.0188443\pi$
$13$			1.53833i			0.118333i	−0.998248	+	0.0591665i	$0.981156\pi$
$14$		−	13.3195i		−	0.951393i
$15$		0			0
$16$	4.00000			0.250000
$17$	−11.0873			−0.652192			−0.326096	−	0.945337i	$-0.605733\pi$
$17$	−11.0873			−0.652192			−0.326096	+	0.945337i	$0.605733\pi$
$18$		0			0
$19$	7.09220			0.373274			0.186637	−	0.982429i	$-0.440241\pi$
$19$	7.09220			0.373274			0.186637	+	0.982429i	$0.440241\pi$
$20$	3.36347	−	9.41738i	0.168174	−	0.470869i
$21$		0			0
$22$		−	3.32989i		−	0.151358i
$23$	8.19590			0.356344			0.178172	−	0.983999i	$-0.442982\pi$
$23$	8.19590			0.356344			0.178172	+	0.983999i	$0.442982\pi$
$24$		0			0
$25$	−19.3435	−	15.8376i	−0.773741	−	0.633502i
$26$			2.17553i			0.0836741i
$27$		0			0
$28$		−	18.8366i		−	0.672737i
$29$		−	17.9495i		−	0.618949i	−0.950908	−	0.309474i	$-0.899847\pi$
$29$		−	17.9495i		−	0.618949i	0.950908	−	0.309474i	$-0.100153\pi$
$30$		0			0
$31$	−58.7165			−1.89408			−0.947041	−	0.321114i	$-0.895943\pi$
$31$	−58.7165			−1.89408			−0.947041	+	0.321114i	$0.895943\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	−15.6798			−0.461169
$35$	−44.3479	−	15.8391i	−1.26708	−	0.452546i
$36$		0			0
$37$		−	20.7943i		−	0.562007i	−0.959707	−	0.281004i	$-0.909333\pi$
$37$		−	20.7943i		−	0.562007i	0.959707	−	0.281004i	$-0.0906673\pi$
$38$	10.0299			0.263944
$39$		0			0
$40$	4.75667	−	13.3182i	0.118917	−	0.332955i
$41$			48.9075i			1.19287i	0.802663	+	0.596433i	$0.203416\pi$
$41$			48.9075i			1.19287i	−0.802663	+	0.596433i	$0.796584\pi$
$42$		0			0
$43$		−	3.55509i		−	0.0826765i	−0.999145	−	0.0413382i	$-0.986838\pi$
$43$		−	3.55509i		−	0.0826765i	0.999145	−	0.0413382i	$-0.0131621\pi$
$44$		−	4.70917i		−	0.107027i
$45$		0			0
$46$	11.5908			0.251973
$47$	69.7706			1.48448			0.742241	−	0.670133i	$-0.233763\pi$
$47$	69.7706			1.48448			0.742241	+	0.670133i	$0.233763\pi$
$48$		0			0
$49$	−39.7046			−0.810299
$50$	−27.3559	−	22.3977i	−0.547117	−	0.447954i
$51$		0			0
$52$			3.07666i			0.0591665i
$53$	−69.0100			−1.30208			−0.651038	−	0.759045i	$-0.725666\pi$
$53$	−69.0100			−1.30208			−0.651038	+	0.759045i	$0.725666\pi$
$54$		0			0
$55$	−11.0870	−	3.95979i	−0.201582	−	0.0719962i
$56$		−	26.6390i		−	0.475697i
$57$		0			0
$58$		−	25.3844i		−	0.437663i
$59$		−	39.3443i		−	0.666852i	−0.942776	−	0.333426i	$-0.891795\pi$
$59$		−	39.3443i		−	0.666852i	0.942776	−	0.333426i	$-0.108205\pi$
$60$		0			0
$61$	115.668			1.89620			0.948099	−	0.317975i	$-0.103003\pi$
$61$	115.668			1.89620			0.948099	+	0.317975i	$0.103003\pi$
$62$	−83.0377			−1.33932
$63$		0			0
$64$	8.00000			0.125000
$65$	7.24352	+	2.58707i	0.111439	+	0.0398010i
$66$		0			0
$67$		−	102.550i		−	1.53060i	−0.643674	−	0.765300i	$-0.722591\pi$
$67$		−	102.550i		−	1.53060i	0.643674	−	0.765300i	$-0.277409\pi$
$68$	−22.1745			−0.326096
$69$		0			0
$70$	−62.7174	−	22.3999i	−0.895963	−	0.319999i
$71$			102.667i			1.44601i	0.690842	+	0.723006i	$0.257240\pi$
$71$			102.667i			1.44601i	−0.690842	+	0.723006i	$0.742760\pi$
$72$		0			0
$73$		−	120.613i		−	1.65223i	−0.563503	−	0.826114i	$-0.690547\pi$
$73$		−	120.613i		−	1.65223i	0.563503	−	0.826114i	$-0.309453\pi$
$74$		−	29.4075i		−	0.397399i
$75$		0			0
$76$	14.1844			0.186637
$77$	−22.1762			−0.288003
$78$		0			0
$79$	−18.2602			−0.231142			−0.115571	−	0.993299i	$-0.536870\pi$
$79$	−18.2602			−0.231142			−0.115571	+	0.993299i	$0.536870\pi$
$80$	6.72695	−	18.8348i	0.0840868	−	0.235435i
$81$		0			0
$82$			69.1657i			0.843484i
$83$	161.941			1.95109			0.975546	−	0.219794i	$-0.0705385\pi$
$83$	161.941			1.95109			0.975546	+	0.219794i	$0.0705385\pi$
$84$		0			0
$85$	−18.6459	+	52.2065i	−0.219363	+	0.614194i
$86$		−	5.02765i		−	0.0584611i
$87$		0			0
$88$		−	6.65977i		−	0.0756792i
$89$			88.2337i			0.991390i	0.868497	+	0.495695i	$0.165087\pi$
$89$			88.2337i			0.991390i	−0.868497	+	0.495695i	$0.834913\pi$
$90$		0			0
$91$	14.4885			0.159214
$92$	16.3918			0.178172
$93$		0			0
$94$	98.6706			1.04969
$95$	11.9272	−	33.3950i	0.125550	−	0.351526i
$96$		0			0
$97$		−	140.620i		−	1.44969i	−0.688911	−	0.724846i	$-0.741911\pi$
$97$		−	140.620i		−	1.44969i	0.688911	−	0.724846i	$-0.258089\pi$
$98$	−56.1508			−0.572968
$99$		0			0
$100$	−38.6870	−	31.6751i	−0.386870	−	0.316751i
$101$		−	54.9707i		−	0.544265i	−0.962260	−	0.272132i	$-0.912271\pi$
$101$		−	54.9707i		−	0.544265i	0.962260	−	0.272132i	$-0.0877288\pi$
$102$		0			0
$103$			14.2571i			0.138419i	0.997602	+	0.0692094i	$0.0220477\pi$
$103$			14.2571i			0.138419i	−0.997602	+	0.0692094i	$0.977952\pi$
$104$			4.35105i			0.0418371i
$105$		0			0
$106$	−97.5949			−0.920706
$107$	81.9655			0.766033			0.383017	−	0.923741i	$-0.374885\pi$
$107$	81.9655			0.766033			0.383017	+	0.923741i	$0.374885\pi$
$108$		0			0
$109$	−112.498			−1.03209			−0.516044	−	0.856562i	$-0.672596\pi$
$109$	−112.498			−1.03209			−0.516044	+	0.856562i	$0.672596\pi$
$110$	−15.6794	−	5.59999i	−0.142540	−	0.0509090i
$111$		0			0
$112$		−	37.6733i		−	0.336368i
$113$	89.8034			0.794721			0.397360	−	0.917663i	$-0.369926\pi$
$113$	89.8034			0.794721			0.397360	+	0.917663i	$0.369926\pi$
$114$		0			0
$115$	13.7833	−	38.5920i	0.119855	−	0.335582i
$116$		−	35.8990i		−	0.309474i
$117$		0			0
$118$		−	55.6412i		−	0.471536i
$119$			104.423i			0.877507i
$120$		0			0
$121$	115.456			0.954181
$122$	163.579			1.34081
$123$		0			0
$124$	−117.433			−0.947041
$125$	−107.105	+	64.4481i	−0.856839	+	0.515585i
$126$		0			0
$127$			88.3594i			0.695743i	0.937542	+	0.347872i	$0.113095\pi$
$127$			88.3594i			0.695743i	−0.937542	+	0.347872i	$0.886905\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	10.2439	+	3.65866i	0.0787991	+	0.0281436i
$131$			173.738i			1.32625i	0.748510	+	0.663123i	$0.230769\pi$
$131$			173.738i			1.32625i	−0.748510	+	0.663123i	$0.769231\pi$
$132$		0			0
$133$		−	66.7966i		−	0.502230i
$134$		−	145.028i		−	1.08230i
$135$		0			0
$136$	−31.3595			−0.230585
$137$	108.958			0.795314			0.397657	−	0.917534i	$-0.369823\pi$
$137$	108.958			0.795314			0.397657	+	0.917534i	$0.369823\pi$
$138$		0			0
$139$	−133.311			−0.959075			−0.479537	−	0.877521i	$-0.659196\pi$
$139$	−133.311			−0.959075			−0.479537	+	0.877521i	$0.659196\pi$
$140$	−88.6958	−	31.6782i	−0.633542	−	0.226273i
$141$		0			0
$142$			145.193i			1.02248i
$143$	3.62213			0.0253296
$144$		0			0
$145$	−84.5187	−	30.1864i	−0.582888	−	0.208182i
$146$		−	170.572i		−	1.16830i
$147$		0			0
$148$		−	41.5885i		−	0.281004i
$149$			124.051i			0.832554i	0.909238	+	0.416277i	$0.136665\pi$
$149$			124.051i			0.832554i	−0.909238	+	0.416277i	$0.863335\pi$
$150$		0			0
$151$	13.6383			0.0903199			0.0451600	−	0.998980i	$-0.485620\pi$
$151$	13.6383			0.0903199			0.0451600	+	0.998980i	$0.485620\pi$
$152$	20.0598			0.131972
$153$		0			0
$154$	−31.3619			−0.203649
$155$	−98.7457	+	276.478i	−0.637069	+	1.78373i
$156$		0			0
$157$		−	246.154i		−	1.56786i	−0.620851	−	0.783929i	$-0.713213\pi$
$157$		−	246.154i		−	1.56786i	0.620851	−	0.783929i	$-0.286787\pi$
$158$	−25.8238			−0.163442
$159$		0			0
$160$	9.51334	−	26.6364i	0.0594584	−	0.166477i
$161$		−	77.1916i		−	0.479451i
$162$		0			0
$163$			56.4039i			0.346036i	0.984919	+	0.173018i	$0.0553519\pi$
$163$			56.4039i			0.346036i	−0.984919	+	0.173018i	$0.944648\pi$
$164$			97.8151i			0.596433i
$165$		0			0
$166$	229.019			1.37963
$167$	99.3751			0.595061			0.297530	−	0.954712i	$-0.403837\pi$
$167$	99.3751			0.595061			0.297530	+	0.954712i	$0.403837\pi$
$168$		0			0
$169$	166.634			0.985997
$170$	−26.3692	+	73.8311i	−0.155113	+	0.434301i
$171$		0			0
$172$		−	7.11018i		−	0.0413382i
$173$	−285.581			−1.65076			−0.825378	−	0.564580i	$-0.809038\pi$
$173$	−285.581			−1.65076			−0.825378	+	0.564580i	$0.809038\pi$
$174$		0			0
$175$	−149.163	+	182.183i	−0.852360	+	1.04105i
$176$		−	9.41834i		−	0.0535133i
$177$		0			0
$178$			124.781i			0.701019i
$179$			180.570i			1.00877i	0.863479	+	0.504385i	$0.168281\pi$
$179$			180.570i			1.00877i	−0.863479	+	0.504385i	$0.831719\pi$
$180$		0			0
$181$	87.7217			0.484650			0.242325	−	0.970195i	$-0.422090\pi$
$181$	87.7217			0.484650			0.242325	+	0.970195i	$0.422090\pi$
$182$	20.4898			0.112581
$183$		0			0
$184$	23.1815			0.125986
$185$	−97.9137	−	34.9705i	−0.529264	−	0.189030i
$186$		0			0
$187$			26.1059i			0.139604i
$188$	139.541			0.742241
$189$		0			0
$190$	16.8676	−	47.2276i	0.0887770	−	0.248567i
$191$		−	61.9072i		−	0.324122i	−0.986781	−	0.162061i	$-0.948186\pi$
$191$		−	61.9072i		−	0.324122i	0.986781	−	0.162061i	$-0.0518141\pi$
$192$		0			0
$193$			224.779i			1.16466i	0.812953	+	0.582329i	$0.197858\pi$
$193$			224.779i			1.16466i	−0.812953	+	0.582329i	$0.802142\pi$
$194$		−	198.867i		−	1.02509i
$195$		0			0
$196$	−79.4093			−0.405149
$197$	260.281			1.32122			0.660612	−	0.750728i	$-0.270297\pi$
$197$	260.281			1.32122			0.660612	+	0.750728i	$0.270297\pi$
$198$		0			0
$199$	−27.2102			−0.136735			−0.0683673	−	0.997660i	$-0.521779\pi$
$199$	−27.2102			−0.136735			−0.0683673	+	0.997660i	$0.521779\pi$
$200$	−54.7117	−	44.7954i	−0.273559	−	0.223977i
$201$		0			0
$202$		−	77.7403i		−	0.384853i
$203$	−169.054			−0.832779
$204$		0			0
$205$	230.290	+	82.2496i	1.12337	+	0.401218i
$206$			20.1626i			0.0978769i
$207$		0			0
$208$			6.15332i			0.0295833i
$209$		−	16.6992i		−	0.0799004i
$210$		0			0
$211$	122.504			0.580589			0.290295	−	0.956937i	$-0.406247\pi$
$211$	122.504			0.580589			0.290295	+	0.956937i	$0.406247\pi$
$212$	−138.020			−0.651038
$213$		0			0
$214$	115.917			0.541667
$215$	−16.7398	−	5.97872i	−0.0778596	−	0.0278080i
$216$		0			0
$217$			553.011i			2.54844i
$218$	−159.096			−0.729797
$219$		0			0
$220$	−22.1740	−	7.91958i	−0.100791	−	0.0359981i
$221$		−	17.0559i		−	0.0771759i
$222$		0			0
$223$			276.247i			1.23878i	0.785085	+	0.619388i	$0.212619\pi$
$223$			276.247i			1.23878i	−0.785085	+	0.619388i	$0.787381\pi$
$224$		−	53.2780i		−	0.237848i
$225$		0			0
$226$	127.001			0.561952
$227$	−2.70773			−0.0119283			−0.00596417	−	0.999982i	$-0.501898\pi$
$227$	−2.70773			−0.0119283			−0.00596417	+	0.999982i	$0.501898\pi$
$228$		0			0
$229$	−209.636			−0.915441			−0.457720	−	0.889096i	$-0.651334\pi$
$229$	−209.636			−0.915441			−0.457720	+	0.889096i	$0.651334\pi$
$230$	19.4926	−	54.5773i	0.0847504	−	0.237292i
$231$		0			0
$232$		−	50.7689i		−	0.218831i
$233$	183.458			0.787375			0.393688	−	0.919244i	$-0.371199\pi$
$233$	183.458			0.787375			0.393688	+	0.919244i	$0.371199\pi$
$234$		0			0
$235$	117.336	−	328.528i	0.499301	−	1.39799i
$236$		−	78.6886i		−	0.333426i
$237$		0			0
$238$			147.677i			0.620491i
$239$			57.9858i			0.242618i	0.992615	+	0.121309i	$0.0387093\pi$
$239$			57.9858i			0.242618i	−0.992615	+	0.121309i	$0.961291\pi$
$240$		0			0
$241$	157.040			0.651618			0.325809	−	0.945436i	$-0.394363\pi$
$241$	157.040			0.651618			0.325809	+	0.945436i	$0.394363\pi$
$242$	163.279			0.674708
$243$		0			0
$244$	231.336			0.948099
$245$	−66.7727	+	186.957i	−0.272542	+	0.763089i
$246$		0			0
$247$			10.9102i			0.0441706i
$248$	−166.075			−0.669659
$249$		0			0
$250$	−151.469	+	91.1433i	−0.605876	+	0.364573i
$251$		−	254.026i		−	1.01206i	−0.862517	−	0.506029i	$-0.831113\pi$
$251$		−	254.026i		−	1.01206i	0.862517	−	0.506029i	$-0.168887\pi$
$252$		0			0
$253$		−	19.2979i		−	0.0762764i
$254$			124.959i			0.491965i
$255$		0			0
$256$	16.0000			0.0625000
$257$	264.751			1.03016			0.515080	−	0.857142i	$-0.327762\pi$
$257$	264.751			1.03016			0.515080	+	0.857142i	$0.327762\pi$
$258$		0			0
$259$	−195.847			−0.756166
$260$	14.4870	+	5.17413i	0.0557194	+	0.0199005i
$261$		0			0
$262$			245.703i			0.937798i
$263$	−178.270			−0.677833			−0.338916	−	0.940816i	$-0.610060\pi$
$263$	−178.270			−0.677833			−0.338916	+	0.940816i	$0.610060\pi$
$264$		0			0
$265$	−116.057	+	324.947i	−0.437950	+	1.22621i
$266$		−	94.4647i		−	0.355130i
$267$		0			0
$268$		−	205.100i		−	0.765300i
$269$		−	270.345i		−	1.00500i	−0.864577	−	0.502501i	$-0.832413\pi$
$269$		−	270.345i		−	1.00500i	0.864577	−	0.502501i	$-0.167587\pi$
$270$		0			0
$271$	389.617			1.43770			0.718850	−	0.695165i	$-0.244669\pi$
$271$	389.617			1.43770			0.718850	+	0.695165i	$0.244669\pi$
$272$	−44.3490			−0.163048
$273$		0			0
$274$	154.090			0.562372
$275$	−37.2909	+	45.5460i	−0.135603	+	0.165622i
$276$		0			0
$277$			483.477i			1.74540i	0.488252	+	0.872702i	$0.337635\pi$
$277$			483.477i			1.74540i	−0.488252	+	0.872702i	$0.662365\pi$
$278$	−188.531			−0.678168
$279$		0			0
$280$	−125.435	−	44.7998i	−0.447982	−	0.159999i
$281$		−	274.852i		−	0.978121i	−0.872250	−	0.489061i	$-0.837340\pi$
$281$		−	274.852i		−	0.978121i	0.872250	−	0.489061i	$-0.162660\pi$
$282$		0			0
$283$			462.554i			1.63447i	0.576307	+	0.817233i	$0.304493\pi$
$283$			462.554i			1.63447i	−0.576307	+	0.817233i	$0.695507\pi$
$284$			205.334i			0.723006i
$285$		0			0
$286$	5.12246			0.0179107
$287$	460.627			1.60497
$288$		0			0
$289$	−166.073			−0.574646
$290$	−119.527	−	42.6899i	−0.412164	−	0.147207i
$291$		0			0
$292$		−	241.225i		−	0.826114i
$293$	−61.0380			−0.208321			−0.104160	−	0.994561i	$-0.533216\pi$
$293$	−61.0380			−0.208321			−0.104160	+	0.994561i	$0.533216\pi$
$294$		0			0
$295$	−185.260	−	66.1667i	−0.628000	−	0.224294i
$296$		−	58.8151i		−	0.198700i
$297$		0			0
$298$			175.434i			0.588705i
$299$			12.6080i			0.0421672i
$300$		0			0
$301$	−33.4829			−0.111239
$302$	19.2875			0.0638658
$303$		0			0
$304$	28.3688			0.0933185
$305$	194.523	−	544.645i	0.637781	−	1.78572i
$306$		0			0
$307$			108.831i			0.354498i	0.984166	+	0.177249i	$0.0567198\pi$
$307$			108.831i			0.354498i	−0.984166	+	0.177249i	$0.943280\pi$
$308$	−44.3524			−0.144001
$309$		0			0
$310$	−139.648	+	390.999i	−0.450476	+	1.26129i
$311$			252.816i			0.812914i	0.913670	+	0.406457i	$0.133236\pi$
$311$			252.816i			0.812914i	−0.913670	+	0.406457i	$0.866764\pi$
$312$		0			0
$313$		−	440.634i		−	1.40778i	−0.710311	−	0.703888i	$-0.751446\pi$
$313$		−	440.634i		−	1.40778i	0.710311	−	0.703888i	$-0.248554\pi$
$314$		−	348.114i		−	1.10864i
$315$		0			0
$316$	−36.5204			−0.115571
$317$	188.211			0.593725			0.296862	−	0.954920i	$-0.404060\pi$
$317$	188.211			0.593725			0.296862	+	0.954920i	$0.404060\pi$
$318$		0			0
$319$	−42.2636			−0.132488
$320$	13.4539	−	37.6695i	0.0420434	−	0.117717i
$321$		0			0
$322$		−	109.165i		−	0.339023i
$323$	−78.6331			−0.243446
$324$		0			0
$325$	24.3634	−	29.7567i	0.0749643	−	0.0915592i
$326$			79.7672i			0.244685i
$327$		0			0
$328$			138.331i			0.421742i
$329$		−	657.122i		−	1.99733i
$330$		0			0
$331$	162.562			0.491124			0.245562	−	0.969381i	$-0.421027\pi$
$331$	162.562			0.491124			0.245562	+	0.969381i	$0.421027\pi$
$332$	323.881			0.975546
$333$		0			0
$334$	140.538			0.420771
$335$	−482.877	−	172.462i	−1.44142	−	0.514813i
$336$		0			0
$337$		−	230.450i		−	0.683829i	−0.939731	−	0.341915i	$-0.888925\pi$
$337$		−	230.450i		−	0.683829i	0.939731	−	0.341915i	$-0.111075\pi$
$338$	235.655			0.697205
$339$		0			0
$340$	−37.2917	+	104.413i	−0.109681	+	0.307097i
$341$			138.253i			0.405434i
$342$		0			0
$343$		−	87.5467i		−	0.255238i
$344$		−	10.0553i		−	0.0292305i
$345$		0			0
$346$	−403.872			−1.16726
$347$	574.856			1.65664			0.828322	−	0.560252i	$-0.189296\pi$
$347$	574.856			1.65664			0.828322	+	0.560252i	$0.189296\pi$
$348$		0			0
$349$	−49.4784			−0.141772			−0.0708860	−	0.997484i	$-0.522583\pi$
$349$	−49.4784			−0.141772			−0.0708860	+	0.997484i	$0.522583\pi$
$350$	−210.948	+	257.646i	−0.602710	+	0.736132i
$351$		0			0
$352$		−	13.3195i		−	0.0378396i
$353$	−402.400			−1.13994			−0.569971	−	0.821665i	$-0.693046\pi$
$353$	−402.400			−1.13994			−0.569971	+	0.821665i	$0.693046\pi$
$354$		0			0
$355$	483.426	+	172.659i	1.36176	+	0.486362i
$356$			176.467i			0.495695i
$357$		0			0
$358$			255.364i			0.713308i
$359$			562.625i			1.56720i	0.621266	+	0.783600i	$0.286619\pi$
$359$			562.625i			1.56720i	−0.621266	+	0.783600i	$0.713381\pi$
$360$		0			0
$361$	−310.701			−0.860667
$362$	124.057			0.342700
$363$		0			0
$364$	28.9770			0.0796070
$365$	−567.927	−	202.839i	−1.55597	−	0.555722i
$366$		0			0
$367$			187.323i			0.510416i	0.966886	+	0.255208i	$0.0821439\pi$
$367$			187.323i			0.510416i	−0.966886	+	0.255208i	$0.917856\pi$
$368$	32.7836			0.0890859
$369$		0			0
$370$	−138.471	−	49.4557i	−0.374246	−	0.133664i
$371$			649.958i			1.75191i
$372$		0			0
$373$		−	552.930i		−	1.48239i	−0.671292	−	0.741193i	$-0.734260\pi$
$373$		−	552.930i		−	1.48239i	0.671292	−	0.741193i	$-0.265740\pi$
$374$			36.9193i			0.0987147i
$375$		0			0
$376$	197.341			0.524844
$377$	27.6123			0.0732421
$378$		0			0
$379$	447.398			1.18047			0.590235	−	0.807232i	$-0.299035\pi$
$379$	447.398			1.18047			0.590235	+	0.807232i	$0.299035\pi$
$380$	23.8544	−	66.7900i	0.0627748	−	0.175763i
$381$		0			0
$382$		−	87.5500i		−	0.229189i
$383$	−561.513			−1.46609			−0.733045	−	0.680180i	$-0.761902\pi$
$383$	−561.513			−1.46609			−0.733045	+	0.680180i	$0.761902\pi$
$384$		0			0
$385$	−37.2946	+	104.421i	−0.0968690	+	0.271223i
$386$			317.885i			0.823537i
$387$		0			0
$388$		−	281.240i		−	0.724846i
$389$		−	430.919i		−	1.10776i	−0.832596	−	0.553881i	$-0.813146\pi$
$389$		−	430.919i		−	1.10776i	0.832596	−	0.553881i	$-0.186854\pi$
$390$		0			0
$391$	−90.8701			−0.232404
$392$	−112.302			−0.286484
$393$		0			0
$394$	368.093			0.934246
$395$	−30.7089	+	85.9817i	−0.0777440	+	0.217675i
$396$		0			0
$397$			656.459i			1.65355i	0.562533	+	0.826775i	$0.309827\pi$
$397$			656.459i			1.65355i	−0.562533	+	0.826775i	$0.690173\pi$
$398$	−38.4810			−0.0966860
$399$		0			0
$400$	−77.3741	−	63.3502i	−0.193435	−	0.158376i
$401$		−	268.170i		−	0.668752i	−0.942440	−	0.334376i	$-0.891474\pi$
$401$		−	268.170i		−	0.668752i	0.942440	−	0.334376i	$-0.108526\pi$
$402$		0			0
$403$		−	90.3254i		−	0.224133i
$404$		−	109.941i		−	0.272132i
$405$		0			0
$406$	−239.079			−0.588864
$407$	−48.9619			−0.120299
$408$		0			0
$409$	307.328			0.751412			0.375706	−	0.926739i	$-0.377400\pi$
$409$	307.328			0.751412			0.375706	+	0.926739i	$0.377400\pi$
$410$	325.680	+	116.319i	0.794341	+	0.283704i
$411$		0			0
$412$			28.5143i			0.0692094i
$413$	−370.557			−0.897232
$414$		0			0
$415$	272.342	−	762.529i	0.656245	−	1.83742i
$416$			8.70211i			0.0209185i
$417$		0			0
$418$		−	23.6162i		−	0.0564981i
$419$			321.629i			0.767612i	0.923414	+	0.383806i	$0.125387\pi$
$419$			321.629i			0.767612i	−0.923414	+	0.383806i	$0.874613\pi$
$420$		0			0
$421$	321.773			0.764307			0.382154	−	0.924099i	$-0.375183\pi$
$421$	321.773			0.764307			0.382154	+	0.924099i	$0.375183\pi$
$422$	173.247			0.410539
$423$		0			0
$424$	−195.190			−0.460353
$425$	214.467	+	175.595i	0.504627	+	0.413165i
$426$		0			0
$427$		−	1089.40i		−	2.55128i
$428$	163.931			0.383017
$429$		0			0
$430$	−23.6737	−	8.45519i	−0.0550550	−	0.0196632i
$431$			90.0447i			0.208920i	0.994529	+	0.104460i	$0.0333115\pi$
$431$			90.0447i			0.208920i	−0.994529	+	0.104460i	$0.966689\pi$
$432$		0			0
$433$			22.6191i			0.0522382i	0.999659	+	0.0261191i	$0.00831491\pi$
$433$			22.6191i			0.0522382i	−0.999659	+	0.0261191i	$0.991685\pi$
$434$			782.075i			1.80202i
$435$		0			0
$436$	−224.995			−0.516044
$437$	58.1270			0.133014
$438$		0			0
$439$	−639.315			−1.45630			−0.728149	−	0.685419i	$-0.759619\pi$
$439$	−639.315			−1.45630			−0.728149	+	0.685419i	$0.759619\pi$
$440$	−31.3588	−	11.2000i	−0.0712700	−	0.0254545i
$441$		0			0
$442$		−	24.1206i		−	0.0545716i
$443$	440.084			0.993418			0.496709	−	0.867917i	$-0.334542\pi$
$443$	440.084			0.993418			0.496709	+	0.867917i	$0.334542\pi$
$444$		0			0
$445$	415.465	+	148.386i	0.933630	+	0.333451i
$446$			390.672i			0.875947i
$447$		0			0
$448$		−	75.3465i		−	0.168184i
$449$			281.927i			0.627899i	0.949440	+	0.313949i	$0.101652\pi$
$449$			281.927i			0.627899i	−0.949440	+	0.313949i	$0.898348\pi$
$450$		0			0
$451$	115.157			0.255337
$452$	179.607			0.397360
$453$		0			0
$454$	−3.82931			−0.00843461
$455$	24.3658	−	68.2217i	0.0535512	−	0.149938i
$456$		0			0
$457$		−	371.693i		−	0.813333i	−0.913577	−	0.406667i	$-0.866691\pi$
$457$		−	371.693i		−	0.813333i	0.913577	−	0.406667i	$-0.133309\pi$
$458$	−296.470			−0.647314
$459$		0			0
$460$	27.5667	−	77.1839i	0.0599276	−	0.167791i
$461$			221.907i			0.481361i	0.970604	+	0.240680i	$0.0773705\pi$
$461$			221.907i			0.481361i	−0.970604	+	0.240680i	$0.922629\pi$
$462$		0			0
$463$			357.946i			0.773102i	0.922268	+	0.386551i	$0.126334\pi$
$463$			357.946i			0.773102i	−0.922268	+	0.386551i	$0.873666\pi$
$464$		−	71.7981i		−	0.154737i
$465$		0			0
$466$	259.449			0.556758
$467$	−392.199			−0.839826			−0.419913	−	0.907564i	$-0.637939\pi$
$467$	−392.199			−0.839826			−0.419913	+	0.907564i	$0.637939\pi$
$468$		0			0
$469$	−965.850			−2.05938
$470$	165.938	−	464.609i	0.353059	−	0.988530i
$471$		0			0
$472$		−	111.282i		−	0.235768i
$473$	−8.37076			−0.0176972
$474$		0			0
$475$	−137.188	−	112.323i	−0.288817	−	0.236470i
$476$			208.847i			0.438753i
$477$		0			0
$478$			82.0043i			0.171557i
$479$		−	804.986i		−	1.68056i	−0.542156	−	0.840278i	$-0.682392\pi$
$479$		−	804.986i		−	1.68056i	0.542156	−	0.840278i	$-0.317608\pi$
$480$		0			0
$481$	31.9884			0.0665040
$482$	222.088			0.460763
$483$		0			0
$484$	230.912			0.477091
$485$	−662.136	−	236.486i	−1.36523	−	0.487600i
$486$		0			0
$487$			477.427i			0.980342i	0.871626	+	0.490171i	$0.163066\pi$
$487$			477.427i			0.980342i	−0.871626	+	0.490171i	$0.836934\pi$
$488$	327.159			0.670407
$489$		0			0
$490$	−94.4309	+	264.397i	−0.192716	+	0.539585i
$491$			359.383i			0.731941i	0.930626	+	0.365970i	$0.119263\pi$
$491$			359.383i			0.731941i	−0.930626	+	0.365970i	$0.880737\pi$
$492$		0			0
$493$			199.011i			0.403673i
$494$			15.4293i			0.0312334i
$495$		0			0
$496$	−234.866			−0.473520
$497$	966.948			1.94557
$498$		0			0
$499$	388.105			0.777766			0.388883	−	0.921287i	$-0.372861\pi$
$499$	388.105			0.777766			0.388883	+	0.921287i	$0.372861\pi$
$500$	−214.210	+	128.896i	−0.428419	+	0.257792i
$501$		0			0
$502$		−	359.248i		−	0.715633i
$503$	−23.5000			−0.0467197			−0.0233598	−	0.999727i	$-0.507436\pi$
$503$	−23.5000			−0.0467197			−0.0233598	+	0.999727i	$0.507436\pi$
$504$		0			0
$505$	−258.840	−	92.4463i	−0.512555	−	0.183062i
$506$		−	27.2914i		−	0.0539356i
$507$		0			0
$508$			176.719i			0.347872i
$509$			159.137i			0.312646i	0.987706	+	0.156323i	$0.0499641\pi$
$509$			159.137i			0.312646i	−0.987706	+	0.156323i	$0.950036\pi$
$510$		0			0
$511$	−1135.97			−2.22303
$512$	22.6274			0.0441942
$513$		0			0
$514$	374.415			0.728434
$515$	67.1324	+	23.9767i	0.130354	+	0.0465568i
$516$		0			0
$517$		−	164.281i		−	0.317758i
$518$	−276.969			−0.534690
$519$		0			0
$520$	20.4878	+	7.31733i	0.0393996	+	0.0140718i
$521$		−	591.045i		−	1.13444i	−0.823565	−	0.567222i	$-0.808018\pi$
$521$		−	591.045i		−	1.13444i	0.823565	−	0.567222i	$-0.191982\pi$
$522$		0			0
$523$		−	542.921i		−	1.03809i	−0.854747	−	0.519045i	$-0.826288\pi$
$523$		−	542.921i		−	1.03809i	0.854747	−	0.519045i	$-0.173712\pi$
$524$			347.476i			0.663123i
$525$		0			0
$526$	−252.112			−0.479300
$527$	651.005			1.23530
$528$		0			0
$529$	−461.827			−0.873019
$530$	−164.129	+	459.544i	−0.309677	+	0.867064i
$531$		0			0
$532$		−	133.593i		−	0.251115i
$533$	−75.2359			−0.141156
$534$		0			0
$535$	137.844	−	385.950i	0.257653	−	0.721402i
$536$		−	290.056i		−	0.541149i
$537$		0			0
$538$		−	382.326i		−	0.710643i
$539$			93.4879i			0.173447i
$540$		0			0
$541$	−744.623			−1.37638			−0.688192	−	0.725529i	$-0.741595\pi$
$541$	−744.623			−1.37638			−0.688192	+	0.725529i	$0.741595\pi$
$542$	551.002			1.01661
$543$		0			0
$544$	−62.7190			−0.115292
$545$	−189.191	+	529.717i	−0.347140	+	0.971957i
$546$		0			0
$547$			126.537i			0.231329i	0.993288	+	0.115664i	$0.0368997\pi$
$547$			126.537i			0.231329i	−0.993288	+	0.115664i	$0.963100\pi$
$548$	217.916			0.397657
$549$		0			0
$550$	−52.7372	+	64.4117i	−0.0958859	+	0.117112i
$551$		−	127.302i		−	0.231037i
$552$		0			0
$553$			171.980i			0.310995i
$554$			683.740i			1.23419i
$555$		0			0
$556$	−266.623			−0.479537
$557$	−363.331			−0.652301			−0.326150	−	0.945318i	$-0.605752\pi$
$557$	−363.331			−0.652301			−0.326150	+	0.945318i	$0.605752\pi$
$558$		0			0
$559$	5.46890			0.00978336
$560$	−177.392	−	63.3565i	−0.316771	−	0.113137i
$561$		0			0
$562$		−	388.700i		−	0.691636i
$563$	305.690			0.542967			0.271484	−	0.962443i	$-0.412486\pi$
$563$	305.690			0.542967			0.271484	+	0.962443i	$0.412486\pi$
$564$		0			0
$565$	151.026	−	422.856i	0.267302	−	0.748419i
$566$			654.150i			1.15574i
$567$		0			0
$568$			290.386i			0.511242i
$569$		−	908.817i		−	1.59722i	−0.601851	−	0.798609i	$-0.705570\pi$
$569$		−	908.817i		−	1.59722i	0.601851	−	0.798609i	$-0.294430\pi$
$570$		0			0
$571$	420.505			0.736436			0.368218	−	0.929739i	$-0.379968\pi$
$571$	420.505			0.736436			0.368218	+	0.929739i	$0.379968\pi$
$572$	7.24426			0.0126648
$573$		0			0
$574$	651.424			1.13489
$575$	−158.538	−	129.803i	−0.275718	−	0.225744i
$576$		0			0
$577$		−	488.343i		−	0.846348i	−0.906048	−	0.423174i	$-0.860916\pi$
$577$		−	488.343i		−	0.846348i	0.906048	−	0.423174i	$-0.139084\pi$
$578$	−234.862			−0.406336
$579$		0			0
$580$	−169.037	−	60.3727i	−0.291444	−	0.104091i
$581$		−	1525.21i		−	2.62514i
$582$		0			0
$583$			162.490i			0.278713i
$584$		−	341.144i		−	0.584151i
$585$		0			0
$586$	−86.3208			−0.147305
$587$	688.435			1.17280			0.586402	−	0.810021i	$-0.300544\pi$
$587$	688.435			1.17280			0.586402	+	0.810021i	$0.300544\pi$
$588$		0			0
$589$	−416.430			−0.707011
$590$	−261.997	−	93.5739i	−0.444063	−	0.158600i
$591$		0			0
$592$		−	83.1771i		−	0.140502i
$593$	−532.286			−0.897616			−0.448808	−	0.893628i	$-0.648151\pi$
$593$	−532.286			−0.897616			−0.448808	+	0.893628i	$0.648151\pi$
$594$		0			0
$595$	491.697	+	175.612i	0.826381	+	0.295147i
$596$			248.101i			0.416277i
$597$		0			0
$598$			17.8304i			0.0298167i
$599$		−	139.317i		−	0.232583i	−0.993215	−	0.116292i	$-0.962899\pi$
$599$		−	139.317i		−	0.232583i	0.993215	−	0.116292i	$-0.0371007\pi$
$600$		0			0
$601$	−12.1666			−0.0202440			−0.0101220	−	0.999949i	$-0.503222\pi$
$601$	−12.1666			−0.0202440			−0.0101220	+	0.999949i	$0.503222\pi$
$602$	−47.3520			−0.0786579
$603$		0			0
$604$	27.2766			0.0451600
$605$	194.166	−	543.646i	0.320936	−	0.898589i
$606$		0			0
$607$		−	371.452i		−	0.611948i	−0.952040	−	0.305974i	$-0.901018\pi$
$607$		−	371.452i		−	0.611948i	0.952040	−	0.305974i	$-0.0989820\pi$
$608$	40.1196			0.0659861
$609$		0			0
$610$	275.097	−	770.245i	0.450979	−	1.26270i
$611$			107.330i			0.175663i
$612$		0			0
$613$		−	656.136i		−	1.07037i	−0.844736	−	0.535184i	$-0.820242\pi$
$613$		−	656.136i		−	1.07037i	0.844736	−	0.535184i	$-0.179758\pi$
$614$			153.910i			0.250668i
$615$		0			0
$616$	−62.7238			−0.101824
$617$	−529.364			−0.857964			−0.428982	−	0.903313i	$-0.641128\pi$
$617$	−529.364			−0.857964			−0.428982	+	0.903313i	$0.641128\pi$
$618$		0			0
$619$	−402.138			−0.649658			−0.324829	−	0.945773i	$-0.605307\pi$
$619$	−402.138			−0.649658			−0.324829	+	0.945773i	$0.605307\pi$
$620$	−197.491	+	552.956i	−0.318535	+	0.891864i
$621$		0			0
$622$			357.536i			0.574817i
$623$	831.013			1.33389
$624$		0			0
$625$	123.344	+	612.708i	0.197350	+	0.980333i
$626$		−	623.151i		−	0.995448i
$627$		0			0
$628$		−	492.307i		−	0.783929i
$629$			230.551i			0.366536i
$630$		0			0
$631$	576.351			0.913393			0.456696	−	0.889623i	$-0.349033\pi$
$631$	576.351			0.913393			0.456696	+	0.889623i	$0.349033\pi$
$632$	−51.6477			−0.0817210
$633$		0			0
$634$	266.170			0.419827
$635$	416.057	+	148.597i	0.655208	+	0.234011i
$636$		0			0
$637$		−	61.0788i		−	0.0958851i
$638$	−59.7698			−0.0936831
$639$		0			0
$640$	19.0267	−	53.2727i	0.0297292	−	0.0832387i
$641$			191.106i			0.298137i	0.988827	+	0.149069i	$0.0476275\pi$
$641$			191.106i			0.298137i	−0.988827	+	0.149069i	$0.952372\pi$
$642$		0			0
$643$		−	879.614i		−	1.36798i	−0.729490	−	0.683992i	$-0.760242\pi$
$643$		−	879.614i		−	1.36798i	0.729490	−	0.683992i	$-0.239758\pi$
$644$		−	154.383i		−	0.239725i
$645$		0			0
$646$	−111.204			−0.172142
$647$	272.480			0.421144			0.210572	−	0.977578i	$-0.432467\pi$
$647$	272.480			0.421144			0.210572	+	0.977578i	$0.432467\pi$
$648$		0			0
$649$	−92.6395			−0.142742
$650$	34.4550	−	42.0824i	0.0530077	−	0.0647421i
$651$		0			0
$652$			112.808i			0.173018i
$653$	354.531			0.542926			0.271463	−	0.962449i	$-0.412492\pi$
$653$	354.531			0.542926			0.271463	+	0.962449i	$0.412492\pi$
$654$		0			0
$655$	818.080	+	292.182i	1.24898	+	0.446079i
$656$			195.630i			0.298217i
$657$		0			0
$658$		−	929.311i		−	1.41233i
$659$			256.429i			0.389119i	0.980891	+	0.194559i	$0.0623277\pi$
$659$			256.429i			0.389119i	−0.980891	+	0.194559i	$0.937672\pi$
$660$		0			0
$661$	806.565			1.22022			0.610109	−	0.792317i	$-0.291125\pi$
$661$	806.565			1.22022			0.610109	+	0.792317i	$0.291125\pi$
$662$	229.897			0.347277
$663$		0			0
$664$	458.037			0.689815
$665$	−314.524	−	112.334i	−0.472969	−	0.168924i
$666$		0			0
$667$		−	147.112i		−	0.220558i
$668$	198.750			0.297530
$669$		0			0
$670$	−682.891	−	243.899i	−1.01924	−	0.364028i
$671$		−	272.350i		−	0.405887i
$672$		0			0
$673$		−	276.517i		−	0.410872i	−0.978671	−	0.205436i	$-0.934139\pi$
$673$		−	276.517i		−	0.410872i	0.978671	−	0.205436i	$-0.0658613\pi$
$674$		−	325.906i		−	0.483540i
$675$		0			0
$676$	333.267			0.492999
$677$	−1072.68			−1.58446			−0.792228	−	0.610225i	$-0.791079\pi$
$677$	−1072.68			−1.58446			−0.792228	+	0.610225i	$0.791079\pi$
$678$		0			0
$679$	−1324.40			−1.95052
$680$	−52.7384	+	147.662i	−0.0775565	+	0.217150i
$681$		0			0
$682$			195.519i			0.286685i
$683$	−553.261			−0.810046			−0.405023	−	0.914307i	$-0.632736\pi$
$683$	−553.261			−0.810046			−0.405023	+	0.914307i	$0.632736\pi$
$684$		0			0
$685$	183.239	−	513.050i	0.267502	−	0.748978i
$686$		−	123.810i		−	0.180481i
$687$		0			0
$688$		−	14.2204i		−	0.0206691i
$689$		−	106.160i		−	0.154079i
$690$		0			0
$691$	−1252.61			−1.81275			−0.906375	−	0.422474i	$-0.861162\pi$
$691$	−1252.61			−1.81275			−0.906375	+	0.422474i	$0.861162\pi$
$692$	−571.162			−0.825378
$693$		0			0
$694$	812.969			1.17142
$695$	−224.195	+	627.722i	−0.322582	+	0.903197i
$696$		0			0
$697$		−	542.251i		−	0.777978i
$698$	−69.9730			−0.100248
$699$		0			0
$700$	−298.326	+	364.367i	−0.426180	+	0.520524i
$701$		−	727.608i		−	1.03796i	−0.854787	−	0.518979i	$-0.826312\pi$
$701$		−	727.608i		−	1.03796i	0.854787	−	0.518979i	$-0.173688\pi$
$702$		0			0
$703$		−	147.477i		−	0.209783i
$704$		−	18.8367i		−	0.0267566i
$705$		0			0
$706$	−569.079			−0.806061
$707$	−517.732			−0.732294
$708$		0			0
$709$	640.817			0.903833			0.451916	−	0.892060i	$-0.350741\pi$
$709$	640.817			0.903833			0.451916	+	0.892060i	$0.350741\pi$
$710$	683.668	+	244.176i	0.962913	+	0.343910i
$711$		0			0
$712$			249.563i			0.350509i
$713$	−481.235			−0.674944
$714$		0			0
$715$	6.09147	−	17.0555i	0.00851953	−	0.0238538i
$716$			361.140i			0.504385i
$717$		0			0
$718$			795.672i			1.10818i
$719$			738.347i			1.02691i	0.858117	+	0.513454i	$0.171634\pi$
$719$			738.347i			1.02691i	−0.858117	+	0.513454i	$0.828366\pi$
$720$		0			0
$721$	134.278			0.186239
$722$	−439.397			−0.608583
$723$		0			0
$724$	175.443			0.242325
$725$	−284.276	+	347.207i	−0.392105	+	0.478906i
$726$		0			0
$727$		−	685.756i		−	0.943268i	−0.881794	−	0.471634i	$-0.843664\pi$
$727$		−	685.756i		−	0.943268i	0.881794	−	0.471634i	$-0.156336\pi$
$728$	40.9796			0.0562907
$729$		0			0
$730$	−803.171	−	286.857i	−1.10023	−	0.392955i
$731$			39.4162i			0.0539209i
$732$		0			0
$733$			147.012i			0.200563i	0.994959	+	0.100281i	$0.0319743\pi$
$733$			147.012i			0.200563i	−0.994959	+	0.100281i	$0.968026\pi$
$734$			264.914i			0.360919i
$735$		0			0
$736$	46.3630			0.0629932
$737$	−241.463			−0.327630
$738$		0			0
$739$	34.9060			0.0472341			0.0236171	−	0.999721i	$-0.492482\pi$
$739$	34.9060			0.0472341			0.0236171	+	0.999721i	$0.492482\pi$
$740$	−195.827	−	69.9410i	−0.264632	−	0.0945148i
$741$		0			0
$742$			919.179i			1.23879i
$743$	−280.930			−0.378103			−0.189051	−	0.981967i	$-0.560541\pi$
$743$	−280.930			−0.378103			−0.189051	+	0.981967i	$0.560541\pi$
$744$		0			0
$745$	584.116	+	208.620i	0.784048	+	0.280027i
$746$		−	781.961i		−	1.04821i
$747$		0			0
$748$			52.2118i			0.0698018i
$749$		−	771.977i		−	1.03068i
$750$		0			0
$751$	−1314.60			−1.75047			−0.875236	−	0.483696i	$-0.839294\pi$
$751$	−1314.60			−1.75047			−0.875236	+	0.483696i	$0.839294\pi$
$752$	279.083			0.371120
$753$		0			0
$754$	39.0497			0.0517900
$755$	22.9360	−	64.2186i	0.0303789	−	0.0850577i
$756$		0			0
$757$			670.545i			0.885793i	0.896573	+	0.442897i	$0.146049\pi$
$757$			670.545i			0.885793i	−0.896573	+	0.442897i	$0.853951\pi$
$758$	632.716			0.834718
$759$		0			0
$760$	33.7353	−	94.4553i	0.0443885	−	0.124283i
$761$		−	451.825i		−	0.593725i	−0.954920	−	0.296863i	$-0.904060\pi$
$761$		−	451.825i		−	0.593725i	0.954920	−	0.296863i	$-0.0959403\pi$
$762$		0			0
$763$			1059.54i			1.38865i
$764$		−	123.814i		−	0.162061i
$765$		0			0
$766$	−794.099			−1.03668
$767$	60.5245			0.0789107
$768$		0			0
$769$	−157.865			−0.205286			−0.102643	−	0.994718i	$-0.532730\pi$
$769$	−157.865			−0.205286			−0.102643	+	0.994718i	$0.532730\pi$
$770$	−52.7425	+	147.673i	−0.0684967	+	0.191784i
$771$		0			0
$772$			449.558i			0.582329i
$773$	408.029			0.527852			0.263926	−	0.964543i	$-0.414983\pi$
$773$	408.029			0.527852			0.263926	+	0.964543i	$0.414983\pi$
$774$		0			0
$775$	1135.78	+	929.926i	1.46553	+	1.19990i
$776$		−	397.734i		−	0.512543i
$777$		0			0
$778$		−	609.412i		−	0.783306i
$779$			346.862i			0.445266i
$780$		0			0
$781$	241.738			0.309523
$782$	−128.510			−0.164335
$783$		0			0
$784$	−158.819			−0.202575
$785$	−1159.06	−	413.965i	−1.47651	−	0.527344i
$786$		0			0
$787$			374.041i			0.475274i	0.971354	+	0.237637i	$0.0763729\pi$
$787$			374.041i			0.475274i	−0.971354	+	0.237637i	$0.923627\pi$
$788$	520.562			0.660612
$789$		0			0
$790$	−43.4289	+	121.597i	−0.0549733	+	0.153920i
$791$		−	845.797i		−	1.06928i
$792$		0			0
$793$			177.936i			0.224383i
$794$			928.373i			1.16924i
$795$		0			0
$796$	−54.4204			−0.0683673
$797$	−539.275			−0.676631			−0.338316	−	0.941033i	$-0.609857\pi$
$797$	−539.275			−0.676631			−0.338316	+	0.941033i	$0.609857\pi$
$798$		0			0
$799$	−773.565			−0.968167
$800$	−109.423	−	89.5907i	−0.136779	−	0.111988i
$801$		0			0
$802$		−	379.249i		−	0.472879i
$803$	−283.993			−0.353664
$804$		0			0
$805$	−363.471	−	129.816i	−0.451517	−	0.161262i
$806$		−	127.739i		−	0.158486i
$807$		0			0
$808$		−	155.481i		−	0.192427i
$809$			220.597i			0.272679i	0.990662	+	0.136339i	$0.0435338\pi$
$809$			220.597i			0.272679i	−0.990662	+	0.136339i	$0.956466\pi$
$810$		0			0
$811$	−502.813			−0.619992			−0.309996	−	0.950738i	$-0.600328\pi$
$811$	−502.813			−0.619992			−0.309996	+	0.950738i	$0.600328\pi$
$812$	−338.108			−0.416390
$813$		0			0
$814$	−69.2425			−0.0850645
$815$	265.589	+	94.8566i	0.325876	+	0.116388i
$816$		0			0
$817$		−	25.2134i		−	0.0308610i
$818$	434.627			0.531329
$819$		0			0
$820$	460.581	+	164.499i	0.561684	+	0.200609i
$821$			560.654i			0.682891i	0.939902	+	0.341446i	$0.110916\pi$
$821$			560.654i			0.682891i	−0.939902	+	0.341446i	$0.889084\pi$
$822$		0			0
$823$			1089.52i			1.32384i	0.749575	+	0.661919i	$0.230258\pi$
$823$			1089.52i			1.32384i	−0.749575	+	0.661919i	$0.769742\pi$
$824$			40.3253i			0.0489384i
$825$		0			0
$826$	−524.047			−0.634439
$827$	963.697			1.16529			0.582646	−	0.812726i	$-0.302017\pi$
$827$	963.697			1.16529			0.582646	+	0.812726i	$0.302017\pi$
$828$		0			0
$829$	−273.552			−0.329978			−0.164989	−	0.986295i	$-0.552759\pi$
$829$	−273.552			−0.329978			−0.164989	+	0.986295i	$0.552759\pi$
$830$	385.149	−	1078.38i	0.464035	−	1.29925i
$831$		0			0
$832$			12.3066i			0.0147916i
$833$	440.216			0.528470
$834$		0			0
$835$	167.123	−	467.927i	0.200147	−	0.560391i
$836$		−	33.3984i		−	0.0399502i
$837$		0			0
$838$			454.853i			0.542784i
$839$		−	558.574i		−	0.665762i	−0.942969	−	0.332881i	$-0.891979\pi$
$839$		−	558.574i		−	0.665762i	0.942969	−	0.332881i	$-0.108021\pi$
$840$		0			0
$841$	518.815			0.616902
$842$	455.056			0.540447
$843$		0			0
$844$	245.009			0.290295
$845$	280.234	−	784.626i	0.331638	−	0.928551i
$846$		0			0
$847$		−	1087.40i		−	1.28383i
$848$	−276.040			−0.325519
$849$		0			0
$850$	303.302	+	248.329i	0.356826	+	0.292152i
$851$		−	170.428i		−	0.200268i
$852$		0			0
$853$		−	921.750i		−	1.08060i	−0.841473	−	0.540299i	$-0.818311\pi$
$853$		−	921.750i		−	1.08060i	0.841473	−	0.540299i	$-0.181689\pi$
$854$		−	1540.64i		−	1.80403i
$855$		0			0
$856$	231.834			0.270834
$857$	−202.666			−0.236483			−0.118241	−	0.992985i	$-0.537726\pi$
$857$	−202.666			−0.236483			−0.118241	+	0.992985i	$0.537726\pi$
$858$		0			0
$859$	1327.28			1.54514			0.772572	−	0.634928i	$-0.218970\pi$
$859$	1327.28			1.54514			0.772572	+	0.634928i	$0.218970\pi$
$860$	−33.4796	−	11.9574i	−0.0389298	−	0.0139040i
$861$		0			0
$862$			127.342i			0.147729i
$863$	983.885			1.14008			0.570038	−	0.821619i	$-0.306929\pi$
$863$	983.885			1.14008			0.570038	+	0.821619i	$0.306929\pi$
$864$		0			0
$865$	−480.272	+	1344.71i	−0.555227	+	1.55458i
$866$			31.9883i			0.0369380i
$867$		0			0
$868$			1106.02i			1.27422i
$869$			42.9952i			0.0494767i
$870$		0			0
$871$	157.756			0.181121
$872$	−318.191			−0.364899
$873$		0			0
$874$	82.2040			0.0940549
$875$	606.992	+	1008.75i	0.693705	+	1.15285i
$876$		0			0
$877$			1306.91i			1.49021i	0.666950	+	0.745103i	$0.267600\pi$
$877$			1306.91i			1.49021i	−0.666950	+	0.745103i	$0.732400\pi$
$878$	−904.128			−1.02976
$879$		0			0
$880$	−44.3480	−	15.8392i	−0.0503955	−	0.0179990i
$881$			642.541i			0.729331i	0.931139	+	0.364666i	$0.118817\pi$
$881$			642.541i			0.729331i	−0.931139	+	0.364666i	$0.881183\pi$
$882$		0			0
$883$			955.440i			1.08204i	0.841010	+	0.541019i	$0.181961\pi$
$883$			955.440i			1.08204i	−0.841010	+	0.541019i	$0.818039\pi$
$884$		−	34.1117i		−	0.0385879i
$885$		0			0
$886$	622.373			0.702452
$887$	−865.363			−0.975606			−0.487803	−	0.872954i	$-0.662202\pi$
$887$	−865.363			−0.975606			−0.487803	+	0.872954i	$0.662202\pi$
$888$		0			0
$889$	832.196			0.936104
$890$	587.557	+	209.849i	0.660176	+	0.235786i
$891$		0			0
$892$			552.494i			0.619388i
$893$	494.828			0.554118
$894$		0			0
$895$	850.248	+	303.671i	0.949998	+	0.339297i
$896$		−	106.556i		−	0.118924i
$897$		0			0
$898$			398.704i			0.443991i
$899$			1053.93i			1.17234i
$900$		0			0
$901$	765.132			0.849203
$902$	162.857			0.180550
$903$		0			0
$904$	254.002			0.280976
$905$	147.525	−	413.054i	0.163011	−	0.456414i
$906$		0			0
$907$			771.360i			0.850452i	0.905087	+	0.425226i	$0.139805\pi$
$907$			771.360i			0.850452i	−0.905087	+	0.425226i	$0.860195\pi$
$908$	−5.41547			−0.00596417
$909$		0			0
$910$	34.4584	−	96.4801i	0.0378664	−	0.106022i
$911$			1306.04i			1.43363i	0.697261	+	0.716817i	$0.254402\pi$
$911$			1306.04i			1.43363i	−0.697261	+	0.716817i	$0.745598\pi$
$912$		0			0
$913$		−	381.303i		−	0.417638i
$914$		−	525.654i		−	0.575113i
$915$		0			0
$916$	−419.272			−0.457720
$917$	1636.32			1.78443
$918$		0			0
$919$	1455.53			1.58382			0.791909	−	0.610640i	$-0.209088\pi$
$919$	1455.53			1.58382			0.791909	+	0.610640i	$0.209088\pi$
$920$	38.9852	−	109.155i	0.0423752	−	0.118646i
$921$		0			0
$922$			313.824i			0.340373i
$923$	−157.935			−0.171111
$924$		0			0
$925$	−329.330	+	402.234i	−0.356033	+	0.434848i
$926$			506.213i			0.546666i
$927$		0			0
$928$		−	101.538i		−	0.109416i
$929$		−	209.956i		−	0.226002i	−0.993595	−	0.113001i	$-0.963954\pi$
$929$		−	209.956i		−	0.226002i	0.993595	−	0.113001i	$-0.0360463\pi$
$930$		0			0
$931$	−281.593			−0.302463
$932$	366.917			0.393688
$933$		0			0
$934$	−554.653			−0.593847
$935$	122.925	+	43.9032i	0.131470	+	0.0469553i
$936$		0			0
$937$		−	890.537i		−	0.950413i	−0.879874	−	0.475206i	$-0.842373\pi$
$937$		−	890.537i		−	0.950413i	0.879874	−	0.475206i	$-0.157627\pi$
$938$	−1365.92			−1.45620
$939$		0			0
$940$	234.672	−	657.057i	0.249651	−	0.698996i
$941$			1081.00i			1.14878i	0.818583	+	0.574389i	$0.194760\pi$
$941$			1081.00i			1.14878i	−0.818583	+	0.574389i	$0.805240\pi$
$942$		0			0
$943$			400.841i			0.425070i
$944$		−	157.377i		−	0.166713i
$945$		0			0
$946$	−11.8380			−0.0125138
$947$	−902.214			−0.952707			−0.476354	−	0.879254i	$-0.658042\pi$
$947$	−902.214			−0.952707			−0.476354	+	0.879254i	$0.658042\pi$
$948$		0			0
$949$	185.542			0.195513
$950$	−194.013	−	158.849i	−0.204225	−	0.167209i
$951$		0			0
$952$			295.354i			0.310245i
$953$	931.541			0.977483			0.488742	−	0.872429i	$-0.337456\pi$
$953$	931.541			0.977483			0.488742	+	0.872429i	$0.337456\pi$
$954$		0			0
$955$	−291.502	−	104.112i	−0.305238	−	0.109017i
$956$			115.972i			0.121309i
$957$		0			0
$958$		−	1138.42i		−	1.18833i
$959$		−	1026.20i		−	1.07007i
$960$		0			0
$961$	2486.63			2.58754
$962$	45.2385			0.0470255
$963$		0			0
$964$	314.080			0.325809
$965$	1058.41	+	378.019i	1.09680	+	0.391730i
$966$		0			0
$967$			1522.98i			1.57496i	0.616341	+	0.787479i	$0.288614\pi$
$967$			1522.98i			1.57496i	−0.616341	+	0.787479i	$0.711386\pi$
$968$	326.559			0.337354
$969$		0			0
$970$	−936.402	−	334.442i	−0.965363	−	0.344785i
$971$			11.0650i			0.0113955i	0.999984	+	0.00569775i	$0.00181366\pi$
$971$			11.0650i			0.0113955i	−0.999984	+	0.00569775i	$0.998186\pi$
$972$		0			0
$973$			1255.57i			1.29041i
$974$			675.183i			0.693207i
$975$		0			0
$976$	462.672			0.474050
$977$	547.166			0.560047			0.280024	−	0.959993i	$-0.409658\pi$
$977$	547.166			0.560047			0.280024	+	0.959993i	$0.409658\pi$
$978$		0			0
$979$	207.754			0.212210
$980$	−133.545	+	373.914i	−0.136271	+	0.381545i
$981$		0			0
$982$			508.244i			0.517560i
$983$	−1406.05			−1.43036			−0.715181	−	0.698939i	$-0.753656\pi$
$983$	−1406.05			−1.43036			−0.715181	+	0.698939i	$0.753656\pi$
$984$		0			0
$985$	437.724	−	1225.58i	0.444390	−	1.24425i
$986$			281.444i			0.285440i
$987$		0			0
$988$			21.8203i			0.0220853i
$989$		−	29.1372i		−	0.0294612i
$990$		0			0
$991$	−700.700			−0.707063			−0.353532	−	0.935423i	$-0.615019\pi$
$991$	−700.700			−0.707063			−0.353532	+	0.935423i	$0.615019\pi$
$992$	−332.151			−0.334829
$993$		0			0
$994$	1367.47			1.37573
$995$	−45.7604	+	128.124i	−0.0459903	+	0.128768i
$996$		0			0
$997$		−	544.164i		−	0.545801i	−0.962042	−	0.272901i	$-0.912017\pi$
$997$		−	544.164i		−	0.545801i	0.962042	−	0.272901i	$-0.0879830\pi$
$998$	548.864			0.549964
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.c.809.21	yes	24
3.2	odd	2	inner	810.3.b.c.809.4	yes	24
5.4	even	2	inner	810.3.b.c.809.3	✓	24
9.2	odd	6		810.3.j.g.539.10		24
9.4	even	3		810.3.j.h.269.5		24
9.5	odd	6		810.3.j.h.269.10		24
9.7	even	3		810.3.j.g.539.5		24
15.14	odd	2	inner	810.3.b.c.809.22	yes	24
45.4	even	6		810.3.j.g.269.10		24
45.14	odd	6		810.3.j.g.269.5		24
45.29	odd	6		810.3.j.h.539.5		24
45.34	even	6		810.3.j.h.539.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.3.b.c.809.3	✓	24	5.4	even	2	inner
810.3.b.c.809.4	yes	24	3.2	odd	2	inner
810.3.b.c.809.21	yes	24	1.1	even	1	trivial
810.3.b.c.809.22	yes	24	15.14	odd	2	inner
810.3.j.g.269.5		24	45.14	odd	6
810.3.j.g.269.10		24	45.4	even	6
810.3.j.g.539.5		24	9.7	even	3
810.3.j.g.539.10		24	9.2	odd	6
810.3.j.h.269.5		24	9.4	even	3
810.3.j.h.269.10		24	9.5	odd	6
810.3.j.h.539.5		24	45.29	odd	6
810.3.j.h.539.10		24	45.34	even	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(1.68174 - 4.70869i) q^{5} -9.41831i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(1.68174 - 4.70869i) q^{5} -9.41831i q^{7} +2.82843 q^{8} +(2.37833 - 6.65909i) q^{10} -2.35458i q^{11} +1.53833i q^{13} -13.3195i q^{14} +4.00000 q^{16} -11.0873 q^{17} +7.09220 q^{19} +(3.36347 - 9.41738i) q^{20} -3.32989i q^{22} +8.19590 q^{23} +(-19.3435 - 15.8376i) q^{25} +2.17553i q^{26} -18.8366i q^{28} -17.9495i q^{29} -58.7165 q^{31} +5.65685 q^{32} -15.6798 q^{34} +(-44.3479 - 15.8391i) q^{35} -20.7943i q^{37} +10.0299 q^{38} +(4.75667 - 13.3182i) q^{40} +48.9075i q^{41} -3.55509i q^{43} -4.70917i q^{44} +11.5908 q^{46} +69.7706 q^{47} -39.7046 q^{49} +(-27.3559 - 22.3977i) q^{50} +3.07666i q^{52} -69.0100 q^{53} +(-11.0870 - 3.95979i) q^{55} -26.6390i q^{56} -25.3844i q^{58} -39.3443i q^{59} +115.668 q^{61} -83.0377 q^{62} +8.00000 q^{64} +(7.24352 + 2.58707i) q^{65} -102.550i q^{67} -22.1745 q^{68} +(-62.7174 - 22.3999i) q^{70} +102.667i q^{71} -120.613i q^{73} -29.4075i q^{74} +14.1844 q^{76} -22.1762 q^{77} -18.2602 q^{79} +(6.72695 - 18.8348i) q^{80} +69.1657i q^{82} +161.941 q^{83} +(-18.6459 + 52.2065i) q^{85} -5.02765i q^{86} -6.65977i q^{88} +88.2337i q^{89} +14.4885 q^{91} +16.3918 q^{92} +98.6706 q^{94} +(11.9272 - 33.3950i) q^{95} -140.620i q^{97} -56.1508 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4}+O(q^{10}) \) 24 * q + 48 * q^4	\( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

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