LMFDB - Embedded newform 531.3.c.c.235.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,3,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	531.c (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:	$14.4687020375$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + \cdots + 570861 $ x^20 + 60x^18 + 1522x^16 + 21286x^14 + 179593x^12 + 941588x^10 + 3057025x^8 + 5962230x^6 + 6517782x^4 + 3428244*x^2 + 570861
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3 $
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.11
Root			$0.537675i$ of defining polynomial
Character	$\chi$	$=$	531.235
Dual form			531.3.c.c.235.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.537675i q^{2} +3.71091 q^{4} +0.803210 q^{5} +5.11098 q^{7} +4.14596i q^{8} +O(q^{10})$	\(q+0.537675i q^{2} +3.71091 q^{4} +0.803210 q^{5} +5.11098 q^{7} +4.14596i q^{8} +0.431866i q^{10} -17.7567i q^{11} -24.6407i q^{13} +2.74805i q^{14} +12.6144 q^{16} -18.2784 q^{17} +28.5669 q^{19} +2.98064 q^{20} +9.54732 q^{22} +11.8668i q^{23} -24.3549 q^{25} +13.2487 q^{26} +18.9664 q^{28} -9.01975 q^{29} -4.94288i q^{31} +23.3663i q^{32} -9.82784i q^{34} +4.10519 q^{35} +39.7950i q^{37} +15.3597i q^{38} +3.33008i q^{40} +38.0744 q^{41} -19.2551i q^{43} -65.8933i q^{44} -6.38050 q^{46} -65.6973i q^{47} -22.8779 q^{49} -13.0950i q^{50} -91.4394i q^{52} +40.1506 q^{53} -14.2623i q^{55} +21.1899i q^{56} -4.84970i q^{58} +(53.3901 + 25.1097i) q^{59} +110.438i q^{61} +2.65767 q^{62} +37.8943 q^{64} -19.7917i q^{65} -30.7950i q^{67} -67.8294 q^{68} +2.20726i q^{70} +95.0078 q^{71} +71.2671i q^{73} -21.3968 q^{74} +106.009 q^{76} -90.7539i q^{77} -13.3580 q^{79} +10.1320 q^{80} +20.4717i q^{82} +142.514i q^{83} -14.6814 q^{85} +10.3530 q^{86} +73.6185 q^{88} +128.582i q^{89} -125.938i q^{91} +44.0366i q^{92} +35.3238 q^{94} +22.9452 q^{95} -97.1513i q^{97} -12.3009i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) $ 20 * q - 40 * q^4 - 8 * q^7	$ 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$			0.537675i			0.268838i	0.990925	+	0.134419i	$0.0429168\pi$
$2$			0.537675i			0.268838i	−0.990925	+	0.134419i	$0.957083\pi$
$3$		0			0
$3$		0			0
$4$	3.71091			0.927726
$5$	0.803210			0.160642			0.0803210	−	0.996769i	$-0.474405\pi$
$5$	0.803210			0.160642			0.0803210	+	0.996769i	$0.474405\pi$
$6$		0			0
$7$	5.11098			0.730140			0.365070	−	0.930980i	$-0.381045\pi$
$7$	5.11098			0.730140			0.365070	+	0.930980i	$0.381045\pi$
$8$			4.14596i			0.518245i
$9$		0			0
$10$			0.431866i			0.0431866i
$11$		−	17.7567i		−	1.61424i	−0.590386	−	0.807121i	$-0.701024\pi$
$11$		−	17.7567i		−	1.61424i	0.590386	−	0.807121i	$-0.298976\pi$
$12$		0			0
$13$		−	24.6407i		−	1.89544i	−0.319101	−	0.947721i	$-0.603381\pi$
$13$		−	24.6407i		−	1.89544i	0.319101	−	0.947721i	$-0.396619\pi$
$14$			2.74805i			0.196289i
$15$		0			0
$16$	12.6144			0.788402
$17$	−18.2784			−1.07520			−0.537600	−	0.843200i	$-0.680669\pi$
$17$	−18.2784			−1.07520			−0.537600	+	0.843200i	$0.680669\pi$
$18$		0			0
$19$	28.5669			1.50352			0.751760	−	0.659437i	$-0.229205\pi$
$19$	28.5669			1.50352			0.751760	+	0.659437i	$0.229205\pi$
$20$	2.98064			0.149032
$21$		0			0
$22$	9.54732			0.433969
$23$			11.8668i			0.515949i	0.966152	+	0.257974i	$0.0830550\pi$
$23$			11.8668i			0.515949i	−0.966152	+	0.257974i	$0.916945\pi$
$24$		0			0
$25$	−24.3549			−0.974194
$26$	13.2487			0.509566
$27$		0			0
$28$	18.9664			0.677370
$29$	−9.01975			−0.311026			−0.155513	−	0.987834i	$-0.549703\pi$
$29$	−9.01975			−0.311026			−0.155513	+	0.987834i	$0.549703\pi$
$30$		0			0
$31$		−	4.94288i		−	0.159448i	−0.996817	−	0.0797239i	$-0.974596\pi$
$31$		−	4.94288i		−	0.159448i	0.996817	−	0.0797239i	$-0.0254039\pi$
$32$			23.3663i			0.730198i
$33$		0			0
$34$		−	9.82784i		−	0.289054i
$35$	4.10519			0.117291
$36$		0			0
$37$			39.7950i			1.07554i	0.843091	+	0.537770i	$0.180733\pi$
$37$			39.7950i			1.07554i	−0.843091	+	0.537770i	$0.819267\pi$
$38$			15.3597i			0.404203i
$39$		0			0
$40$			3.33008i			0.0832520i
$41$	38.0744			0.928644			0.464322	−	0.885667i	$-0.346298\pi$
$41$	38.0744			0.928644			0.464322	+	0.885667i	$0.346298\pi$
$42$		0			0
$43$		−	19.2551i		−	0.447793i	−0.974613	−	0.223896i	$-0.928122\pi$
$43$		−	19.2551i		−	0.447793i	0.974613	−	0.223896i	$-0.0718778\pi$
$44$		−	65.8933i		−	1.49758i
$45$		0			0
$46$	−6.38050			−0.138706
$47$		−	65.6973i		−	1.39782i	−0.715212	−	0.698908i	$-0.753670\pi$
$47$		−	65.6973i		−	1.39782i	0.715212	−	0.698908i	$-0.246330\pi$
$48$		0			0
$49$	−22.8779			−0.466896
$50$		−	13.0950i		−	0.261900i
$51$		0			0
$52$		−	91.4394i		−	1.75845i
$53$	40.1506			0.757559			0.378779	−	0.925487i	$-0.376344\pi$
$53$	40.1506			0.757559			0.378779	+	0.925487i	$0.376344\pi$
$54$		0			0
$55$		−	14.2623i		−	0.259315i
$56$			21.1899i			0.378392i
$57$		0			0
$58$		−	4.84970i		−	0.0836155i
$59$	53.3901	+	25.1097i	0.904917	+	0.425589i
$59$	53.3901	+	25.1097i	0.904917	+	0.425589i
$60$		0			0
$61$			110.438i			1.81047i	0.424916	+	0.905233i	$0.360303\pi$
$61$			110.438i			1.81047i	−0.424916	+	0.905233i	$0.639697\pi$
$62$	2.65767			0.0428656
$63$		0			0
$64$	37.8943			0.592098
$65$		−	19.7917i		−	0.304487i
$66$		0			0
$67$		−	30.7950i		−	0.459627i	−0.973235	−	0.229814i	$-0.926188\pi$
$67$		−	30.7950i		−	0.459627i	0.973235	−	0.229814i	$-0.0738117\pi$
$68$	−67.8294			−0.997491
$69$		0			0
$70$			2.20726i			0.0315323i
$71$	95.0078			1.33814			0.669069	−	0.743200i	$-0.266693\pi$
$71$	95.0078			1.33814			0.669069	+	0.743200i	$0.266693\pi$
$72$		0			0
$73$			71.2671i			0.976262i	0.872770	+	0.488131i	$0.162321\pi$
$73$			71.2671i			0.976262i	−0.872770	+	0.488131i	$0.837679\pi$
$74$	−21.3968			−0.289146
$75$		0			0
$76$	106.009			1.39486
$77$		−	90.7539i		−	1.17862i
$78$		0			0
$79$	−13.3580			−0.169089			−0.0845446	−	0.996420i	$-0.526944\pi$
$79$	−13.3580			−0.169089			−0.0845446	+	0.996420i	$0.526944\pi$
$80$	10.1320			0.126651
$81$		0			0
$82$			20.4717i			0.249654i
$83$			142.514i			1.71704i	0.512783	+	0.858518i	$0.328614\pi$
$83$			142.514i			1.71704i	−0.512783	+	0.858518i	$0.671386\pi$
$84$		0			0
$85$	−14.6814			−0.172722
$86$	10.3530			0.120384
$87$		0			0
$88$	73.6185			0.836574
$89$			128.582i			1.44474i	0.691506	+	0.722371i	$0.256948\pi$
$89$			128.582i			1.44474i	−0.691506	+	0.722371i	$0.743052\pi$
$90$		0			0
$91$		−	125.938i		−	1.38394i
$92$			44.0366i			0.478659i
$93$		0			0
$94$	35.3238			0.375786
$95$	22.9452			0.241528
$96$		0			0
$97$		−	97.1513i		−	1.00156i	−0.865575	−	0.500780i	$-0.833047\pi$
$97$		−	97.1513i		−	1.00156i	0.865575	−	0.500780i	$-0.166953\pi$
$98$		−	12.3009i		−	0.125519i
$99$		0			0
$100$	−90.3786			−0.903786
$101$		−	100.698i		−	0.997012i	−0.866886	−	0.498506i	$-0.833882\pi$
$101$		−	100.698i		−	0.997012i	0.866886	−	0.498506i	$-0.166118\pi$
$102$		0			0
$103$			118.731i			1.15273i	0.817194	+	0.576363i	$0.195528\pi$
$103$			118.731i			1.15273i	−0.817194	+	0.576363i	$0.804472\pi$
$104$	102.160			0.982304
$105$		0			0
$106$			21.5880i			0.203660i
$107$	−158.040			−1.47701			−0.738503	−	0.674250i	$-0.764467\pi$
$107$	−158.040			−1.47701			−0.738503	+	0.674250i	$0.764467\pi$
$108$		0			0
$109$		−	88.9732i		−	0.816268i	−0.912922	−	0.408134i	$-0.866180\pi$
$109$		−	88.9732i		−	0.816268i	0.912922	−	0.408134i	$-0.133820\pi$
$110$	7.66850			0.0697137
$111$		0			0
$112$	64.4721			0.575644
$113$		−	158.257i		−	1.40050i	−0.713895	−	0.700252i	$-0.753071\pi$
$113$		−	158.257i		−	1.40050i	0.713895	−	0.700252i	$-0.246929\pi$
$114$		0			0
$115$			9.53155i			0.0828830i
$116$	−33.4714			−0.288547
$117$		0			0
$118$	−13.5009	+	28.7065i	−0.114414	+	0.243276i
$119$	−93.4205			−0.785046
$120$		0			0
$121$	−194.299			−1.60578
$122$	−59.3800			−0.486721
$123$		0			0
$124$		−	18.3426i		−	0.147924i
$125$	−39.6423			−0.317138
$126$		0			0
$127$	−101.633			−0.800259			−0.400129	−	0.916459i	$-0.631035\pi$
$127$	−101.633			−0.800259			−0.400129	+	0.916459i	$0.631035\pi$
$128$			113.840i			0.889376i
$129$		0			0
$130$	10.6415			0.0818577
$131$			86.1937i			0.657967i	0.944336	+	0.328984i	$0.106706\pi$
$131$			86.1937i			0.657967i	−0.944336	+	0.328984i	$0.893294\pi$
$132$		0			0
$133$	146.005			1.09778
$134$	16.5577			0.123565
$135$		0			0
$136$		−	75.7816i		−	0.557217i
$137$	174.098			1.27079			0.635395	−	0.772188i	$-0.280837\pi$
$137$	174.098			1.27079			0.635395	+	0.772188i	$0.280837\pi$
$138$		0			0
$139$	166.414			1.19722			0.598612	−	0.801039i	$-0.295719\pi$
$139$	166.414			1.19722			0.598612	+	0.801039i	$0.295719\pi$
$140$	15.2340			0.108814
$141$		0			0
$142$			51.0834i			0.359742i
$143$	−437.537			−3.05970
$144$		0			0
$145$	−7.24475			−0.0499638
$146$	−38.3186			−0.262456
$147$		0			0
$148$			147.676i			0.997808i
$149$		−	27.1832i		−	0.182438i	−0.995831	−	0.0912188i	$-0.970924\pi$
$149$		−	27.1832i		−	0.182438i	0.995831	−	0.0912188i	$-0.0290763\pi$
$150$		0			0
$151$			94.4601i			0.625563i	0.949825	+	0.312782i	$0.101261\pi$
$151$			94.4601i			0.625563i	−0.949825	+	0.312782i	$0.898739\pi$
$152$			118.437i			0.779192i
$153$		0			0
$154$	48.7961			0.316858
$155$		−	3.97017i		−	0.0256140i
$156$		0			0
$157$			13.0757i			0.0832847i	0.999133	+	0.0416424i	$0.0132590\pi$
$157$			13.0757i			0.0832847i	−0.999133	+	0.0416424i	$0.986741\pi$
$158$		−	7.18229i		−	0.0454575i
$159$		0			0
$160$			18.7681i			0.117300i
$161$			60.6511i			0.376715i
$162$		0			0
$163$	−31.6216			−0.193998			−0.0969989	−	0.995284i	$-0.530924\pi$
$163$	−31.6216			−0.193998			−0.0969989	+	0.995284i	$0.530924\pi$
$164$	141.290			0.861527
$165$		0			0
$166$	−76.6263			−0.461604
$167$	−142.242			−0.851751			−0.425875	−	0.904782i	$-0.640034\pi$
$167$	−142.242			−0.851751			−0.425875	+	0.904782i	$0.640034\pi$
$168$		0			0
$169$	−438.166			−2.59270
$170$		−	7.89382i		−	0.0464342i
$171$		0			0
$172$		−	71.4538i		−	0.415429i
$173$			202.500i			1.17052i	0.810845	+	0.585261i	$0.199008\pi$
$173$			202.500i			1.17052i	−0.810845	+	0.585261i	$0.800992\pi$
$174$		0			0
$175$	−124.477			−0.711298
$176$		−	223.990i		−	1.27267i
$177$		0			0
$178$	−69.1354			−0.388401
$179$			190.907i			1.06652i	0.845952	+	0.533259i	$0.179033\pi$
$179$			190.907i			1.06652i	−0.845952	+	0.533259i	$0.820967\pi$
$180$		0			0
$181$	−123.971			−0.684922			−0.342461	−	0.939532i	$-0.611261\pi$
$181$	−123.971			−0.684922			−0.342461	+	0.939532i	$0.611261\pi$
$182$	67.7139			0.372054
$183$		0			0
$184$	−49.1994			−0.267388
$185$			31.9637i			0.172777i
$186$		0			0
$187$			324.563i			1.73563i
$188$		−	243.797i		−	1.29679i
$189$		0			0
$190$			12.3371i			0.0649319i
$191$		−	125.336i		−	0.656211i	−0.944641	−	0.328105i	$-0.893590\pi$
$191$		−	125.336i		−	0.656211i	0.944641	−	0.328105i	$-0.106410\pi$
$192$		0			0
$193$	−108.008			−0.559626			−0.279813	−	0.960055i	$-0.590272\pi$
$193$	−108.008			−0.559626			−0.279813	+	0.960055i	$0.590272\pi$
$194$	52.2358			0.269257
$195$		0			0
$196$	−84.8978			−0.433152
$197$	132.587			0.673031			0.336516	−	0.941678i	$-0.390752\pi$
$197$	132.587			0.673031			0.336516	+	0.941678i	$0.390752\pi$
$198$		0			0
$199$	−229.429			−1.15291			−0.576455	−	0.817129i	$-0.695564\pi$
$199$	−229.429			−1.15291			−0.576455	+	0.817129i	$0.695564\pi$
$200$		−	100.974i		−	0.504872i
$201$		0			0
$202$	54.1429			0.268034
$203$	−46.0997			−0.227092
$204$		0			0
$205$	30.5817			0.149179
$206$	−63.8386			−0.309896
$207$		0			0
$208$		−	310.829i		−	1.49437i
$209$		−	507.253i		−	2.42705i
$210$		0			0
$211$			270.274i			1.28092i	0.767991	+	0.640461i	$0.221257\pi$
$211$			270.274i			1.28092i	−0.767991	+	0.640461i	$0.778743\pi$
$212$	148.995			0.702807
$213$		0			0
$214$		−	84.9741i		−	0.397075i
$215$		−	15.4659i		−	0.0719343i
$216$		0			0
$217$		−	25.2630i		−	0.116419i
$218$	47.8387			0.219444
$219$		0			0
$220$		−	52.9261i		−	0.240573i
$221$			450.393i			2.03798i
$222$		0			0
$223$	37.2371			0.166982			0.0834912	−	0.996509i	$-0.473393\pi$
$223$	37.2371			0.166982			0.0834912	+	0.996509i	$0.473393\pi$
$224$			119.425i			0.533146i
$225$		0			0
$226$	85.0909			0.376508
$227$		−	157.805i		−	0.695175i	−0.937648	−	0.347587i	$-0.887001\pi$
$227$		−	157.805i		−	0.695175i	0.937648	−	0.347587i	$-0.112999\pi$
$228$		0			0
$229$			304.839i			1.33118i	0.746319	+	0.665588i	$0.231819\pi$
$229$			304.839i			1.33118i	−0.746319	+	0.665588i	$0.768181\pi$
$230$	−5.12488			−0.0222821
$231$		0			0
$232$		−	37.3956i		−	0.161188i
$233$			163.385i			0.701221i	0.936521	+	0.350611i	$0.114026\pi$
$233$			163.385i			0.701221i	−0.936521	+	0.350611i	$0.885974\pi$
$234$		0			0
$235$		−	52.7687i		−	0.224548i
$236$	198.126	+	93.1798i	0.839515	+	0.394830i
$237$		0			0
$238$		−	50.2299i		−	0.211050i
$239$	235.307			0.984549			0.492275	−	0.870440i	$-0.336166\pi$
$239$	235.307			0.984549			0.492275	+	0.870440i	$0.336166\pi$
$240$		0			0
$241$	71.2458			0.295626			0.147813	−	0.989015i	$-0.452777\pi$
$241$	71.2458			0.295626			0.147813	+	0.989015i	$0.452777\pi$
$242$		−	104.470i		−	0.431694i
$243$		0			0
$244$			409.826i			1.67962i
$245$	−18.3758			−0.0750031
$246$		0			0
$247$		−	703.909i		−	2.84983i
$248$	20.4930			0.0826331
$249$		0			0
$250$		−	21.3147i		−	0.0852587i
$251$	−415.223			−1.65427			−0.827137	−	0.562001i	$-0.810032\pi$
$251$	−415.223			−1.65427			−0.827137	+	0.562001i	$0.810032\pi$
$252$		0			0
$253$	210.715			0.832866
$254$		−	54.6455i		−	0.215140i
$255$		0			0
$256$	90.3680			0.353000
$257$	−383.556			−1.49244			−0.746218	−	0.665701i	$-0.768132\pi$
$257$	−383.556			−1.49244			−0.746218	+	0.665701i	$0.768132\pi$
$258$		0			0
$259$			203.391i			0.785295i
$260$		−	73.4450i		−	0.282481i
$261$		0			0
$262$	−46.3442			−0.176886
$263$	176.978			0.672921			0.336460	−	0.941698i	$-0.390770\pi$
$263$	176.978			0.672921			0.336460	+	0.941698i	$0.390770\pi$
$264$		0			0
$265$	32.2494			0.121696
$266$			78.5031i			0.295124i
$267$		0			0
$268$		−	114.277i		−	0.426408i
$269$		−	151.733i		−	0.564063i	−0.959405	−	0.282031i	$-0.908992\pi$
$269$		−	151.733i		−	0.564063i	0.959405	−	0.282031i	$-0.0910082\pi$
$270$		0			0
$271$	316.355			1.16736			0.583680	−	0.811984i	$-0.301612\pi$
$271$	316.355			1.16736			0.583680	+	0.811984i	$0.301612\pi$
$272$	−230.572			−0.847690
$273$		0			0
$274$			93.6083i			0.341636i
$275$			432.461i			1.57259i
$276$		0			0
$277$	122.572			0.442499			0.221250	−	0.975217i	$-0.428986\pi$
$277$	122.572			0.442499			0.221250	+	0.975217i	$0.428986\pi$
$278$			89.4768i			0.321859i
$279$		0			0
$280$			17.0200i			0.0607856i
$281$	188.817			0.671945			0.335973	−	0.941872i	$-0.390935\pi$
$281$	188.817			0.671945			0.335973	+	0.941872i	$0.390935\pi$
$282$		0			0
$283$		−	326.692i		−	1.15439i	−0.816606	−	0.577195i	$-0.804147\pi$
$283$		−	326.692i		−	1.15439i	0.816606	−	0.577195i	$-0.195853\pi$
$284$	352.565			1.24143
$285$		0			0
$286$		−	235.253i		−	0.822563i
$287$	194.597			0.678039
$288$		0			0
$289$	45.0999			0.156055
$290$		−	3.89532i		−	0.0134322i
$291$		0			0
$292$			264.466i			0.905704i
$293$	−61.3922			−0.209530			−0.104765	−	0.994497i	$-0.533409\pi$
$293$	−61.3922			−0.209530			−0.104765	+	0.994497i	$0.533409\pi$
$294$		0			0
$295$	42.8834	+	20.1684i	0.145368	+	0.0683674i
$296$	−164.989			−0.557394
$297$		0			0
$298$	14.6157			0.0490461
$299$	292.407			0.977950
$300$		0			0
$301$		−	98.4124i		−	0.326951i
$302$	−50.7888			−0.168175
$303$		0			0
$304$	360.355			1.18538
$305$			88.7052i			0.290837i
$306$		0			0
$307$	−274.766			−0.895003			−0.447501	−	0.894283i	$-0.647686\pi$
$307$	−274.766			−0.895003			−0.447501	+	0.894283i	$0.647686\pi$
$308$		−	336.779i		−	1.09344i
$309$		0			0
$310$	2.13466			0.00688601
$311$	−488.255			−1.56995			−0.784976	−	0.619526i	$-0.787325\pi$
$311$	−488.255			−1.56995			−0.784976	+	0.619526i	$0.787325\pi$
$312$		0			0
$313$		−	244.585i		−	0.781421i	−0.920514	−	0.390711i	$-0.872229\pi$
$313$		−	244.585i		−	0.781421i	0.920514	−	0.390711i	$-0.127771\pi$
$314$	−7.03048			−0.0223901
$315$		0			0
$316$	−49.5704			−0.156868
$317$	329.194			1.03847			0.519234	−	0.854632i	$-0.326217\pi$
$317$	329.194			1.03847			0.519234	+	0.854632i	$0.326217\pi$
$318$		0			0
$319$			160.161i			0.502071i
$320$	30.4370			0.0951157
$321$		0			0
$322$	−32.6106			−0.101275
$323$	−522.157			−1.61658
$324$		0			0
$325$			600.121i			1.84653i
$326$		−	17.0022i		−	0.0521539i
$327$		0			0
$328$			157.855i			0.481265i
$329$		−	335.778i		−	1.02060i
$330$		0			0
$331$	397.945			1.20225			0.601126	−	0.799154i	$-0.294719\pi$
$331$	397.945			1.20225			0.601126	+	0.799154i	$0.294719\pi$
$332$			528.856i			1.59294i
$333$		0			0
$334$		−	76.4802i		−	0.228983i
$335$		−	24.7349i		−	0.0738354i
$336$		0			0
$337$		−	120.588i		−	0.357829i	−0.983865	−	0.178915i	$-0.942741\pi$
$337$		−	120.588i		−	0.357829i	0.983865	−	0.178915i	$-0.0572586\pi$
$338$		−	235.591i		−	0.697015i
$339$		0			0
$340$	−54.4812			−0.160239
$341$	−87.7692			−0.257388
$342$		0			0
$343$	−367.366			−1.07104
$344$	79.8309			0.232067
$345$		0			0
$346$	−108.879			−0.314680
$347$		−	170.990i		−	0.492767i	−0.969172	−	0.246384i	$-0.920758\pi$
$347$		−	170.990i		−	0.492767i	0.969172	−	0.246384i	$-0.0792423\pi$
$348$		0			0
$349$		−	294.342i		−	0.843388i	−0.906738	−	0.421694i	$-0.861436\pi$
$349$		−	294.342i		−	0.843388i	0.906738	−	0.421694i	$-0.138564\pi$
$350$		−	66.9283i		−	0.191224i
$351$		0			0
$352$	414.908			1.17872
$353$			165.480i			0.468783i	0.972142	+	0.234391i	$0.0753098\pi$
$353$			165.480i			0.468783i	−0.972142	+	0.234391i	$0.924690\pi$
$354$		0			0
$355$	76.3112			0.214961
$356$			477.156i			1.34032i
$357$		0			0
$358$	−102.646			−0.286720
$359$	−231.500			−0.644845			−0.322423	−	0.946596i	$-0.604497\pi$
$359$	−231.500			−0.644845			−0.322423	+	0.946596i	$0.604497\pi$
$360$		0			0
$361$	455.067			1.26057
$362$		−	66.6561i		−	0.184133i
$363$		0			0
$364$		−	467.345i		−	1.28391i
$365$			57.2425i			0.156829i
$366$		0			0
$367$			23.9385i			0.0652276i	0.999468	+	0.0326138i	$0.0103831\pi$
$367$			23.9385i			0.0652276i	−0.999468	+	0.0326138i	$0.989617\pi$
$368$			149.693i			0.406775i
$369$		0			0
$370$	−17.1861			−0.0464490
$371$	205.209			0.553124
$372$		0			0
$373$	279.182			0.748476			0.374238	−	0.927333i	$-0.377904\pi$
$373$	279.182			0.748476			0.374238	+	0.927333i	$0.377904\pi$
$374$	−174.510			−0.466604
$375$		0			0
$376$	272.379			0.724412
$377$			222.253i			0.589531i
$378$		0			0
$379$	89.9356			0.237297			0.118649	−	0.992936i	$-0.462144\pi$
$379$	89.9356			0.237297			0.118649	+	0.992936i	$0.462144\pi$
$380$	85.1475			0.224072
$381$		0			0
$382$	67.3902			0.176414
$383$	143.916			0.375761			0.187880	−	0.982192i	$-0.439838\pi$
$383$	143.916			0.375761			0.187880	+	0.982192i	$0.439838\pi$
$384$		0			0
$385$		−	72.8944i		−	0.189336i
$386$		−	58.0731i		−	0.150448i
$387$		0			0
$388$		−	360.519i		−	0.929173i
$389$	182.927			0.470250			0.235125	−	0.971965i	$-0.424450\pi$
$389$	182.927			0.470250			0.235125	+	0.971965i	$0.424450\pi$
$390$		0			0
$391$		−	216.907i		−	0.554748i
$392$		−	94.8510i		−	0.241967i
$393$		0			0
$394$			71.2888i			0.180936i
$395$	−10.7293			−0.0271628
$396$		0			0
$397$		−	170.243i		−	0.428823i	−0.976743	−	0.214412i	$-0.931217\pi$
$397$		−	170.243i		−	0.428823i	0.976743	−	0.214412i	$-0.0687834\pi$
$398$		−	123.358i		−	0.309945i
$399$		0			0
$400$	−307.223			−0.768057
$401$		−	512.169i		−	1.27723i	−0.769527	−	0.638614i	$-0.779508\pi$
$401$		−	512.169i		−	1.27723i	0.769527	−	0.638614i	$-0.220492\pi$
$402$		0			0
$403$	−121.796			−0.302224
$404$		−	373.682i		−	0.924954i
$405$		0			0
$406$		−	24.7867i		−	0.0610510i
$407$	706.627			1.73618
$408$		0			0
$409$			91.1229i			0.222794i	0.993776	+	0.111397i	$0.0355326\pi$
$409$			91.1229i			0.222794i	−0.993776	+	0.111397i	$0.964467\pi$
$410$			16.4430i			0.0401050i
$411$		0			0
$412$			440.598i			1.06941i
$413$	272.875	+	128.335i	0.660715	+	0.310739i
$414$		0			0
$415$			114.469i			0.275828i
$416$	575.763			1.38405
$417$		0			0
$418$	272.737			0.652481
$419$			90.7750i			0.216647i	0.994116	+	0.108323i	$0.0345482\pi$
$419$			90.7750i			0.216647i	−0.994116	+	0.108323i	$0.965452\pi$
$420$		0			0
$421$		−	504.157i		−	1.19752i	−0.800927	−	0.598762i	$-0.795660\pi$
$421$		−	504.157i		−	1.19752i	0.800927	−	0.598762i	$-0.204340\pi$
$422$	−145.320			−0.344360
$423$		0			0
$424$			166.463i			0.392601i
$425$	445.168			1.04745
$426$		0			0
$427$			564.448i			1.32189i
$428$	−586.470			−1.37026
$429$		0			0
$430$	8.31562			0.0193387
$431$		−	335.908i		−	0.779369i	−0.920948	−	0.389685i	$-0.872584\pi$
$431$		−	335.908i		−	0.779369i	0.920948	−	0.389685i	$-0.127416\pi$
$432$		0			0
$433$	165.667			0.382603			0.191302	−	0.981531i	$-0.438729\pi$
$433$	165.667			0.382603			0.191302	+	0.981531i	$0.438729\pi$
$434$	13.5833			0.0312979
$435$		0			0
$436$		−	330.171i		−	0.757273i
$437$			338.998i			0.775739i
$438$		0			0
$439$	−159.958			−0.364369			−0.182185	−	0.983264i	$-0.558317\pi$
$439$	−159.958			−0.364369			−0.182185	+	0.983264i	$0.558317\pi$
$440$	59.1311			0.134389
$441$		0			0
$442$	−242.165			−0.547885
$443$		−	65.1418i		−	0.147047i	−0.997293	−	0.0735235i	$-0.976576\pi$
$443$		−	65.1418i		−	0.147047i	0.997293	−	0.0735235i	$-0.0234244\pi$
$444$		0			0
$445$			103.278i			0.232086i
$446$			20.0215i			0.0448912i
$447$		0			0
$448$	193.677			0.432314
$449$	371.170			0.826658			0.413329	−	0.910582i	$-0.364366\pi$
$449$	371.170			0.826658			0.413329	+	0.910582i	$0.364366\pi$
$450$		0			0
$451$		−	676.074i		−	1.49906i
$452$		−	587.277i		−	1.29929i
$453$		0			0
$454$	84.8477			0.186889
$455$		−	101.155i		−	0.222318i
$456$		0			0
$457$			169.801i			0.371555i	0.982592	+	0.185777i	$0.0594803\pi$
$457$			169.801i			0.371555i	−0.982592	+	0.185777i	$0.940520\pi$
$458$	−163.905			−0.357870
$459$		0			0
$460$			35.3707i			0.0768928i
$461$	657.621			1.42651			0.713255	−	0.700905i	$-0.247220\pi$
$461$	657.621			1.42651			0.713255	+	0.700905i	$0.247220\pi$
$462$		0			0
$463$			729.046i			1.57461i	0.616562	+	0.787306i	$0.288525\pi$
$463$			729.046i			1.57461i	−0.616562	+	0.787306i	$0.711475\pi$
$464$	−113.779			−0.245214
$465$		0			0
$466$	−87.8478			−0.188515
$467$			144.936i			0.310355i	0.987887	+	0.155178i	$0.0495950\pi$
$467$			144.936i			0.310355i	−0.987887	+	0.155178i	$0.950405\pi$
$468$		0			0
$469$		−	157.393i		−	0.335592i
$470$	28.3725			0.0603669
$471$		0			0
$472$	−104.104	+	221.353i	−0.220559	+	0.468969i
$473$	−341.906			−0.722846
$474$		0			0
$475$	−695.742			−1.46472
$476$	−346.675			−0.728308
$477$		0			0
$478$			126.519i			0.264684i
$479$	73.1977			0.152814			0.0764068	−	0.997077i	$-0.475655\pi$
$479$	73.1977			0.152814			0.0764068	+	0.997077i	$0.475655\pi$
$480$		0			0
$481$	980.578			2.03862
$482$			38.3071i			0.0794753i
$483$		0			0
$484$	−721.026			−1.48972
$485$		−	78.0328i		−	0.160892i
$486$		0			0
$487$	−54.7261			−0.112374			−0.0561869	−	0.998420i	$-0.517894\pi$
$487$	−54.7261			−0.112374			−0.0561869	+	0.998420i	$0.517894\pi$
$488$	−457.873			−0.938265
$489$		0			0
$490$		−	9.88020i		−	0.0201637i
$491$	−588.790			−1.19916			−0.599582	−	0.800313i	$-0.704667\pi$
$491$	−588.790			−1.19916			−0.599582	+	0.800313i	$0.704667\pi$
$492$		0			0
$493$	164.867			0.334415
$494$	378.474			0.766143
$495$		0			0
$496$		−	62.3517i		−	0.125709i
$497$	485.583			0.977028
$498$		0			0
$499$	368.842			0.739162			0.369581	−	0.929199i	$-0.379501\pi$
$499$	368.842			0.739162			0.369581	+	0.929199i	$0.379501\pi$
$500$	−147.109			−0.294218
$501$		0			0
$502$		−	223.255i		−	0.444731i
$503$		−	74.8276i		−	0.148763i	−0.997230	−	0.0743813i	$-0.976302\pi$
$503$		−	74.8276i		−	0.148763i	0.997230	−	0.0743813i	$-0.0236982\pi$
$504$		0			0
$505$		−	80.8818i		−	0.160162i
$506$			113.296i			0.223906i
$507$		0			0
$508$	−377.150			−0.742421
$509$		−	853.153i		−	1.67614i	−0.545566	−	0.838068i	$-0.683685\pi$
$509$		−	853.153i		−	1.67614i	0.545566	−	0.838068i	$-0.316315\pi$
$510$		0			0
$511$			364.245i			0.712808i
$512$			503.949i			0.984276i
$513$		0			0
$514$		−	206.229i		−	0.401223i
$515$			95.3657i			0.185176i
$516$		0			0
$517$	−1166.57			−2.25641
$518$	−109.359			−0.211117
$519$		0			0
$520$	82.0556			0.157799
$521$	−168.377			−0.323180			−0.161590	−	0.986858i	$-0.551662\pi$
$521$	−168.377			−0.323180			−0.161590	+	0.986858i	$0.551662\pi$
$522$		0			0
$523$	−213.899			−0.408986			−0.204493	−	0.978868i	$-0.565554\pi$
$523$	−213.899			−0.408986			−0.204493	+	0.978868i	$0.565554\pi$
$524$			319.857i			0.610413i
$525$		0			0
$526$			95.1568i			0.180906i
$527$			90.3480i			0.171438i
$528$		0			0
$529$	388.179			0.733797
$530$			17.3397i			0.0327164i
$531$		0			0
$532$	541.810			1.01844
$533$		−	938.181i		−	1.76019i
$534$		0			0
$535$	−126.939			−0.237269
$536$	127.675			0.238200
$537$		0			0
$538$	81.5830			0.151641
$539$			406.235i			0.753684i
$540$		0			0
$541$			236.317i			0.436816i	0.975858	+	0.218408i	$0.0700863\pi$
$541$			236.317i			0.436816i	−0.975858	+	0.218408i	$0.929914\pi$
$542$			170.096i			0.313831i
$543$		0			0
$544$		−	427.099i		−	0.785109i
$545$		−	71.4641i		−	0.131127i
$546$		0			0
$547$	−466.310			−0.852487			−0.426243	−	0.904609i	$-0.640163\pi$
$547$	−466.310			−0.852487			−0.426243	+	0.904609i	$0.640163\pi$
$548$	646.062			1.17894
$549$		0			0
$550$	−232.524			−0.422770
$551$	−257.666			−0.467634
$552$		0			0
$553$	−68.2726			−0.123459
$554$			65.9041i			0.118960i
$555$		0			0
$556$	617.547			1.11070
$557$	−1045.31			−1.87668			−0.938339	−	0.345717i	$-0.887636\pi$
$557$	−1045.31			−1.87668			−0.938339	+	0.345717i	$0.887636\pi$
$558$		0			0
$559$	−474.460			−0.848765
$560$	51.7846			0.0924725
$561$		0			0
$562$			101.522i			0.180644i
$563$			1039.19i			1.84581i	0.385026	+	0.922906i	$0.374192\pi$
$563$			1039.19i			1.84581i	−0.385026	+	0.922906i	$0.625808\pi$
$564$		0			0
$565$		−	127.114i		−	0.224980i
$566$	175.654			0.310343
$567$		0			0
$568$			393.899i			0.693484i
$569$			206.318i			0.362597i	0.983428	+	0.181298i	$0.0580300\pi$
$569$			206.318i			0.362597i	−0.983428	+	0.181298i	$0.941970\pi$
$570$		0			0
$571$		−	144.089i		−	0.252345i	−0.992008	−	0.126173i	$-0.959731\pi$
$571$		−	144.089i		−	0.252345i	0.992008	−	0.126173i	$-0.0402693\pi$
$572$	−1623.66			−2.83857
$573$		0			0
$574$			104.630i			0.182283i
$575$		−	289.015i		−	0.502634i
$576$		0			0
$577$	−596.647			−1.03405			−0.517025	−	0.855970i	$-0.672961\pi$
$577$	−596.647			−1.03405			−0.517025	+	0.855970i	$0.672961\pi$
$578$			24.2491i			0.0419535i
$579$		0			0
$580$	−26.8846			−0.0463527
$581$			728.386i			1.25368i
$582$		0			0
$583$		−	712.941i		−	1.22288i
$584$	−295.471			−0.505943
$585$		0			0
$586$		−	33.0091i		−	0.0563295i
$587$		−	177.665i		−	0.302666i	−0.988483	−	0.151333i	$-0.951643\pi$
$587$		−	177.665i		−	0.302666i	0.988483	−	0.151333i	$-0.0483566\pi$
$588$		0			0
$589$		−	141.203i		−	0.239733i
$590$	−10.8440	+	23.0574i	−0.0183797	+	0.0390803i
$591$		0			0
$592$			501.992i			0.847959i
$593$	297.828			0.502240			0.251120	−	0.967956i	$-0.419201\pi$
$593$	297.828			0.502240			0.251120	+	0.967956i	$0.419201\pi$
$594$		0			0
$595$	−75.0362			−0.126111
$596$		−	100.874i		−	0.169252i
$597$		0			0
$598$			157.220i			0.262910i
$599$	665.978			1.11182			0.555908	−	0.831244i	$-0.312371\pi$
$599$	665.978			1.11182			0.555908	+	0.831244i	$0.312371\pi$
$600$		0			0
$601$			892.749i			1.48544i	0.669603	+	0.742720i	$0.266464\pi$
$601$			892.749i			1.48544i	−0.669603	+	0.742720i	$0.733536\pi$
$602$	52.9139			0.0878968
$603$		0			0
$604$			350.532i			0.580351i
$605$	−156.063			−0.257955
$606$		0			0
$607$	−507.085			−0.835395			−0.417697	−	0.908586i	$-0.637163\pi$
$607$	−507.085			−0.835395			−0.417697	+	0.908586i	$0.637163\pi$
$608$			667.503i			1.09787i
$609$		0			0
$610$	−47.6946			−0.0781879
$611$	−1618.83			−2.64948
$612$		0			0
$613$			16.5526i			0.0270026i	0.999909	+	0.0135013i	$0.00429774\pi$
$613$			16.5526i			0.0270026i	−0.999909	+	0.0135013i	$0.995702\pi$
$614$		−	147.735i		−	0.240610i
$615$		0			0
$616$	376.262			0.610816
$617$	357.976			0.580188			0.290094	−	0.956998i	$-0.406313\pi$
$617$	357.976			0.580188			0.290094	+	0.956998i	$0.406313\pi$
$618$		0			0
$619$	−484.085			−0.782044			−0.391022	−	0.920381i	$-0.627878\pi$
$619$	−484.085			−0.782044			−0.391022	+	0.920381i	$0.627878\pi$
$620$		−	14.7329i		−	0.0237628i
$621$		0			0
$622$		−	262.523i		−	0.422062i
$623$			657.180i			1.05486i
$624$		0			0
$625$	577.030			0.923248
$626$	131.507			0.210076
$627$		0			0
$628$			48.5227i			0.0772654i
$629$		−	727.389i		−	1.15642i
$630$		0			0
$631$	517.329			0.819856			0.409928	−	0.912118i	$-0.365554\pi$
$631$	517.329			0.819856			0.409928	+	0.912118i	$0.365554\pi$
$632$		−	55.3820i		−	0.0876297i
$633$		0			0
$634$			177.000i			0.279179i
$635$	−81.6325			−0.128555
$636$		0			0
$637$			563.729i			0.884974i
$638$	−86.1145			−0.134976
$639$		0			0
$640$			91.4375i			0.142871i
$641$	412.313			0.643234			0.321617	−	0.946870i	$-0.395774\pi$
$641$	412.313			0.643234			0.321617	+	0.946870i	$0.395774\pi$
$642$		0			0
$643$	77.3753			0.120335			0.0601674	−	0.998188i	$-0.480837\pi$
$643$	77.3753			0.120335			0.0601674	+	0.998188i	$0.480837\pi$
$644$			225.070i			0.349488i
$645$		0			0
$646$		−	280.751i		−	0.434599i
$647$	−1058.14			−1.63546			−0.817728	−	0.575605i	$-0.804767\pi$
$647$	−1058.14			−1.63546			−0.817728	+	0.575605i	$0.804767\pi$
$648$		0			0
$649$	445.865	−	948.030i	0.687003	−	1.46075i
$650$	−322.671			−0.496416
$651$		0			0
$652$	−117.345			−0.179977
$653$	399.017			0.611052			0.305526	−	0.952184i	$-0.401168\pi$
$653$	399.017			0.611052			0.305526	+	0.952184i	$0.401168\pi$
$654$		0			0
$655$			69.2316i			0.105697i
$656$	480.287			0.732145
$657$		0			0
$658$	180.539			0.274376
$659$		−	309.391i		−	0.469486i	−0.972057	−	0.234743i	$-0.924575\pi$
$659$		−	309.391i		−	0.469486i	0.972057	−	0.234743i	$-0.0754249\pi$
$660$		0			0
$661$	646.215			0.977633			0.488816	−	0.872387i	$-0.337429\pi$
$661$	646.215			0.977633			0.488816	+	0.872387i	$0.337429\pi$
$662$			213.965i			0.323210i
$663$		0			0
$664$	−590.858			−0.889846
$665$	117.272			0.176349
$666$		0			0
$667$		−	107.036i		−	0.160473i
$668$	−527.848			−0.790191
$669$		0			0
$670$	13.2993			0.0198497
$671$	1961.02			2.92253
$672$		0			0
$673$		−	940.573i		−	1.39758i	−0.715326	−	0.698791i	$-0.753722\pi$
$673$		−	940.573i		−	1.39758i	0.715326	−	0.698791i	$-0.246278\pi$
$674$	64.8374			0.0961979
$675$		0			0
$676$	−1625.99			−2.40531
$677$	−1185.19			−1.75065			−0.875327	−	0.483531i	$-0.839354\pi$
$677$	−1185.19			−1.75065			−0.875327	+	0.483531i	$0.839354\pi$
$678$		0			0
$679$		−	496.538i		−	0.731278i
$680$		−	60.8685i		−	0.0895125i
$681$		0			0
$682$		−	47.1913i		−	0.0691955i
$683$			126.072i			0.184586i	0.995732	+	0.0922928i	$0.0294196\pi$
$683$			126.072i			0.184586i	−0.995732	+	0.0922928i	$0.970580\pi$
$684$		0			0
$685$	139.837			0.204142
$686$		−	197.524i		−	0.287936i
$687$		0			0
$688$		−	242.892i		−	0.353041i
$689$		−	989.341i		−	1.43591i
$690$		0			0
$691$			1168.97i			1.69171i	0.533410	+	0.845857i	$0.320910\pi$
$691$			1168.97i			1.69171i	−0.533410	+	0.845857i	$0.679090\pi$
$692$			751.459i			1.08592i
$693$		0			0
$694$	91.9372			0.132474
$695$	133.665			0.192324
$696$		0			0
$697$	−695.939			−0.998478
$698$	158.261			0.226734
$699$		0			0
$700$	−461.923			−0.659890
$701$			747.168i			1.06586i	0.846159	+	0.532930i	$0.178909\pi$
$701$			747.168i			1.06586i	−0.846159	+	0.532930i	$0.821091\pi$
$702$		0			0
$703$			1136.82i			1.61710i
$704$		−	672.876i		−	0.955789i
$705$		0			0
$706$	−88.9747			−0.126027
$707$		−	514.666i		−	0.727958i
$708$		0			0
$709$	1148.98			1.62056			0.810282	−	0.586040i	$-0.199314\pi$
$709$	1148.98			1.62056			0.810282	+	0.586040i	$0.199314\pi$
$710$			41.0307i			0.0577897i
$711$		0			0
$712$	−533.096			−0.748731
$713$	58.6563			0.0822669
$714$		0			0
$715$	−351.434			−0.491516
$716$			708.437i			0.989437i
$717$		0			0
$718$		−	124.472i		−	0.173359i
$719$			392.721i			0.546204i	0.961985	+	0.273102i	$0.0880496\pi$
$719$			392.721i			0.546204i	−0.961985	+	0.273102i	$0.911950\pi$
$720$		0			0
$721$			606.830i			0.841650i
$722$			244.678i			0.338889i
$723$		0			0
$724$	−460.044			−0.635421
$725$	219.675			0.303000
$726$		0			0
$727$	1029.86			1.41658			0.708292	−	0.705920i	$-0.249466\pi$
$727$	1029.86			1.41658			0.708292	+	0.705920i	$0.249466\pi$
$728$	522.135			0.717219
$729$		0			0
$730$	−30.7779			−0.0421615
$731$			351.952i			0.481467i
$732$		0			0
$733$	−679.097			−0.926462			−0.463231	−	0.886238i	$-0.653310\pi$
$733$	−679.097			−0.926462			−0.463231	+	0.886238i	$0.653310\pi$
$734$	−12.8712			−0.0175356
$735$		0			0
$736$	−277.284			−0.376745
$737$	−546.817			−0.741950
$738$		0			0
$739$			297.353i			0.402372i	0.979553	+	0.201186i	$0.0644795\pi$
$739$			297.353i			0.402372i	−0.979553	+	0.201186i	$0.935521\pi$
$740$			118.614i			0.160290i
$741$		0			0
$742$			110.336i			0.148701i
$743$	−592.272			−0.797136			−0.398568	−	0.917139i	$-0.630493\pi$
$743$	−592.272			−0.797136			−0.398568	+	0.917139i	$0.630493\pi$
$744$		0			0
$745$		−	21.8338i		−	0.0293071i
$746$			150.109i			0.201219i
$747$		0			0
$748$			1204.42i			1.61019i
$749$	−807.737			−1.07842
$750$		0			0
$751$			546.455i			0.727637i	0.931470	+	0.363818i	$0.118527\pi$
$751$			546.455i			0.727637i	−0.931470	+	0.363818i	$0.881473\pi$
$752$		−	828.735i		−	1.10204i
$753$		0			0
$754$	−119.500			−0.158488
$755$			75.8712i			0.100492i
$756$		0			0
$757$	−392.628			−0.518663			−0.259331	−	0.965788i	$-0.583502\pi$
$757$	−392.628			−0.518663			−0.259331	+	0.965788i	$0.583502\pi$
$758$			48.3561i			0.0637944i
$759$		0			0
$760$			95.1300i			0.125171i
$761$	526.584			0.691963			0.345982	−	0.938241i	$-0.387546\pi$
$761$	526.584			0.691963			0.345982	+	0.938241i	$0.387546\pi$
$762$		0			0
$763$		−	454.740i		−	0.595989i
$764$		−	465.111i		−	0.608784i
$765$		0			0
$766$			77.3803i			0.101019i
$767$	618.722	−	1315.57i	0.806678	−	1.71522i
$768$		0			0
$769$		−	1298.20i		−	1.68817i	−0.536212	−	0.844084i	$-0.680145\pi$
$769$		−	1298.20i		−	1.68817i	0.536212	−	0.844084i	$-0.319855\pi$
$770$	39.1935			0.0509007
$771$		0			0
$772$	−400.807			−0.519179
$773$			1105.65i			1.43034i	0.698951	+	0.715170i	$0.253651\pi$
$773$			1105.65i			1.43034i	−0.698951	+	0.715170i	$0.746349\pi$
$774$		0			0
$775$			120.383i			0.155333i
$776$	402.786			0.519054
$777$		0			0
$778$			98.3554i			0.126421i
$779$	1087.67			1.39623
$780$		0			0
$781$		−	1687.02i		−	2.16008i
$782$	116.625			0.149137
$783$		0			0
$784$	−288.592			−0.368102
$785$			10.5025i			0.0133790i
$786$		0			0
$787$	597.635			0.759384			0.379692	−	0.925113i	$-0.376030\pi$
$787$	597.635			0.759384			0.379692	+	0.925113i	$0.376030\pi$
$788$	492.018			0.624389
$789$		0			0
$790$		−	5.76889i		−	0.00730239i
$791$		−	808.848i		−	1.02256i
$792$		0			0
$793$	2721.28			3.43163
$794$	91.5354			0.115284
$795$		0			0
$796$	−851.389			−1.06958
$797$		−	974.030i		−	1.22212i	−0.791584	−	0.611060i	$-0.790743\pi$
$797$		−	974.030i		−	1.22212i	0.791584	−	0.611060i	$-0.209257\pi$
$798$		0			0
$799$			1200.84i			1.50293i
$800$		−	569.083i		−	0.711354i
$801$		0			0
$802$	275.380			0.343367
$803$	1265.47			1.57592
$804$		0			0
$805$			48.7155i			0.0605162i
$806$		−	65.4869i		−	0.0812492i
$807$		0			0
$808$	417.491			0.516697
$809$			971.146i			1.20043i	0.799840	+	0.600214i	$0.204918\pi$
$809$			971.146i			1.20043i	−0.799840	+	0.600214i	$0.795082\pi$
$810$		0			0
$811$		−	66.0398i		−	0.0814301i	−0.999171	−	0.0407151i	$-0.987036\pi$
$811$		−	66.0398i		−	0.0814301i	0.999171	−	0.0407151i	$-0.0129636\pi$
$812$	−171.072			−0.210680
$813$		0			0
$814$			379.936i			0.466752i
$815$	−25.3988			−0.0311642
$816$		0			0
$817$		−	550.058i		−	0.673266i
$818$	−48.9945			−0.0598955
$819$		0			0
$820$	113.486			0.138397
$821$			420.533i			0.512220i	0.966648	+	0.256110i	$0.0824410\pi$
$821$			420.533i			0.512220i	−0.966648	+	0.256110i	$0.917559\pi$
$822$		0			0
$823$		−	490.370i		−	0.595832i	−0.954592	−	0.297916i	$-0.903708\pi$
$823$		−	490.370i		−	0.595832i	0.954592	−	0.297916i	$-0.0962916\pi$
$824$	−492.253			−0.597395
$825$		0			0
$826$	−69.0027	+	146.718i	−0.0835384	+	0.177625i
$827$	−884.674			−1.06974			−0.534870	−	0.844935i	$-0.679639\pi$
$827$	−884.674			−1.06974			−0.534870	+	0.844935i	$0.679639\pi$
$828$		0			0
$829$	379.340			0.457587			0.228794	−	0.973475i	$-0.426522\pi$
$829$	379.340			0.457587			0.228794	+	0.973475i	$0.426522\pi$
$830$	−61.5470			−0.0741530
$831$		0			0
$832$		−	933.742i		−	1.12229i
$833$	418.172			0.502007
$834$		0			0
$835$	−114.250			−0.136827
$836$		−	1882.37i		−	2.25163i
$837$		0			0
$838$	−48.8075			−0.0582428
$839$		−	59.4763i		−	0.0708896i	−0.999372	−	0.0354448i	$-0.988715\pi$
$839$		−	59.4763i		−	0.0708896i	0.999372	−	0.0354448i	$-0.0112848\pi$
$840$		0			0
$841$	−759.644			−0.903263
$842$	271.073			0.321939
$843$		0			0
$844$			1002.96i			1.18834i
$845$	−351.939			−0.416496
$846$		0			0
$847$	−993.059			−1.17244
$848$	506.478			0.597261
$849$		0			0
$850$			239.356i			0.281595i
$851$	−472.240			−0.554924
$852$		0			0
$853$	18.7626			0.0219960			0.0109980	−	0.999940i	$-0.496499\pi$
$853$	18.7626			0.0219960			0.0109980	+	0.999940i	$0.496499\pi$
$854$	−303.490			−0.355374
$855$		0			0
$856$		−	655.227i		−	0.765452i
$857$			1345.88i			1.57046i	0.619204	+	0.785230i	$0.287456\pi$
$857$			1345.88i			1.57046i	−0.619204	+	0.785230i	$0.712544\pi$
$858$		0			0
$859$		−	1534.35i		−	1.78620i	−0.449854	−	0.893102i	$-0.648524\pi$
$859$		−	1534.35i		−	1.78620i	0.449854	−	0.893102i	$-0.351476\pi$
$860$		−	57.3924i		−	0.0667354i
$861$		0			0
$862$	180.610			0.209524
$863$			301.276i			0.349103i	0.984648	+	0.174551i	$0.0558475\pi$
$863$			301.276i			0.349103i	−0.984648	+	0.174551i	$0.944152\pi$
$864$		0			0
$865$			162.650i			0.188035i
$866$			89.0751i			0.102858i
$867$		0			0
$868$		−	93.7485i		−	0.108005i
$869$			237.194i			0.272951i
$870$		0			0
$871$	−758.812			−0.871197
$872$	368.880			0.423027
$873$		0			0
$874$	−182.271			−0.208548
$875$	−202.611			−0.231555
$876$		0			0
$877$	−88.0368			−0.100384			−0.0501920	−	0.998740i	$-0.515983\pi$
$877$	−88.0368			−0.100384			−0.0501920	+	0.998740i	$0.515983\pi$
$878$		−	86.0055i		−	0.0979561i
$879$		0			0
$880$		−	179.911i		−	0.204445i
$881$			282.027i			0.320121i	0.987107	+	0.160060i	$0.0511689\pi$
$881$			282.027i			0.320121i	−0.987107	+	0.160060i	$0.948831\pi$
$882$		0			0
$883$	1417.42			1.60524			0.802619	−	0.596493i	$-0.203440\pi$
$883$	1417.42			1.60524			0.802619	+	0.596493i	$0.203440\pi$
$884$			1671.37i			1.89069i
$885$		0			0
$886$	35.0252			0.0395318
$887$			9.10283i			0.0102625i	0.999987	+	0.00513125i	$0.00163333\pi$
$887$			9.10283i			0.0102625i	−0.999987	+	0.00513125i	$0.998367\pi$
$888$		0			0
$889$	−519.443			−0.584301
$890$	−55.5302			−0.0623935
$891$		0			0
$892$	138.183			0.154914
$893$		−	1876.77i		−	2.10164i
$894$		0			0
$895$			153.338i			0.171328i
$896$			581.834i			0.649369i
$897$		0			0
$898$			199.569i			0.222237i
$899$			44.5836i			0.0495924i
$900$		0			0
$901$	−733.889			−0.814527
$902$	363.508			0.403003
$903$		0			0
$904$	656.128			0.725805
$905$	−99.5747			−0.110027
$906$		0			0
$907$	−443.496			−0.488970			−0.244485	−	0.969653i	$-0.578619\pi$
$907$	−443.496			−0.488970			−0.244485	+	0.969653i	$0.578619\pi$
$908$		−	585.598i		−	0.644932i
$909$		0			0
$910$	54.3884			0.0597675
$911$	−114.969			−0.126201			−0.0631003	−	0.998007i	$-0.520099\pi$
$911$	−114.969			−0.126201			−0.0631003	+	0.998007i	$0.520099\pi$
$912$		0			0
$913$	2530.57			2.77171
$914$	−91.2976			−0.0998880
$915$		0			0
$916$			1131.23i			1.23497i
$917$			440.534i			0.480408i
$918$		0			0
$919$		−	1066.64i		−	1.16066i	−0.814383	−	0.580328i	$-0.802924\pi$
$919$		−	1066.64i		−	1.16066i	0.814383	−	0.580328i	$-0.197076\pi$
$920$	−39.5174			−0.0429537
$921$		0			0
$922$			353.586i			0.383499i
$923$		−	2341.06i		−	2.53636i
$924$		0			0
$925$		−	969.202i		−	1.04779i
$926$	−391.990			−0.423315
$927$		0			0
$928$		−	210.758i		−	0.227110i
$929$			842.876i			0.907294i	0.891182	+	0.453647i	$0.149877\pi$
$929$			842.876i			0.907294i	−0.891182	+	0.453647i	$0.850123\pi$
$930$		0			0
$931$	−653.551			−0.701988
$932$			606.305i			0.650541i
$933$		0			0
$934$	−77.9285			−0.0834352
$935$			260.693i			0.278816i
$936$		0			0
$937$			1628.55i			1.73805i	0.494772	+	0.869023i	$0.335252\pi$
$937$			1628.55i			1.73805i	−0.494772	+	0.869023i	$0.664748\pi$
$938$	84.6262			0.0902198
$939$		0			0
$940$		−	195.820i		−	0.208319i
$941$			216.768i			0.230359i	0.993345	+	0.115179i	$0.0367443\pi$
$941$			216.768i			0.230359i	−0.993345	+	0.115179i	$0.963256\pi$
$942$		0			0
$943$			451.822i			0.479132i
$944$	673.486	+	316.745i	0.713439	+	0.335535i
$945$		0			0
$946$		−	183.835i		−	0.194328i
$947$	−945.701			−0.998629			−0.499314	−	0.866421i	$-0.666415\pi$
$947$	−945.701			−0.998629			−0.499314	+	0.866421i	$0.666415\pi$
$948$		0			0
$949$	1756.07			1.85045
$950$		−	374.083i		−	0.393772i
$951$		0			0
$952$		−	387.318i		−	0.406847i
$953$	−106.234			−0.111473			−0.0557365	−	0.998446i	$-0.517751\pi$
$953$	−106.234			−0.111473			−0.0557365	+	0.998446i	$0.517751\pi$
$954$		0			0
$955$		−	100.671i		−	0.105415i
$956$	873.203			0.913392
$957$		0			0
$958$			39.3566i			0.0410821i
$959$	889.811			0.927853
$960$		0			0
$961$	936.568			0.974576
$962$			527.233i			0.548059i
$963$		0			0
$964$	264.386			0.274260
$965$	−86.7529			−0.0898994
$966$		0			0
$967$			898.516i			0.929179i	0.885526	+	0.464590i	$0.153798\pi$
$967$			898.516i			0.929179i	−0.885526	+	0.464590i	$0.846202\pi$
$968$		−	805.557i		−	0.832187i
$969$		0			0
$970$	41.9563			0.0432540
$971$	−96.7025			−0.0995907			−0.0497953	−	0.998759i	$-0.515857\pi$
$971$	−96.7025			−0.0995907			−0.0497953	+	0.998759i	$0.515857\pi$
$972$		0			0
$973$	850.539			0.874141
$974$		−	29.4249i		−	0.0302103i
$975$		0			0
$976$			1393.12i			1.42738i
$977$		−	1230.50i		−	1.25946i	−0.776812	−	0.629732i	$-0.783165\pi$
$977$		−	1230.50i		−	1.25946i	0.776812	−	0.629732i	$-0.216835\pi$
$978$		0			0
$979$	2283.19			2.33216
$980$	−68.1907			−0.0695824
$981$		0			0
$982$		−	316.578i		−	0.322381i
$983$			1521.39i			1.54770i	0.633368	+	0.773851i	$0.281672\pi$
$983$			1521.39i			1.54770i	−0.633368	+	0.773851i	$0.718328\pi$
$984$		0			0
$985$	106.495			0.108117
$986$			88.6447i			0.0899034i
$987$		0			0
$988$		−	2612.14i		−	2.64387i
$989$	228.497			0.231038
$990$		0			0
$991$		−	1725.31i		−	1.74098i	−0.492186	−	0.870490i	$-0.663802\pi$
$991$		−	1725.31i		−	1.74098i	0.492186	−	0.870490i	$-0.336198\pi$
$992$	115.497			0.116428
$993$		0			0
$994$			261.086i			0.262662i
$995$	−184.280			−0.185206
$996$		0			0
$997$	−1404.90			−1.40912			−0.704561	−	0.709643i	$-0.748856\pi$
$997$	−1404.90			−1.40912			−0.704561	+	0.709643i	$0.748856\pi$
$998$			198.317i			0.198715i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.3.c.c.235.11		20
3.2	odd	2		177.3.c.a.58.10	✓	20
59.58	odd	2	inner	531.3.c.c.235.10		20
177.176	even	2		177.3.c.a.58.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.3.c.a.58.10	✓	20	3.2	odd	2
177.3.c.a.58.11	yes	20	177.176	even	2
531.3.c.c.235.10		20	59.58	odd	2	inner
531.3.c.c.235.11		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.537675i q^{2} +3.71091 q^{4} +0.803210 q^{5} +5.11098 q^{7} +4.14596i q^{8} +O(q^{10})\)	\(q+0.537675i q^{2} +3.71091 q^{4} +0.803210 q^{5} +5.11098 q^{7} +4.14596i q^{8} +0.431866i q^{10} -17.7567i q^{11} -24.6407i q^{13} +2.74805i q^{14} +12.6144 q^{16} -18.2784 q^{17} +28.5669 q^{19} +2.98064 q^{20} +9.54732 q^{22} +11.8668i q^{23} -24.3549 q^{25} +13.2487 q^{26} +18.9664 q^{28} -9.01975 q^{29} -4.94288i q^{31} +23.3663i q^{32} -9.82784i q^{34} +4.10519 q^{35} +39.7950i q^{37} +15.3597i q^{38} +3.33008i q^{40} +38.0744 q^{41} -19.2551i q^{43} -65.8933i q^{44} -6.38050 q^{46} -65.6973i q^{47} -22.8779 q^{49} -13.0950i q^{50} -91.4394i q^{52} +40.1506 q^{53} -14.2623i q^{55} +21.1899i q^{56} -4.84970i q^{58} +(53.3901 + 25.1097i) q^{59} +110.438i q^{61} +2.65767 q^{62} +37.8943 q^{64} -19.7917i q^{65} -30.7950i q^{67} -67.8294 q^{68} +2.20726i q^{70} +95.0078 q^{71} +71.2671i q^{73} -21.3968 q^{74} +106.009 q^{76} -90.7539i q^{77} -13.3580 q^{79} +10.1320 q^{80} +20.4717i q^{82} +142.514i q^{83} -14.6814 q^{85} +10.3530 q^{86} +73.6185 q^{88} +128.582i q^{89} -125.938i q^{91} +44.0366i q^{92} +35.3238 q^{94} +22.9452 q^{95} -97.1513i q^{97} -12.3009i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} - 8 q^{7}+O(q^{10}) \) 20 * q - 40 * q^4 - 8 * q^7	\( 20 q - 40 q^{4} - 8 q^{7} - 8 q^{16} - 16 q^{17} - 60 q^{19} + 164 q^{20} + 40 q^{22} + 100 q^{25} + 156 q^{26} + 200 q^{28} + 60 q^{29} + 32 q^{35} - 28 q^{41} + 180 q^{46} + 284 q^{49} + 8 q^{53} + 152 q^{59} + 8 q^{62} + 204 q^{64} - 384 q^{68} - 92 q^{71} - 104 q^{74} + 120 q^{76} - 420 q^{79} - 376 q^{80} - 348 q^{85} - 232 q^{86} - 212 q^{88} + 152 q^{94} - 788 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 - 8 * q^7 - 8 * q^16 - 16 * q^17 - 60 * q^19 + 164 * q^20 + 40 * q^22 + 100 * q^25 + 156 * q^26 + 200 * q^28 + 60 * q^29 + 32 * q^35 - 28 * q^41 + 180 * q^46 + 284 * q^49 + 8 * q^53 + 152 * q^59 + 8 * q^62 + 204 * q^64 - 384 * q^68 - 92 * q^71 - 104 * q^74 + 120 * q^76 - 420 * q^79 - 376 * q^80 - 348 * q^85 - 232 * q^86 - 212 * q^88 + 152 * q^94 - 788 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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