LMFDB - Embedded newform 387.3.b.c.343.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,3,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, 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("387.343");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	387.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:	$10.5449862307$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 20x^{4} + 121x^{2} + 214 $ x^6 + 20x^4 + 121x^2 + 214
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			343.1
Root			$-3.18991i$ of defining polynomial
Character	$\chi$	$=$	387.343
Dual form			387.3.b.c.343.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.18991i q^{2} -6.17554 q^{4} -7.66434i q^{5} -10.1297i q^{7} +6.93980i q^{8} +O(q^{10})$	\(q-3.18991i q^{2} -6.17554 q^{4} -7.66434i q^{5} -10.1297i q^{7} +6.93980i q^{8} -24.4486 q^{10} +2.43517 q^{11} +18.1491 q^{13} -32.3129 q^{14} -2.56483 q^{16} +1.13567 q^{17} -12.3269i q^{19} +47.3314i q^{20} -7.76798i q^{22} +24.2847 q^{23} -33.7420 q^{25} -57.8939i q^{26} +62.5565i q^{28} +35.5218i q^{29} -12.9823 q^{31} +35.9408i q^{32} -3.62270i q^{34} -77.6375 q^{35} +41.5116i q^{37} -39.3216 q^{38} +53.1889 q^{40} +65.0667 q^{41} +(23.6064 + 35.9408i) q^{43} -15.0385 q^{44} -77.4662i q^{46} -51.0402 q^{47} -53.6110 q^{49} +107.634i q^{50} -112.080 q^{52} -56.2026 q^{53} -18.6640i q^{55} +70.2981 q^{56} +113.312 q^{58} +88.8355 q^{59} -65.1393i q^{61} +41.4123i q^{62} +104.389 q^{64} -139.101i q^{65} +45.3618 q^{67} -7.01339 q^{68} +247.657i q^{70} -63.7983i q^{71} +8.80247i q^{73} +132.418 q^{74} +76.1251i q^{76} -24.6676i q^{77} +31.8496 q^{79} +19.6577i q^{80} -207.557i q^{82} +68.4740 q^{83} -8.70417i q^{85} +(114.648 - 75.3023i) q^{86} +16.8996i q^{88} +5.90432i q^{89} -183.845i q^{91} -149.971 q^{92} +162.814i q^{94} -94.4773 q^{95} -100.657 q^{97} +171.015i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4}+O(q^{10}) $ 6 * q - 16 * q^4	$ 6 q - 16 q^{4} - 2 q^{10} - 38 q^{11} + 30 q^{13} - 36 q^{14} - 68 q^{16} + 20 q^{17} + 80 q^{23} - 84 q^{25} - 112 q^{31} - 208 q^{35} - 170 q^{38} + 206 q^{40} + 172 q^{41} + 10 q^{43} + 36 q^{44} - 30 q^{47} - 6 q^{49} - 120 q^{52} + 110 q^{53} + 264 q^{56} + 430 q^{58} + 12 q^{59} + 100 q^{64} - 70 q^{67} + 50 q^{68} + 50 q^{74} + 178 q^{79} - 10 q^{83} + 372 q^{86} - 150 q^{92} + 130 q^{95} - 380 q^{97}+O(q^{100}) $ 6 * q - 16 * q^4 - 2 * q^10 - 38 * q^11 + 30 * q^13 - 36 * q^14 - 68 * q^16 + 20 * q^17 + 80 * q^23 - 84 * q^25 - 112 * q^31 - 208 * q^35 - 170 * q^38 + 206 * q^40 + 172 * q^41 + 10 * q^43 + 36 * q^44 - 30 * q^47 - 6 * q^49 - 120 * q^52 + 110 * q^53 + 264 * q^56 + 430 * q^58 + 12 * q^59 + 100 * q^64 - 70 * q^67 + 50 * q^68 + 50 * q^74 + 178 * q^79 - 10 * q^83 + 372 * q^86 - 150 * q^92 + 130 * q^95 - 380 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	3.18991i		−	1.59496i	−0.603348	−	0.797478i	$-0.706167\pi$
$2$		−	3.18991i		−	1.59496i	0.603348	−	0.797478i	$-0.293833\pi$
$3$		0			0
$3$		0			0
$4$	−6.17554			−1.54389
$5$		−	7.66434i		−	1.53287i	−0.642324	−	0.766434i	$-0.722029\pi$
$5$		−	7.66434i		−	1.53287i	0.642324	−	0.766434i	$-0.277971\pi$
$6$		0			0
$7$		−	10.1297i		−	1.44710i	−0.690271	−	0.723551i	$-0.742509\pi$
$7$		−	10.1297i		−	1.44710i	0.690271	−	0.723551i	$-0.257491\pi$
$8$			6.93980i			0.867475i
$9$		0			0
$10$	−24.4486			−2.44486
$11$	2.43517			0.221379			0.110690	−	0.993855i	$-0.464694\pi$
$11$	2.43517			0.221379			0.110690	+	0.993855i	$0.464694\pi$
$12$		0			0
$13$	18.1491			1.39608			0.698041	−	0.716058i	$-0.254055\pi$
$13$	18.1491			1.39608			0.698041	+	0.716058i	$0.254055\pi$
$14$	−32.3129			−2.30806
$15$		0			0
$16$	−2.56483			−0.160302
$17$	1.13567			0.0668042			0.0334021	−	0.999442i	$-0.489366\pi$
$17$	1.13567			0.0668042			0.0334021	+	0.999442i	$0.489366\pi$
$18$		0			0
$19$		−	12.3269i		−	0.648783i	−0.945923	−	0.324391i	$-0.894841\pi$
$19$		−	12.3269i		−	0.648783i	0.945923	−	0.324391i	$-0.105159\pi$
$20$			47.3314i			2.36657i
$21$		0			0
$22$		−	7.76798i		−	0.353090i
$23$	24.2847			1.05586			0.527929	−	0.849288i	$-0.322969\pi$
$23$	24.2847			1.05586			0.527929	+	0.849288i	$0.322969\pi$
$24$		0			0
$25$	−33.7420			−1.34968
$26$		−	57.8939i		−	2.22669i
$27$		0			0
$28$			62.5565i			2.23416i
$29$			35.5218i			1.22489i	0.790513	+	0.612445i	$0.209814\pi$
$29$			35.5218i			1.22489i	−0.790513	+	0.612445i	$0.790186\pi$
$30$		0			0
$31$	−12.9823			−0.418783			−0.209391	−	0.977832i	$-0.567148\pi$
$31$	−12.9823			−0.418783			−0.209391	+	0.977832i	$0.567148\pi$
$32$			35.9408i			1.12315i
$33$		0			0
$34$		−	3.62270i		−	0.106550i
$35$	−77.6375			−2.21821
$36$		0			0
$37$			41.5116i			1.12193i	0.827838	+	0.560967i	$0.189571\pi$
$37$			41.5116i			1.12193i	−0.827838	+	0.560967i	$0.810429\pi$
$38$	−39.3216			−1.03478
$39$		0			0
$40$	53.1889			1.32972
$41$	65.0667			1.58699			0.793496	−	0.608576i	$-0.208259\pi$
$41$	65.0667			1.58699			0.793496	+	0.608576i	$0.208259\pi$
$42$		0			0
$43$	23.6064	+	35.9408i	0.548986	+	0.835832i
$43$	23.6064	+	35.9408i	0.548986	+	0.835832i
$44$	−15.0385			−0.341784
$45$		0			0
$46$		−	77.4662i		−	1.68405i
$47$	−51.0402			−1.08596			−0.542981	−	0.839745i	$-0.682704\pi$
$47$	−51.0402			−1.08596			−0.542981	+	0.839745i	$0.682704\pi$
$48$		0			0
$49$	−53.6110			−1.09410
$50$			107.634i			2.15268i
$51$		0			0
$52$	−112.080			−2.15539
$53$	−56.2026			−1.06043			−0.530214	−	0.847864i	$-0.677888\pi$
$53$	−56.2026			−1.06043			−0.530214	+	0.847864i	$0.677888\pi$
$54$		0			0
$55$		−	18.6640i		−	0.339345i
$56$	70.2981			1.25532
$57$		0			0
$58$	113.312			1.95365
$59$	88.8355			1.50569			0.752843	−	0.658200i	$-0.228682\pi$
$59$	88.8355			1.50569			0.752843	+	0.658200i	$0.228682\pi$
$60$		0			0
$61$		−	65.1393i		−	1.06786i	−0.845529	−	0.533929i	$-0.820715\pi$
$61$		−	65.1393i		−	1.06786i	0.845529	−	0.533929i	$-0.179285\pi$
$62$			41.4123i			0.667940i
$63$		0			0
$64$	104.389			1.63107
$65$		−	139.101i		−	2.14001i
$66$		0			0
$67$	45.3618			0.677042			0.338521	−	0.940959i	$-0.390073\pi$
$67$	45.3618			0.677042			0.338521	+	0.940959i	$0.390073\pi$
$68$	−7.01339			−0.103138
$69$		0			0
$70$			247.657i			3.53796i
$71$		−	63.7983i		−	0.898567i	−0.893389	−	0.449284i	$-0.851679\pi$
$71$		−	63.7983i		−	0.898567i	0.893389	−	0.449284i	$-0.148321\pi$
$72$		0			0
$73$			8.80247i			0.120582i	0.998181	+	0.0602909i	$0.0192028\pi$
$73$			8.80247i			0.120582i	−0.998181	+	0.0602909i	$0.980797\pi$
$74$	132.418			1.78944
$75$		0			0
$76$			76.1251i			1.00165i
$77$		−	24.6676i		−	0.320358i
$78$		0			0
$79$	31.8496			0.403159			0.201580	−	0.979472i	$-0.435393\pi$
$79$	31.8496			0.403159			0.201580	+	0.979472i	$0.435393\pi$
$80$			19.6577i			0.245721i
$81$		0			0
$82$		−	207.557i		−	2.53118i
$83$	68.4740			0.824988			0.412494	−	0.910960i	$-0.364658\pi$
$83$	68.4740			0.824988			0.412494	+	0.910960i	$0.364658\pi$
$84$		0			0
$85$		−	8.70417i		−	0.102402i
$86$	114.648	−	75.3023i	1.33312	−	0.875608i
$87$		0			0
$88$			16.8996i			0.192041i
$89$			5.90432i			0.0663406i	0.999450	+	0.0331703i	$0.0105604\pi$
$89$			5.90432i			0.0663406i	−0.999450	+	0.0331703i	$0.989440\pi$
$90$		0			0
$91$		−	183.845i		−	2.02027i
$92$	−149.971			−1.63012
$93$		0			0
$94$			162.814i			1.73206i
$95$	−94.4773			−0.994498
$96$		0			0
$97$	−100.657			−1.03770			−0.518849	−	0.854866i	$-0.673639\pi$
$97$	−100.657			−1.03770			−0.518849	+	0.854866i	$0.673639\pi$
$98$			171.015i			1.74505i
$99$		0			0
$100$	208.376			2.08376
$101$	−13.7062			−0.135705			−0.0678525	−	0.997695i	$-0.521615\pi$
$101$	−13.7062			−0.135705			−0.0678525	+	0.997695i	$0.521615\pi$
$102$		0			0
$103$	−96.0402			−0.932429			−0.466214	−	0.884672i	$-0.654382\pi$
$103$	−96.0402			−0.932429			−0.466214	+	0.884672i	$0.654382\pi$
$104$			125.951i			1.21107i
$105$		0			0
$106$			179.282i			1.69134i
$107$	2.08036			0.0194427			0.00972133	−	0.999953i	$-0.496906\pi$
$107$	2.08036			0.0194427			0.00972133	+	0.999953i	$0.496906\pi$
$108$		0			0
$109$	−53.1078			−0.487228			−0.243614	−	0.969872i	$-0.578333\pi$
$109$	−53.1078			−0.487228			−0.243614	+	0.969872i	$0.578333\pi$
$110$	−59.5364			−0.541240
$111$		0			0
$112$			25.9810i			0.231973i
$113$			28.8791i			0.255568i	0.991802	+	0.127784i	$0.0407864\pi$
$113$			28.8791i			0.255568i	−0.991802	+	0.127784i	$0.959214\pi$
$114$		0			0
$115$		−	186.126i		−	1.61849i
$116$		−	219.367i		−	1.89109i
$117$		0			0
$118$		−	283.378i		−	2.40150i
$119$		−	11.5040i		−	0.0966725i
$120$		0			0
$121$	−115.070			−0.950991
$122$	−207.789			−1.70319
$123$		0			0
$124$	80.1726			0.646553
$125$			67.0020i			0.536016i
$126$		0			0
$127$	148.673			1.17066			0.585328	−	0.810797i	$-0.300966\pi$
$127$	148.673			1.17066			0.585328	+	0.810797i	$0.300966\pi$
$128$		−	189.227i		−	1.47834i
$129$		0			0
$130$	−443.719			−3.41322
$131$		−	157.214i		−	1.20011i	−0.799960	−	0.600054i	$-0.795146\pi$
$131$		−	157.214i		−	1.20011i	0.799960	−	0.600054i	$-0.204854\pi$
$132$		0			0
$133$	−124.868			−0.938854
$134$		−	144.700i		−	1.07985i
$135$		0			0
$136$			7.88133i			0.0579510i
$137$		−	121.604i		−	0.887618i	−0.896122	−	0.443809i	$-0.853627\pi$
$137$		−	121.604i		−	0.887618i	0.896122	−	0.443809i	$-0.146373\pi$
$138$		0			0
$139$	−27.8992			−0.200714			−0.100357	−	0.994952i	$-0.531998\pi$
$139$	−27.8992			−0.200714			−0.100357	+	0.994952i	$0.531998\pi$
$140$	479.454			3.42467
$141$		0			0
$142$	−203.511			−1.43318
$143$	44.1961			0.309063
$144$		0			0
$145$	272.251			1.87759
$146$	28.0791			0.192323
$147$		0			0
$148$		−	256.357i		−	1.73214i
$149$			212.121i			1.42363i	0.702366	+	0.711816i	$0.252127\pi$
$149$			212.121i			1.42363i	−0.702366	+	0.711816i	$0.747873\pi$
$150$		0			0
$151$			108.847i			0.720842i	0.932790	+	0.360421i	$0.117367\pi$
$151$			108.847i			0.720842i	−0.932790	+	0.360421i	$0.882633\pi$
$152$	85.5460			0.562802
$153$		0			0
$154$	−78.6874			−0.510957
$155$			99.5005i			0.641938i
$156$		0			0
$157$		−	106.772i		−	0.680075i	−0.940412	−	0.340038i	$-0.889560\pi$
$157$		−	106.772i		−	0.680075i	0.940412	−	0.340038i	$-0.110440\pi$
$158$		−	101.597i		−	0.643021i
$159$		0			0
$160$	275.462			1.72164
$161$		−	245.997i		−	1.52793i
$162$		0			0
$163$			202.231i			1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$163$			202.231i			1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$164$	−401.822			−2.45013
$165$		0			0
$166$		−	218.426i		−	1.31582i
$167$	41.0703			0.245930			0.122965	−	0.992411i	$-0.460760\pi$
$167$	41.0703			0.245930			0.122965	+	0.992411i	$0.460760\pi$
$168$		0			0
$169$	160.389			0.949045
$170$	−27.7656			−0.163327
$171$		0			0
$172$	−145.782	−	221.954i	−0.847571	−	1.29043i
$173$	24.1006			0.139310			0.0696549	−	0.997571i	$-0.477810\pi$
$173$	24.1006			0.139310			0.0696549	+	0.997571i	$0.477810\pi$
$174$		0			0
$175$			341.797i			1.95313i
$176$	−6.24580			−0.0354875
$177$		0			0
$178$	18.8343			0.105810
$179$			83.0659i			0.464055i	0.972709	+	0.232028i	$0.0745360\pi$
$179$			83.0659i			0.464055i	−0.972709	+	0.232028i	$0.925464\pi$
$180$		0			0
$181$	−112.273			−0.620291			−0.310145	−	0.950689i	$-0.600378\pi$
$181$	−112.273			−0.620291			−0.310145	+	0.950689i	$0.600378\pi$
$182$	−586.449			−3.22225
$183$		0			0
$184$			168.531i			0.915930i
$185$	318.159			1.71978
$186$		0			0
$187$	2.76556			0.0147891
$188$	315.201			1.67660
$189$		0			0
$190$			301.374i			1.58618i
$191$		−	43.7078i		−	0.228836i	−0.993433	−	0.114418i	$-0.963500\pi$
$191$		−	43.7078i		−	0.228836i	0.993433	−	0.114418i	$-0.0365004\pi$
$192$		0			0
$193$	236.100			1.22332			0.611659	−	0.791122i	$-0.290503\pi$
$193$	236.100			1.22332			0.611659	+	0.791122i	$0.290503\pi$
$194$			321.086i			1.65508i
$195$		0			0
$196$	331.077			1.68917
$197$	−350.134			−1.77733			−0.888665	−	0.458557i	$-0.848366\pi$
$197$	−350.134			−1.77733			−0.888665	+	0.458557i	$0.848366\pi$
$198$		0			0
$199$		−	353.573i		−	1.77675i	−0.459122	−	0.888373i	$-0.651836\pi$
$199$		−	353.573i		−	1.77675i	0.459122	−	0.888373i	$-0.348164\pi$
$200$		−	234.163i		−	1.17081i
$201$		0			0
$202$			43.7216i			0.216444i
$203$	359.826			1.77254
$204$		0			0
$205$		−	498.693i		−	2.43265i
$206$			306.360i			1.48718i
$207$		0			0
$208$	−46.5493			−0.223794
$209$		−	30.0180i		−	0.143627i
$210$		0			0
$211$			281.519i			1.33421i	0.744962	+	0.667107i	$0.232468\pi$
$211$			281.519i			1.33421i	−0.744962	+	0.667107i	$0.767532\pi$
$212$	347.082			1.63718
$213$		0			0
$214$		−	6.63618i		−	0.0310102i
$215$	275.462	−	180.927i	1.28122	−	0.841522i
$216$		0			0
$217$			131.507i			0.606021i
$218$			169.409i			0.777107i
$219$		0			0
$220$			115.260i			0.523910i
$221$	20.6114			0.0932642
$222$		0			0
$223$			95.1942i			0.426880i	0.976956	+	0.213440i	$0.0684668\pi$
$223$			95.1942i			0.426880i	−0.976956	+	0.213440i	$0.931533\pi$
$224$	364.070			1.62531
$225$		0			0
$226$	92.1219			0.407619
$227$		−	106.481i		−	0.469079i	−0.972107	−	0.234540i	$-0.924642\pi$
$227$		−	106.481i		−	0.469079i	0.972107	−	0.234540i	$-0.0753583\pi$
$228$		0			0
$229$	170.352			0.743896			0.371948	−	0.928254i	$-0.378690\pi$
$229$	170.352			0.743896			0.371948	+	0.928254i	$0.378690\pi$
$230$	−593.727			−2.58142
$231$		0			0
$232$	−246.514			−1.06256
$233$		−	55.4970i		−	0.238184i	−0.992883	−	0.119092i	$-0.962002\pi$
$233$		−	55.4970i		−	0.238184i	0.992883	−	0.119092i	$-0.0379984\pi$
$234$		0			0
$235$			391.189i			1.66463i
$236$	−548.608			−2.32461
$237$		0			0
$238$	−36.6969			−0.154188
$239$	20.7695			0.0869017			0.0434509	−	0.999056i	$-0.486165\pi$
$239$	20.7695			0.0869017			0.0434509	+	0.999056i	$0.486165\pi$
$240$		0			0
$241$			303.282i			1.25843i	0.777231	+	0.629216i	$0.216624\pi$
$241$			303.282i			1.25843i	−0.777231	+	0.629216i	$0.783376\pi$
$242$			367.063i			1.51679i
$243$		0			0
$244$			402.271i			1.64865i
$245$			410.893i			1.67711i
$246$		0			0
$247$		−	223.721i		−	0.905754i
$248$		−	90.0943i		−	0.363283i
$249$		0			0
$250$	213.731			0.854922
$251$	7.25467			0.0289031			0.0144515	−	0.999896i	$-0.495400\pi$
$251$	7.25467			0.0289031			0.0144515	+	0.999896i	$0.495400\pi$
$252$		0			0
$253$	59.1375			0.233745
$254$		−	474.255i		−	1.86715i
$255$		0			0
$256$	−186.065			−0.726816
$257$			316.956i			1.23329i	0.787241	+	0.616645i	$0.211509\pi$
$257$			316.956i			1.23329i	−0.787241	+	0.616645i	$0.788491\pi$
$258$		0			0
$259$	420.500			1.62355
$260$			859.022i			3.30393i
$261$		0			0
$262$	−501.499			−1.91412
$263$			229.845i			0.873934i	0.899478	+	0.436967i	$0.143947\pi$
$263$			229.845i			0.873934i	−0.899478	+	0.436967i	$0.856053\pi$
$264$		0			0
$265$			430.756i			1.62549i
$266$			398.317i			1.49743i
$267$		0			0
$268$	−280.134			−1.04528
$269$	−285.119			−1.05992			−0.529961	−	0.848022i	$-0.677794\pi$
$269$	−285.119			−1.05992			−0.529961	+	0.848022i	$0.677794\pi$
$270$		0			0
$271$	78.3270			0.289030			0.144515	−	0.989503i	$-0.453838\pi$
$271$	78.3270			0.289030			0.144515	+	0.989503i	$0.453838\pi$
$272$	−2.91281			−0.0107088
$273$		0			0
$274$	−387.905			−1.41571
$275$	−82.1677			−0.298791
$276$		0			0
$277$			119.631i			0.431880i	0.976407	+	0.215940i	$0.0692815\pi$
$277$			119.631i			0.431880i	−0.976407	+	0.215940i	$0.930718\pi$
$278$			88.9960i			0.320129i
$279$		0			0
$280$		−	538.789i		−	1.92424i
$281$	150.423			0.535314			0.267657	−	0.963514i	$-0.413751\pi$
$281$	150.423			0.535314			0.267657	+	0.963514i	$0.413751\pi$
$282$		0			0
$283$	−261.144			−0.922771			−0.461385	−	0.887200i	$-0.652647\pi$
$283$	−261.144			−0.922771			−0.461385	+	0.887200i	$0.652647\pi$
$284$			393.989i			1.38729i
$285$		0			0
$286$		−	140.982i		−	0.492943i
$287$		−	659.106i		−	2.29654i
$288$		0			0
$289$	−287.710			−0.995537
$290$		−	868.458i		−	2.99468i
$291$		0			0
$292$		−	54.3601i		−	0.186165i
$293$	−358.409			−1.22324			−0.611619	−	0.791152i	$-0.709482\pi$
$293$	−358.409			−1.22324			−0.611619	+	0.791152i	$0.709482\pi$
$294$		0			0
$295$		−	680.865i		−	2.30802i
$296$	−288.082			−0.973250
$297$		0			0
$298$	676.648			2.27063
$299$	440.745			1.47406
$300$		0			0
$301$	364.070	−	239.126i	1.20953	−	0.794438i
$302$	347.213			1.14971
$303$		0			0
$304$			31.6163i			0.104001i
$305$	−499.250			−1.63688
$306$		0			0
$307$	509.730			1.66036			0.830180	−	0.557496i	$-0.188238\pi$
$307$	509.730			1.66036			0.830180	+	0.557496i	$0.188238\pi$
$308$			152.336i			0.494596i
$309$		0			0
$310$	317.398			1.02386
$311$	27.7820			0.0893312			0.0446656	−	0.999002i	$-0.485778\pi$
$311$	27.7820			0.0893312			0.0446656	+	0.999002i	$0.485778\pi$
$312$		0			0
$313$			38.5686i			0.123222i	0.998100	+	0.0616111i	$0.0196239\pi$
$313$			38.5686i			0.123222i	−0.998100	+	0.0616111i	$0.980376\pi$
$314$	−340.593			−1.08469
$315$		0			0
$316$	−196.688			−0.622432
$317$	−85.8945			−0.270961			−0.135480	−	0.990780i	$-0.543258\pi$
$317$	−85.8945			−0.270961			−0.135480	+	0.990780i	$0.543258\pi$
$318$		0			0
$319$			86.5017i			0.271165i
$320$		−	800.069i		−	2.50022i
$321$		0			0
$322$	−784.710			−2.43699
$323$		−	13.9993i		−	0.0433414i
$324$		0			0
$325$	−612.387			−1.88427
$326$	645.098			1.97883
$327$		0			0
$328$			451.549i			1.37668i
$329$			517.022i			1.57150i
$330$		0			0
$331$		−	363.666i		−	1.09869i	−0.835596	−	0.549345i	$-0.814877\pi$
$331$		−	363.666i		−	1.09869i	0.835596	−	0.549345i	$-0.185123\pi$
$332$	−422.864			−1.27369
$333$		0			0
$334$		−	131.011i		−	0.392247i
$335$		−	347.668i		−	1.03782i
$336$		0			0
$337$	8.64293			0.0256467			0.0128233	−	0.999918i	$-0.495918\pi$
$337$	8.64293			0.0256467			0.0128233	+	0.999918i	$0.495918\pi$
$338$		−	511.626i		−	1.51369i
$339$		0			0
$340$			53.7530i			0.158097i
$341$	−31.6140			−0.0927098
$342$		0			0
$343$			46.7084i			0.136176i
$344$	−249.422	+	163.823i	−0.725063	+	0.476231i
$345$		0			0
$346$		−	76.8788i		−	0.222193i
$347$		−	278.736i		−	0.803273i	−0.915799	−	0.401636i	$-0.868442\pi$
$347$		−	278.736i		−	0.803273i	0.915799	−	0.401636i	$-0.131558\pi$
$348$		0			0
$349$			169.755i			0.486403i	0.969976	+	0.243201i	$0.0781976\pi$
$349$			169.755i			0.486403i	−0.969976	+	0.243201i	$0.921802\pi$
$350$	1090.30			3.11515
$351$		0			0
$352$			87.5219i			0.248642i
$353$	−26.8357			−0.0760219			−0.0380109	−	0.999277i	$-0.512102\pi$
$353$	−26.8357			−0.0760219			−0.0380109	+	0.999277i	$0.512102\pi$
$354$		0			0
$355$	−488.971			−1.37738
$356$		−	36.4624i		−	0.102422i
$357$		0			0
$358$	264.973			0.740148
$359$	−169.169			−0.471223			−0.235611	−	0.971847i	$-0.575709\pi$
$359$	−169.169			−0.471223			−0.235611	+	0.971847i	$0.575709\pi$
$360$		0			0
$361$	209.048			0.579081
$362$			358.140i			0.989336i
$363$		0			0
$364$			1135.34i			3.11907i
$365$	67.4651			0.184836
$366$		0			0
$367$	−540.524			−1.47282			−0.736409	−	0.676536i	$-0.763480\pi$
$367$	−540.524			−1.47282			−0.736409	+	0.676536i	$0.763480\pi$
$368$	−62.2862			−0.169256
$369$		0			0
$370$		−	1014.90i		−	2.74297i
$371$			569.316i			1.53455i
$372$		0			0
$373$		−	26.7152i		−	0.0716226i	−0.999359	−	0.0358113i	$-0.988598\pi$
$373$		−	26.7152i		−	0.0716226i	0.999359	−	0.0358113i	$-0.0114015\pi$
$374$		−	8.82188i		−	0.0235879i
$375$		0			0
$376$		−	354.209i		−	0.942044i
$377$			644.688i			1.71005i
$378$		0			0
$379$	315.193			0.831644			0.415822	−	0.909446i	$-0.363494\pi$
$379$	315.193			0.831644			0.415822	+	0.909446i	$0.363494\pi$
$380$	583.449			1.53539
$381$		0			0
$382$	−139.424			−0.364984
$383$			131.099i			0.342295i	0.985245	+	0.171148i	$0.0547475\pi$
$383$			131.099i			0.342295i	−0.985245	+	0.171148i	$0.945252\pi$
$384$		0			0
$385$	−189.061			−0.491066
$386$		−	753.139i		−	1.95114i
$387$		0			0
$388$	621.610			1.60209
$389$			110.050i			0.282904i	0.989945	+	0.141452i	$0.0451771\pi$
$389$			110.050i			0.282904i	−0.989945	+	0.141452i	$0.954823\pi$
$390$		0			0
$391$	27.5795			0.0705358
$392$		−	372.050i		−	0.949106i
$393$		0			0
$394$			1116.90i			2.83476i
$395$		−	244.106i		−	0.617989i
$396$		0			0
$397$	68.3215			0.172094			0.0860472	−	0.996291i	$-0.472576\pi$
$397$	68.3215			0.172094			0.0860472	+	0.996291i	$0.472576\pi$
$398$	−1127.87			−2.83383
$399$		0			0
$400$	86.5426			0.216356
$401$	675.508			1.68456			0.842279	−	0.539042i	$-0.181213\pi$
$401$	675.508			1.68456			0.842279	+	0.539042i	$0.181213\pi$
$402$		0			0
$403$	−235.616			−0.584655
$404$	84.6433			0.209513
$405$		0			0
$406$		−	1147.81i		−	2.82713i
$407$			101.088i			0.248373i
$408$		0			0
$409$		−	1.76834i		−	0.00432356i	−0.999998	−	0.00216178i	$-0.999312\pi$
$409$		−	1.76834i		−	0.00432356i	0.999998	−	0.00216178i	$-0.000688116\pi$
$410$	−1590.79			−3.87997
$411$		0			0
$412$	593.100			1.43956
$413$		−	899.878i		−	2.17888i
$414$		0			0
$415$		−	524.808i		−	1.26460i
$416$			652.291i			1.56801i
$417$		0			0
$418$	−95.7549			−0.229079
$419$			444.028i			1.05973i	0.848081	+	0.529867i	$0.177758\pi$
$419$			444.028i			1.05973i	−0.848081	+	0.529867i	$0.822242\pi$
$420$		0			0
$421$			141.195i			0.335381i	0.985840	+	0.167690i	$0.0536309\pi$
$421$			141.195i			0.335381i	−0.985840	+	0.167690i	$0.946369\pi$
$422$	898.022			2.12801
$423$		0			0
$424$		−	390.035i		−	0.919894i
$425$	−38.3199			−0.0901645
$426$		0			0
$427$	−659.843			−1.54530
$428$	−12.8474			−0.0300173
$429$		0			0
$430$	−577.142	−	878.700i	−1.34219	−	2.04349i
$431$	−302.066			−0.700849			−0.350424	−	0.936591i	$-0.613963\pi$
$431$	−302.066			−0.700849			−0.350424	+	0.936591i	$0.613963\pi$
$432$		0			0
$433$			380.497i			0.878747i	0.898304	+	0.439374i	$0.144800\pi$
$433$			380.497i			0.878747i	−0.898304	+	0.439374i	$0.855200\pi$
$434$	419.495			0.966578
$435$		0			0
$436$	327.970			0.752224
$437$		−	299.355i		−	0.685022i
$438$		0			0
$439$	558.769			1.27282			0.636412	−	0.771350i	$-0.280418\pi$
$439$	558.769			1.27282			0.636412	+	0.771350i	$0.280418\pi$
$440$	129.524			0.294373
$441$		0			0
$442$		−	65.7485i		−	0.148752i
$443$	346.462			0.782082			0.391041	−	0.920373i	$-0.372115\pi$
$443$	346.462			0.782082			0.391041	+	0.920373i	$0.372115\pi$
$444$		0			0
$445$	45.2527			0.101691
$446$	303.661			0.680855
$447$		0			0
$448$		−	1057.43i		−	2.36033i
$449$		−	250.699i		−	0.558350i	−0.960240	−	0.279175i	$-0.909939\pi$
$449$		−	250.699i		−	0.558350i	0.960240	−	0.279175i	$-0.0900609\pi$
$450$		0			0
$451$	158.448			0.351327
$452$		−	178.344i		−	0.394567i
$453$		0			0
$454$	−339.665			−0.748161
$455$	−1409.05			−3.09681
$456$		0			0
$457$			745.356i			1.63098i	0.578773	+	0.815488i	$0.303532\pi$
$457$			745.356i			1.63098i	−0.578773	+	0.815488i	$0.696468\pi$
$458$		−	543.408i		−	1.18648i
$459$		0			0
$460$			1149.43i			2.49876i
$461$	−6.73204			−0.0146031			−0.00730156	−	0.999973i	$-0.502324\pi$
$461$	−6.73204			−0.0146031			−0.00730156	+	0.999973i	$0.502324\pi$
$462$		0			0
$463$			448.198i			0.968031i	0.875059	+	0.484016i	$0.160822\pi$
$463$			448.198i			0.968031i	−0.875059	+	0.484016i	$0.839178\pi$
$464$		−	91.1074i		−	0.196352i
$465$		0			0
$466$	−177.031			−0.379894
$467$			506.048i			1.08361i	0.840503	+	0.541807i	$0.182260\pi$
$467$			506.048i			1.08361i	−0.840503	+	0.541807i	$0.817740\pi$
$468$		0			0
$469$		−	459.502i		−	0.979749i
$470$	1247.86			2.65502
$471$		0			0
$472$			616.500i			1.30614i
$473$	57.4856	+	87.5219i	0.121534	+	0.185036i
$474$		0			0
$475$			415.934i			0.875650i
$476$			71.0436i			0.149251i
$477$		0			0
$478$		−	66.2529i		−	0.138604i
$479$	772.714			1.61318			0.806591	−	0.591110i	$-0.201310\pi$
$479$	772.714			1.61318			0.806591	+	0.591110i	$0.201310\pi$
$480$		0			0
$481$			753.397i			1.56631i
$482$	967.443			2.00714
$483$		0			0
$484$	710.620			1.46822
$485$			771.467i			1.59065i
$486$		0			0
$487$	−762.454			−1.56561			−0.782807	−	0.622265i	$-0.786213\pi$
$487$	−762.454			−1.56561			−0.782807	+	0.622265i	$0.786213\pi$
$488$	452.054			0.926340
$489$		0			0
$490$	1310.71			2.67492
$491$		−	928.398i		−	1.89083i	−0.325868	−	0.945415i	$-0.605657\pi$
$491$		−	928.398i		−	1.89083i	0.325868	−	0.945415i	$-0.394343\pi$
$492$		0			0
$493$			40.3411i			0.0818279i
$494$	−713.651			−1.44464
$495$		0			0
$496$	33.2973			0.0671317
$497$	−646.258			−1.30032
$498$		0			0
$499$		−	669.978i		−	1.34264i	−0.741167	−	0.671320i	$-0.765728\pi$
$499$		−	669.978i		−	1.34264i	0.741167	−	0.671320i	$-0.234272\pi$
$500$		−	413.774i		−	0.827548i
$501$		0			0
$502$		−	23.1418i		−	0.0460991i
$503$		−	274.168i		−	0.545065i	−0.962147	−	0.272532i	$-0.912139\pi$
$503$		−	274.168i		−	0.545065i	0.962147	−	0.272532i	$-0.0878612\pi$
$504$		0			0
$505$			105.049i			0.208018i
$506$		−	188.643i		−	0.372813i
$507$		0			0
$508$	−918.139			−1.80736
$509$	−703.988			−1.38308			−0.691540	−	0.722338i	$-0.743067\pi$
$509$	−703.988			−1.38308			−0.691540	+	0.722338i	$0.743067\pi$
$510$		0			0
$511$	89.1665			0.174494
$512$		−	163.380i		−	0.319101i
$513$		0			0
$514$	1011.06			1.96704
$515$			736.084i			1.42929i
$516$		0			0
$517$	−124.292			−0.240409
$518$		−	1341.36i		−	2.58950i
$519$		0			0
$520$	965.330			1.85640
$521$		−	588.189i		−	1.12896i	−0.825446	−	0.564480i	$-0.809077\pi$
$521$		−	588.189i		−	1.12896i	0.825446	−	0.564480i	$-0.190923\pi$
$522$		0			0
$523$		−	743.907i		−	1.42238i	−0.702998	−	0.711192i	$-0.748156\pi$
$523$		−	743.907i		−	1.42238i	0.702998	−	0.711192i	$-0.251844\pi$
$524$			970.882i			1.85283i
$525$		0			0
$526$	733.184			1.39389
$527$	−14.7436			−0.0279765
$528$		0			0
$529$	60.7485			0.114837
$530$	1374.07			2.59259
$531$		0			0
$532$	771.125			1.44948
$533$	1180.90			2.21557
$534$		0			0
$535$		−	15.9446i		−	0.0298030i
$536$			314.802i			0.587317i
$537$		0			0
$538$			909.506i			1.69053i
$539$	−130.552			−0.242212
$540$		0			0
$541$	−2.77685			−0.00513282			−0.00256641	−	0.999997i	$-0.500817\pi$
$541$	−2.77685			−0.00513282			−0.00256641	+	0.999997i	$0.500817\pi$
$542$		−	249.856i		−	0.460990i
$543$		0			0
$544$			40.8169i			0.0750311i
$545$			407.036i			0.746856i
$546$		0			0
$547$	700.748			1.28108			0.640538	−	0.767927i	$-0.278711\pi$
$547$	700.748			1.28108			0.640538	+	0.767927i	$0.278711\pi$
$548$			750.968i			1.37038i
$549$		0			0
$550$			262.108i			0.476559i
$551$	437.873			0.794688
$552$		0			0
$553$		−	322.627i		−	0.583412i
$554$	381.612			0.688830
$555$		0			0
$556$	172.293			0.309879
$557$	590.822			1.06072			0.530361	−	0.847772i	$-0.322056\pi$
$557$	590.822			1.06072			0.530361	+	0.847772i	$0.322056\pi$
$558$		0			0
$559$	428.434	+	652.291i	0.766429	+	1.16689i
$560$	199.127			0.355584
$561$		0			0
$562$		−	479.837i		−	0.853803i
$563$	306.648			0.544668			0.272334	−	0.962203i	$-0.412205\pi$
$563$	306.648			0.544668			0.272334	+	0.962203i	$0.412205\pi$
$564$		0			0
$565$	221.339			0.391751
$566$			833.027i			1.47178i
$567$		0			0
$568$	442.747			0.779484
$569$	5.67367			0.00997130			0.00498565	−	0.999988i	$-0.498413\pi$
$569$	5.67367			0.00997130			0.00498565	+	0.999988i	$0.498413\pi$
$570$		0			0
$571$		−	811.264i		−	1.42078i	−0.703810	−	0.710389i	$-0.748519\pi$
$571$		−	811.264i		−	1.42078i	0.703810	−	0.710389i	$-0.251481\pi$
$572$	−272.935			−0.477159
$573$		0			0
$574$	−2102.49			−3.66288
$575$	−819.417			−1.42507
$576$		0			0
$577$		−	976.820i		−	1.69293i	−0.532445	−	0.846464i	$-0.678727\pi$
$577$		−	976.820i		−	1.69293i	0.532445	−	0.846464i	$-0.321273\pi$
$578$			917.771i			1.58784i
$579$		0			0
$580$	−1681.30			−2.89879
$581$		−	693.622i		−	1.19384i
$582$		0			0
$583$	−136.863			−0.234757
$584$	−61.0874			−0.104602
$585$		0			0
$586$			1143.29i			1.95101i
$587$		−	681.557i		−	1.16109i	−0.814230	−	0.580543i	$-0.802840\pi$
$587$		−	681.557i		−	1.16109i	0.814230	−	0.580543i	$-0.197160\pi$
$588$		0			0
$589$			160.031i			0.271699i
$590$	−2171.90			−3.68119
$591$		0			0
$592$		−	106.470i		−	0.179848i
$593$			1030.13i			1.73715i	0.495561	+	0.868573i	$0.334962\pi$
$593$			1030.13i			1.73715i	−0.495561	+	0.868573i	$0.665038\pi$
$594$		0			0
$595$	−88.1708			−0.148186
$596$		−	1309.96i		−	2.19793i
$597$		0			0
$598$		−	1405.94i		−	2.35107i
$599$	−1028.30			−1.71669			−0.858345	−	0.513073i	$-0.828507\pi$
$599$	−1028.30			−1.71669			−0.858345	+	0.513073i	$0.828507\pi$
$600$		0			0
$601$			655.529i			1.09073i	0.838198	+	0.545365i	$0.183609\pi$
$601$			655.529i			1.09073i	−0.838198	+	0.545365i	$0.816391\pi$
$602$	−762.790	−	1161.35i	−1.26709	−	1.92915i
$603$		0			0
$604$		−	672.190i		−	1.11290i
$605$			881.935i			1.45774i
$606$		0			0
$607$		−	1016.42i		−	1.67450i	−0.546822	−	0.837249i	$-0.684163\pi$
$607$		−	1016.42i		−	1.67450i	0.546822	−	0.837249i	$-0.315837\pi$
$608$	443.037			0.728680
$609$		0			0
$610$			1592.56i			2.61076i
$611$	−926.332			−1.51609
$612$		0			0
$613$	−1029.51			−1.67946			−0.839728	−	0.543007i	$-0.817286\pi$
$613$	−1029.51			−1.67946			−0.839728	+	0.543007i	$0.817286\pi$
$614$		−	1626.00i		−	2.64820i
$615$		0			0
$616$	171.188			0.277903
$617$	−426.603			−0.691415			−0.345708	−	0.938342i	$-0.612361\pi$
$617$	−426.603			−0.691415			−0.345708	+	0.938342i	$0.612361\pi$
$618$		0			0
$619$	−503.515			−0.813432			−0.406716	−	0.913555i	$-0.633326\pi$
$619$	−503.515			−0.813432			−0.406716	+	0.913555i	$0.633326\pi$
$620$		−	614.470i		−	0.991080i
$621$		0			0
$622$		−	88.6222i		−	0.142479i
$623$	59.8090			0.0960016
$624$		0			0
$625$	−330.025			−0.528041
$626$	123.030			0.196534
$627$		0			0
$628$			659.374i			1.04996i
$629$			47.1436i			0.0749500i
$630$		0			0
$631$		−	299.884i		−	0.475252i	−0.971357	−	0.237626i	$-0.923631\pi$
$631$		−	299.884i		−	0.475252i	0.971357	−	0.237626i	$-0.0763692\pi$
$632$			221.030i			0.349730i
$633$		0			0
$634$			273.996i			0.432170i
$635$		−	1139.48i		−	1.79446i
$636$		0			0
$637$	−972.990			−1.52746
$638$	275.933			0.432497
$639$		0			0
$640$	−1450.30			−2.26610
$641$			724.346i			1.13002i	0.825082	+	0.565012i	$0.191129\pi$
$641$			724.346i			1.13002i	−0.825082	+	0.565012i	$0.808871\pi$
$642$		0			0
$643$	743.353			1.15607			0.578035	−	0.816012i	$-0.303820\pi$
$643$	743.353			1.15607			0.578035	+	0.816012i	$0.303820\pi$
$644$			1519.17i			2.35896i
$645$		0			0
$646$	−44.6565			−0.0691277
$647$			61.7177i			0.0953905i	0.998862	+	0.0476953i	$0.0151876\pi$
$647$			61.7177i			0.0953905i	−0.998862	+	0.0476953i	$0.984812\pi$
$648$		0			0
$649$	216.330			0.333328
$650$			1953.46i			3.00532i
$651$		0			0
$652$		−	1248.88i		−	1.91547i
$653$			422.488i			0.646996i	0.946229	+	0.323498i	$0.104859\pi$
$653$			422.488i			0.646996i	−0.946229	+	0.323498i	$0.895141\pi$
$654$		0			0
$655$	−1204.94			−1.83960
$656$	−166.885			−0.254398
$657$		0			0
$658$	1649.26			2.50647
$659$	620.921			0.942217			0.471108	−	0.882075i	$-0.343854\pi$
$659$	620.921			0.942217			0.471108	+	0.882075i	$0.343854\pi$
$660$		0			0
$661$	496.134			0.750581			0.375291	−	0.926907i	$-0.377543\pi$
$661$	496.134			0.750581			0.375291	+	0.926907i	$0.377543\pi$
$662$	−1160.06			−1.75236
$663$		0			0
$664$			475.196i			0.715656i
$665$			957.027i			1.43914i
$666$		0			0
$667$			862.638i			1.29331i
$668$	−253.631			−0.379687
$669$		0			0
$670$	−1109.03			−1.65527
$671$		−	158.625i		−	0.236402i
$672$		0			0
$673$		−	453.069i		−	0.673209i	−0.941646	−	0.336604i	$-0.890722\pi$
$673$		−	453.069i		−	0.673209i	0.941646	−	0.336604i	$-0.109278\pi$
$674$		−	27.5702i		−	0.0409053i
$675$		0			0
$676$	−990.487			−1.46522
$677$			233.751i			0.345275i	0.984985	+	0.172638i	$0.0552289\pi$
$677$			233.751i			0.345275i	−0.984985	+	0.172638i	$0.944771\pi$
$678$		0			0
$679$			1019.62i			1.50165i
$680$	60.4052			0.0888312
$681$		0			0
$682$			100.846i			0.147868i
$683$	−578.370			−0.846808			−0.423404	−	0.905941i	$-0.639165\pi$
$683$	−578.370			−0.846808			−0.423404	+	0.905941i	$0.639165\pi$
$684$		0			0
$685$	−932.011			−1.36060
$686$	148.996			0.217195
$687$		0			0
$688$	−60.5463	−	92.1819i	−0.0880034	−	0.133985i
$689$	−1020.03			−1.48044
$690$		0			0
$691$			127.474i			0.184477i	0.995737	+	0.0922387i	$0.0294023\pi$
$691$			127.474i			0.184477i	−0.995737	+	0.0922387i	$0.970598\pi$
$692$	−148.834			−0.215078
$693$		0			0
$694$	−889.142			−1.28118
$695$			213.829i			0.307667i
$696$		0			0
$697$	73.8944			0.106018
$698$	541.502			0.775791
$699$		0			0
$700$		−	2110.78i		−	3.01541i
$701$	−126.686			−0.180721			−0.0903606	−	0.995909i	$-0.528802\pi$
$701$	−126.686			−0.180721			−0.0903606	+	0.995909i	$0.528802\pi$
$702$		0			0
$703$	511.708			0.727892
$704$	254.204			0.361085
$705$		0			0
$706$			85.6036i			0.121252i
$707$			138.840i			0.196379i
$708$		0			0
$709$	733.535			1.03461			0.517303	−	0.855803i	$-0.326936\pi$
$709$	733.535			1.03461			0.517303	+	0.855803i	$0.326936\pi$
$710$			1559.78i			2.19687i
$711$		0			0
$712$	−40.9748			−0.0575488
$713$	−315.271			−0.442175
$714$		0			0
$715$		−	338.734i		−	0.473753i
$716$		−	512.977i		−	0.716449i
$717$		0			0
$718$			539.634i			0.751580i
$719$	885.641			1.23177			0.615884	−	0.787837i	$-0.288799\pi$
$719$	885.641			1.23177			0.615884	+	0.787837i	$0.288799\pi$
$720$		0			0
$721$			972.859i			1.34932i
$722$		−	666.846i		−	0.923609i
$723$		0			0
$724$	693.344			0.957658
$725$		−	1198.58i		−	1.65321i
$726$		0			0
$727$		−	244.282i		−	0.336014i	−0.985786	−	0.168007i	$-0.946267\pi$
$727$		−	244.282i		−	0.336014i	0.985786	−	0.168007i	$-0.0537331\pi$
$728$	1275.85			1.75253
$729$		0			0
$730$		−	215.208i		−	0.294805i
$731$	26.8091	+	40.8169i	0.0366746	+	0.0558371i
$732$		0			0
$733$			1317.81i			1.79783i	0.438120	+	0.898917i	$0.355645\pi$
$733$			1317.81i			1.79783i	−0.438120	+	0.898917i	$0.644355\pi$
$734$			1724.23i			2.34908i
$735$		0			0
$736$			872.812i			1.18589i
$737$	110.464			0.149883
$738$		0			0
$739$			126.907i			0.171728i	0.996307	+	0.0858641i	$0.0273651\pi$
$739$			126.907i			0.171728i	−0.996307	+	0.0858641i	$0.972635\pi$
$740$	−1964.80			−2.65514
$741$		0			0
$742$	1816.07			2.44753
$743$		−	483.885i		−	0.651259i	−0.945497	−	0.325629i	$-0.894424\pi$
$743$		−	483.885i		−	0.651259i	0.945497	−	0.325629i	$-0.105576\pi$
$744$		0			0
$745$	1625.77			2.18224
$746$	−85.2192			−0.114235
$747$		0			0
$748$	−17.0788			−0.0228326
$749$		−	21.0735i		−	0.0281355i
$750$		0			0
$751$			11.4506i			0.0152471i	0.999971	+	0.00762354i	$0.00242667\pi$
$751$			11.4506i			0.0152471i	−0.999971	+	0.00762354i	$0.997573\pi$
$752$	130.909			0.174082
$753$		0			0
$754$	2056.50			2.72745
$755$	834.241			1.10495
$756$		0			0
$757$		−	995.145i		−	1.31459i	−0.753633	−	0.657295i	$-0.771701\pi$
$757$		−	995.145i		−	1.31459i	0.753633	−	0.657295i	$-0.228299\pi$
$758$		−	1005.44i		−	1.32644i
$759$		0			0
$760$		−	655.653i		−	0.862701i
$761$			1067.64i			1.40295i	0.712695	+	0.701474i	$0.247474\pi$
$761$			1067.64i			1.40295i	−0.712695	+	0.701474i	$0.752526\pi$
$762$		0			0
$763$			537.967i			0.705068i
$764$			269.919i			0.353297i
$765$		0			0
$766$	418.195			0.545946
$767$	1612.28			2.10206
$768$		0			0
$769$	−361.976			−0.470710			−0.235355	−	0.971909i	$-0.575625\pi$
$769$	−361.976			−0.470710			−0.235355	+	0.971909i	$0.575625\pi$
$770$			603.087i			0.783230i
$771$		0			0
$772$	−1458.05			−1.88866
$773$		−	579.373i		−	0.749513i	−0.927123	−	0.374756i	$-0.877726\pi$
$773$		−	579.373i		−	0.749513i	0.927123	−	0.374756i	$-0.122274\pi$
$774$		0			0
$775$	438.048			0.565224
$776$		−	698.537i		−	0.900176i
$777$		0			0
$778$	351.049			0.451220
$779$		−	802.068i		−	1.02961i
$780$		0			0
$781$		−	155.360i		−	0.198924i
$782$		−	87.9762i		−	0.112502i
$783$		0			0
$784$	137.503			0.175387
$785$	−818.335			−1.04246
$786$		0			0
$787$	−1046.34			−1.32954			−0.664768	−	0.747050i	$-0.731470\pi$
$787$	−1046.34			−1.32954			−0.664768	+	0.747050i	$0.731470\pi$
$788$	2162.27			2.74399
$789$		0			0
$790$	−778.676			−0.985666
$791$	292.537			0.369832
$792$		0			0
$793$		−	1182.22i		−	1.49082i
$794$		−	217.940i		−	0.274483i
$795$		0			0
$796$			2183.50i			2.74309i
$797$	959.219			1.20354			0.601769	−	0.798670i	$-0.294463\pi$
$797$	959.219			1.20354			0.601769	+	0.798670i	$0.294463\pi$
$798$		0			0
$799$	−57.9649			−0.0725468
$800$		−	1212.72i		−	1.51589i
$801$		0			0
$802$		−	2154.81i		−	2.68680i
$803$			21.4355i			0.0266943i
$804$		0			0
$805$	−1885.41			−2.34212
$806$			751.595i			0.932500i
$807$		0			0
$808$		−	95.1183i		−	0.117721i
$809$	−639.565			−0.790562			−0.395281	−	0.918560i	$-0.629353\pi$
$809$	−639.565			−0.790562			−0.395281	+	0.918560i	$0.629353\pi$
$810$		0			0
$811$		−	515.867i		−	0.636087i	−0.948076	−	0.318044i	$-0.896974\pi$
$811$		−	515.867i		−	0.636087i	0.948076	−	0.318044i	$-0.103026\pi$
$812$	−2222.12			−2.73660
$813$		0			0
$814$	322.461			0.396144
$815$	1549.96			1.90180
$816$		0			0
$817$	443.037	−	290.993i	0.542273	−	0.356172i
$818$	−5.64084			−0.00689589
$819$		0			0
$820$			3079.70i			3.75573i
$821$	−1288.57			−1.56951			−0.784756	−	0.619805i	$-0.787212\pi$
$821$	−1288.57			−1.56951			−0.784756	+	0.619805i	$0.787212\pi$
$822$		0			0
$823$	516.675			0.627795			0.313897	−	0.949457i	$-0.398365\pi$
$823$	516.675			0.627795			0.313897	+	0.949457i	$0.398365\pi$
$824$		−	666.499i		−	0.808858i
$825$		0			0
$826$	−2870.53			−3.47522
$827$	−35.6545			−0.0431131			−0.0215565	−	0.999768i	$-0.506862\pi$
$827$	−35.6545			−0.0431131			−0.0215565	+	0.999768i	$0.506862\pi$
$828$		0			0
$829$			617.780i			0.745212i	0.927990	+	0.372606i	$0.121536\pi$
$829$			617.780i			0.745212i	−0.927990	+	0.372606i	$0.878464\pi$
$830$	−1674.09			−2.01698
$831$		0			0
$832$	1894.56			2.27711
$833$	−60.8846			−0.0730907
$834$		0			0
$835$		−	314.776i		−	0.376978i
$836$			185.378i			0.221744i
$837$		0			0
$838$	1416.41			1.69023
$839$			98.1185i			0.116947i	0.998289	+	0.0584735i	$0.0186233\pi$
$839$			98.1185i			0.116947i	−0.998289	+	0.0584735i	$0.981377\pi$
$840$		0			0
$841$	−420.800			−0.500357
$842$	450.401			0.534918
$843$		0			0
$844$		−	1738.53i		−	2.05987i
$845$		−	1229.27i		−	1.45476i
$846$		0			0
$847$			1165.63i			1.37618i
$848$	144.150			0.169988
$849$		0			0
$850$			122.237i			0.143808i
$851$			1008.10i			1.18460i
$852$		0			0
$853$	564.129			0.661347			0.330674	−	0.943745i	$-0.392724\pi$
$853$	564.129			0.661347			0.330674	+	0.943745i	$0.392724\pi$
$854$			2104.84i			2.46468i
$855$		0			0
$856$			14.4373i			0.0168660i
$857$	699.603			0.816340			0.408170	−	0.912906i	$-0.366167\pi$
$857$	699.603			0.816340			0.408170	+	0.912906i	$0.366167\pi$
$858$		0			0
$859$			1395.67i			1.62476i	0.583128	+	0.812380i	$0.301829\pi$
$859$			1395.67i			1.62476i	−0.583128	+	0.812380i	$0.698171\pi$
$860$	−1701.13	+	1117.32i	−1.97806	+	1.29921i
$861$		0			0
$862$			963.564i			1.11782i
$863$		−	768.514i		−	0.890515i	−0.895403	−	0.445257i	$-0.853112\pi$
$863$		−	768.514i		−	0.890515i	0.895403	−	0.445257i	$-0.146888\pi$
$864$		0			0
$865$		−	184.715i		−	0.213543i
$866$	1213.75			1.40156
$867$		0			0
$868$		−	812.125i		−	0.935628i
$869$	77.5591			0.0892510
$870$		0			0
$871$	823.275			0.945206
$872$		−	368.558i		−	0.422658i
$873$		0			0
$874$	−954.916			−1.09258
$875$	678.711			0.775670
$876$		0			0
$877$	1318.08			1.50294			0.751470	−	0.659767i	$-0.229345\pi$
$877$	1318.08			1.50294			0.751470	+	0.659767i	$0.229345\pi$
$878$		−	1782.43i		−	2.03010i
$879$		0			0
$880$			47.8699i			0.0543976i
$881$	691.467			0.784867			0.392433	−	0.919780i	$-0.371633\pi$
$881$	691.467			0.784867			0.392433	+	0.919780i	$0.371633\pi$
$882$		0			0
$883$	−173.332			−0.196299			−0.0981493	−	0.995172i	$-0.531292\pi$
$883$	−173.332			−0.196299			−0.0981493	+	0.995172i	$0.531292\pi$
$884$	−127.287			−0.143989
$885$		0			0
$886$		−	1105.19i		−	1.24739i
$887$		−	142.316i		−	0.160446i	−0.996777	−	0.0802232i	$-0.974437\pi$
$887$		−	142.316i		−	0.160446i	0.996777	−	0.0802232i	$-0.0255633\pi$
$888$		0			0
$889$		−	1506.02i		−	1.69406i
$890$		−	144.352i		−	0.162193i
$891$		0			0
$892$		−	587.876i		−	0.659054i
$893$			629.166i			0.704553i
$894$		0			0
$895$	636.645			0.711335
$896$	−1916.82			−2.13931
$897$		0			0
$898$	−799.708			−0.890543
$899$		−	461.154i		−	0.512963i
$900$		0			0
$901$	−63.8278			−0.0708410
$902$		−	505.437i		−	0.560351i
$903$		0			0
$904$	−200.415			−0.221698
$905$			860.495i			0.950823i
$906$		0			0
$907$	1795.52			1.97963			0.989814	−	0.142370i	$-0.0454721\pi$
$907$	1795.52			1.97963			0.989814	+	0.142370i	$0.0454721\pi$
$908$			657.578i			0.724205i
$909$		0			0
$910$			4494.74i			4.93928i
$911$			365.632i			0.401353i	0.979658	+	0.200676i	$0.0643140\pi$
$911$			365.632i			0.401353i	−0.979658	+	0.200676i	$0.935686\pi$
$912$		0			0
$913$	166.746			0.182635
$914$	2377.62			2.60134
$915$		0			0
$916$	−1052.02			−1.14849
$917$	−1592.53			−1.73668
$918$		0			0
$919$	1083.62			1.17913			0.589566	−	0.807720i	$-0.299299\pi$
$919$	1083.62			1.17913			0.589566	+	0.807720i	$0.299299\pi$
$920$	1291.68			1.40400
$921$		0			0
$922$			21.4746i			0.0232913i
$923$		−	1157.88i		−	1.25447i
$924$		0			0
$925$		−	1400.69i		−	1.51426i
$926$	1429.71			1.54397
$927$		0			0
$928$	−1276.68			−1.37573
$929$			1400.56i			1.50760i	0.657105	+	0.753799i	$0.271781\pi$
$929$			1400.56i			1.50760i	−0.657105	+	0.753799i	$0.728219\pi$
$930$		0			0
$931$			660.856i			0.709835i
$932$			342.724i			0.367730i
$933$		0			0
$934$	1614.25			1.72832
$935$		−	21.1962i		−	0.0226697i
$936$		0			0
$937$		−	846.473i		−	0.903387i	−0.892173	−	0.451693i	$-0.850820\pi$
$937$		−	846.473i		−	0.903387i	0.892173	−	0.451693i	$-0.149180\pi$
$938$	−1465.77			−1.56266
$939$		0			0
$940$		−	2415.81i		−	2.57001i
$941$	1354.63			1.43956			0.719780	−	0.694203i	$-0.244243\pi$
$941$	1354.63			1.43956			0.719780	+	0.694203i	$0.244243\pi$
$942$		0			0
$943$	1580.13			1.67564
$944$	−227.848			−0.241364
$945$		0			0
$946$	279.187	−	183.374i	0.295124	−	0.193841i
$947$	209.288			0.221001			0.110501	−	0.993876i	$-0.464755\pi$
$947$	209.288			0.221001			0.110501	+	0.993876i	$0.464755\pi$
$948$		0			0
$949$			159.757i			0.168342i
$950$	1326.79			1.39662
$951$		0			0
$952$	79.8356			0.0838610
$953$			293.406i			0.307876i	0.988080	+	0.153938i	$0.0491957\pi$
$953$			293.406i			0.307876i	−0.988080	+	0.153938i	$0.950804\pi$
$954$		0			0
$955$	−334.991			−0.350776
$956$	−128.263			−0.134166
$957$		0			0
$958$		−	2464.89i		−	2.57295i
$959$	−1231.81			−1.28447
$960$		0			0
$961$	−792.461			−0.824621
$962$	2403.27			2.49820
$963$		0			0
$964$		−	1872.93i		−	1.94288i
$965$		−	1809.55i		−	1.87518i
$966$		0			0
$967$	−1028.49			−1.06359			−0.531794	−	0.846874i	$-0.678482\pi$
$967$	−1028.49			−1.06359			−0.531794	+	0.846874i	$0.678482\pi$
$968$		−	798.562i		−	0.824961i
$969$		0			0
$970$	2460.91			2.53702
$971$	438.558			0.451656			0.225828	−	0.974167i	$-0.427491\pi$
$971$	438.558			0.451656			0.225828	+	0.974167i	$0.427491\pi$
$972$		0			0
$973$			282.611i			0.290453i
$974$			2432.16i			2.49709i
$975$		0			0
$976$			167.071i			0.171180i
$977$	626.212			0.640954			0.320477	−	0.947256i	$-0.396157\pi$
$977$	626.212			0.640954			0.320477	+	0.947256i	$0.396157\pi$
$978$		0			0
$979$			14.3780i			0.0146864i
$980$		−	2537.49i		−	2.58927i
$981$		0			0
$982$	−2961.51			−3.01579
$983$		−	1720.98i		−	1.75074i	−0.483454	−	0.875370i	$-0.660618\pi$
$983$		−	1720.98i		−	1.75074i	0.483454	−	0.875370i	$-0.339382\pi$
$984$		0			0
$985$			2683.54i			2.72441i
$986$	128.685			0.130512
$987$		0			0
$988$			1381.60i			1.39838i
$989$	573.275	+	872.812i	0.579651	+	0.882520i
$990$		0			0
$991$			753.187i			0.760027i	0.924981	+	0.380014i	$0.124081\pi$
$991$			753.187i			0.760027i	−0.924981	+	0.380014i	$0.875919\pi$
$992$		−	466.593i		−	0.470356i
$993$		0			0
$994$			2061.51i			2.07395i
$995$	−2709.90			−2.72352
$996$		0			0
$997$		−	533.031i		−	0.534635i	−0.963609	−	0.267317i	$-0.913863\pi$
$997$		−	533.031i		−	0.534635i	0.963609	−	0.267317i	$-0.0861372\pi$
$998$	−2137.17			−2.14145
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.3.b.c.343.1		6
3.2	odd	2		43.3.b.b.42.6	yes	6
12.11	even	2		688.3.b.e.257.4		6
43.42	odd	2	inner	387.3.b.c.343.6		6
129.128	even	2		43.3.b.b.42.1	✓	6
516.515	odd	2		688.3.b.e.257.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.b.b.42.1	✓	6	129.128	even	2
43.3.b.b.42.6	yes	6	3.2	odd	2
387.3.b.c.343.1		6	1.1	even	1	trivial
387.3.b.c.343.6		6	43.42	odd	2	inner
688.3.b.e.257.3		6	516.515	odd	2
688.3.b.e.257.4		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.18991i q^{2} -6.17554 q^{4} -7.66434i q^{5} -10.1297i q^{7} +6.93980i q^{8} +O(q^{10})\)	\(q-3.18991i q^{2} -6.17554 q^{4} -7.66434i q^{5} -10.1297i q^{7} +6.93980i q^{8} -24.4486 q^{10} +2.43517 q^{11} +18.1491 q^{13} -32.3129 q^{14} -2.56483 q^{16} +1.13567 q^{17} -12.3269i q^{19} +47.3314i q^{20} -7.76798i q^{22} +24.2847 q^{23} -33.7420 q^{25} -57.8939i q^{26} +62.5565i q^{28} +35.5218i q^{29} -12.9823 q^{31} +35.9408i q^{32} -3.62270i q^{34} -77.6375 q^{35} +41.5116i q^{37} -39.3216 q^{38} +53.1889 q^{40} +65.0667 q^{41} +(23.6064 + 35.9408i) q^{43} -15.0385 q^{44} -77.4662i q^{46} -51.0402 q^{47} -53.6110 q^{49} +107.634i q^{50} -112.080 q^{52} -56.2026 q^{53} -18.6640i q^{55} +70.2981 q^{56} +113.312 q^{58} +88.8355 q^{59} -65.1393i q^{61} +41.4123i q^{62} +104.389 q^{64} -139.101i q^{65} +45.3618 q^{67} -7.01339 q^{68} +247.657i q^{70} -63.7983i q^{71} +8.80247i q^{73} +132.418 q^{74} +76.1251i q^{76} -24.6676i q^{77} +31.8496 q^{79} +19.6577i q^{80} -207.557i q^{82} +68.4740 q^{83} -8.70417i q^{85} +(114.648 - 75.3023i) q^{86} +16.8996i q^{88} +5.90432i q^{89} -183.845i q^{91} -149.971 q^{92} +162.814i q^{94} -94.4773 q^{95} -100.657 q^{97} +171.015i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4}+O(q^{10}) \) 6 * q - 16 * q^4	\( 6 q - 16 q^{4} - 2 q^{10} - 38 q^{11} + 30 q^{13} - 36 q^{14} - 68 q^{16} + 20 q^{17} + 80 q^{23} - 84 q^{25} - 112 q^{31} - 208 q^{35} - 170 q^{38} + 206 q^{40} + 172 q^{41} + 10 q^{43} + 36 q^{44} - 30 q^{47} - 6 q^{49} - 120 q^{52} + 110 q^{53} + 264 q^{56} + 430 q^{58} + 12 q^{59} + 100 q^{64} - 70 q^{67} + 50 q^{68} + 50 q^{74} + 178 q^{79} - 10 q^{83} + 372 q^{86} - 150 q^{92} + 130 q^{95} - 380 q^{97}+O(q^{100}) \) 6 * q - 16 * q^4 - 2 * q^10 - 38 * q^11 + 30 * q^13 - 36 * q^14 - 68 * q^16 + 20 * q^17 + 80 * q^23 - 84 * q^25 - 112 * q^31 - 208 * q^35 - 170 * q^38 + 206 * q^40 + 172 * q^41 + 10 * q^43 + 36 * q^44 - 30 * q^47 - 6 * q^49 - 120 * q^52 + 110 * q^53 + 264 * q^56 + 430 * q^58 + 12 * q^59 + 100 * q^64 - 70 * q^67 + 50 * q^68 + 50 * q^74 + 178 * q^79 - 10 * q^83 + 372 * q^86 - 150 * q^92 + 130 * q^95 - 380 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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