LMFDB - Embedded newform 333.6.c.d.73.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [333,6,Mod(73,333)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(333, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("333.73");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 333 = 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	333.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:	$53.4078119977$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + \cdots + 178006118400 $ x^16 + 390x^14 + 60701x^12 + 4799932x^10 + 203487156x^8 + 4519465040x^6 + 48993644736x^4 + 211923220224*x^2 + 178006118400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			73.7
Root			$-2.85171i$ of defining polynomial
Character	$\chi$	$=$	333.73
Dual form			333.6.c.d.73.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85171i q^{2} +23.8677 q^{4} -38.2713i q^{5} -96.3094 q^{7} -159.319i q^{8} +O(q^{10})$	\(q-2.85171i q^{2} +23.8677 q^{4} -38.2713i q^{5} -96.3094 q^{7} -159.319i q^{8} -109.139 q^{10} +564.497 q^{11} +678.423i q^{13} +274.647i q^{14} +309.435 q^{16} +2074.76i q^{17} +2693.06i q^{19} -913.449i q^{20} -1609.78i q^{22} -3071.93i q^{23} +1660.31 q^{25} +1934.67 q^{26} -2298.69 q^{28} +2966.43i q^{29} -5199.33i q^{31} -5980.62i q^{32} +5916.63 q^{34} +3685.89i q^{35} +(7849.71 + 2779.56i) q^{37} +7679.83 q^{38} -6097.34 q^{40} +6499.88 q^{41} +8586.79i q^{43} +13473.3 q^{44} -8760.27 q^{46} +8498.42 q^{47} -7531.50 q^{49} -4734.72i q^{50} +16192.4i q^{52} -22725.8 q^{53} -21604.0i q^{55} +15343.9i q^{56} +8459.41 q^{58} -23622.9i q^{59} +14456.4i q^{61} -14827.0 q^{62} -7153.11 q^{64} +25964.1 q^{65} -4980.51 q^{67} +49519.8i q^{68} +10511.1 q^{70} +7644.87 q^{71} +76611.5 q^{73} +(7926.51 - 22385.1i) q^{74} +64277.1i q^{76} -54366.4 q^{77} -43134.5i q^{79} -11842.5i q^{80} -18535.8i q^{82} +48677.9 q^{83} +79403.8 q^{85} +24487.1 q^{86} -89935.0i q^{88} -46571.1i q^{89} -65338.5i q^{91} -73320.0i q^{92} -24235.1i q^{94} +103067. q^{95} +135193. i q^{97} +21477.7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 268 q^{4} + 190 q^{7}+O(q^{10}) $ 16 * q - 268 * q^4 + 190 * q^7	$ 16 q - 268 q^{4} + 190 q^{7} - 74 q^{10} + 1110 q^{11} + 2900 q^{16} - 12052 q^{25} - 4902 q^{26} - 16824 q^{28} + 20556 q^{34} - 11400 q^{37} - 12108 q^{38} + 16966 q^{40} - 3918 q^{41} - 125394 q^{44} + 17470 q^{46} - 3822 q^{47} - 32618 q^{49} + 24126 q^{53} - 164718 q^{58} + 81426 q^{62} + 158076 q^{64} - 98976 q^{65} + 23560 q^{67} - 222404 q^{70} + 50046 q^{71} - 196274 q^{73} - 141216 q^{74} + 239574 q^{77} + 215814 q^{83} - 346472 q^{85} - 197640 q^{86} + 132504 q^{95}+O(q^{100}) $ 16 * q - 268 * q^4 + 190 * q^7 - 74 * q^10 + 1110 * q^11 + 2900 * q^16 - 12052 * q^25 - 4902 * q^26 - 16824 * q^28 + 20556 * q^34 - 11400 * q^37 - 12108 * q^38 + 16966 * q^40 - 3918 * q^41 - 125394 * q^44 + 17470 * q^46 - 3822 * q^47 - 32618 * q^49 + 24126 * q^53 - 164718 * q^58 + 81426 * q^62 + 158076 * q^64 - 98976 * q^65 + 23560 * q^67 - 222404 * q^70 + 50046 * q^71 - 196274 * q^73 - 141216 * q^74 + 239574 * q^77 + 215814 * q^83 - 346472 * q^85 - 197640 * q^86 + 132504 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/333\mathbb{Z}\right)^\times$.

$n$	$38$	$298$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	2.85171i		−	0.504117i	−0.967712	−	0.252058i	$-0.918893\pi$
$2$		−	2.85171i		−	0.504117i	0.967712	−	0.252058i	$-0.0811075\pi$
$3$		0			0
$3$		0			0
$4$	23.8677			0.745866
$5$		−	38.2713i		−	0.684618i	−0.939587	−	0.342309i	$-0.888791\pi$
$5$		−	38.2713i		−	0.684618i	0.939587	−	0.342309i	$-0.111209\pi$
$6$		0			0
$7$	−96.3094			−0.742888			−0.371444	−	0.928455i	$-0.621137\pi$
$7$	−96.3094			−0.742888			−0.371444	+	0.928455i	$0.621137\pi$
$8$		−	159.319i		−	0.880120i
$9$		0			0
$10$	−109.139			−0.345127
$11$	564.497			1.40663			0.703315	−	0.710878i	$-0.251702\pi$
$11$	564.497			1.40663			0.703315	+	0.710878i	$0.251702\pi$
$12$		0			0
$13$			678.423i			1.11338i	0.830721	+	0.556688i	$0.187928\pi$
$13$			678.423i			1.11338i	−0.830721	+	0.556688i	$0.812072\pi$
$14$			274.647i			0.374502i
$15$		0			0
$16$	309.435			0.302183
$17$			2074.76i			1.74119i	0.492002	+	0.870594i	$0.336265\pi$
$17$			2074.76i			1.74119i	−0.492002	+	0.870594i	$0.663735\pi$
$18$		0			0
$19$			2693.06i			1.71144i	0.517440	+	0.855719i	$0.326885\pi$
$19$			2693.06i			1.71144i	−0.517440	+	0.855719i	$0.673115\pi$
$20$		−	913.449i		−	0.510634i
$21$		0			0
$22$		−	1609.78i		−	0.709106i
$23$		−	3071.93i		−	1.21085i	−0.795901	−	0.605427i	$-0.793002\pi$
$23$		−	3071.93i		−	1.21085i	0.795901	−	0.605427i	$-0.206998\pi$
$24$		0			0
$25$	1660.31			0.531298
$26$	1934.67			0.561272
$27$		0			0
$28$	−2298.69			−0.554095
$29$			2966.43i			0.654996i	0.944852	+	0.327498i	$0.106206\pi$
$29$			2966.43i			0.654996i	−0.944852	+	0.327498i	$0.893794\pi$
$30$		0			0
$31$		−	5199.33i		−	0.971725i	−0.874035	−	0.485863i	$-0.838506\pi$
$31$		−	5199.33i		−	0.971725i	0.874035	−	0.485863i	$-0.161494\pi$
$32$		−	5980.62i		−	1.03246i
$33$		0			0
$34$	5916.63			0.877762
$35$			3685.89i			0.508595i
$36$		0			0
$37$	7849.71	+	2779.56i	0.942648	+	0.333789i
$37$	7849.71	+	2779.56i	0.942648	+	0.333789i
$38$	7679.83			0.862765
$39$		0			0
$40$	−6097.34			−0.602546
$41$	6499.88			0.603873			0.301937	−	0.953328i	$-0.402367\pi$
$41$	6499.88			0.603873			0.301937	+	0.953328i	$0.402367\pi$
$42$		0			0
$43$			8586.79i			0.708206i	0.935206	+	0.354103i	$0.115214\pi$
$43$			8586.79i			0.708206i	−0.935206	+	0.354103i	$0.884786\pi$
$44$	13473.3			1.04916
$45$		0			0
$46$	−8760.27			−0.610412
$47$	8498.42			0.561169			0.280584	−	0.959829i	$-0.409472\pi$
$47$	8498.42			0.561169			0.280584	+	0.959829i	$0.409472\pi$
$48$		0			0
$49$	−7531.50			−0.448117
$50$		−	4734.72i		−	0.267836i
$51$		0			0
$52$			16192.4i			0.830430i
$53$	−22725.8			−1.11130			−0.555648	−	0.831418i	$-0.687530\pi$
$53$	−22725.8			−1.11130			−0.555648	+	0.831418i	$0.687530\pi$
$54$		0			0
$55$		−	21604.0i		−	0.963004i
$56$			15343.9i			0.653831i
$57$		0			0
$58$	8459.41			0.330195
$59$		−	23622.9i		−	0.883492i	−0.897140	−	0.441746i	$-0.854359\pi$
$59$		−	23622.9i		−	0.883492i	0.897140	−	0.441746i	$-0.145641\pi$
$60$		0			0
$61$			14456.4i			0.497435i	0.968576	+	0.248718i	$0.0800091\pi$
$61$			14456.4i			0.497435i	−0.968576	+	0.248718i	$0.919991\pi$
$62$	−14827.0			−0.489863
$63$		0			0
$64$	−7153.11			−0.218295
$65$	25964.1			0.762238
$66$		0			0
$67$	−4980.51			−0.135546			−0.0677730	−	0.997701i	$-0.521589\pi$
$67$	−4980.51			−0.135546			−0.0677730	+	0.997701i	$0.521589\pi$
$68$			49519.8i			1.29869i
$69$		0			0
$70$	10511.1			0.256391
$71$	7644.87			0.179980			0.0899900	−	0.995943i	$-0.471316\pi$
$71$	7644.87			0.179980			0.0899900	+	0.995943i	$0.471316\pi$
$72$		0			0
$73$	76611.5			1.68262			0.841312	−	0.540550i	$-0.181784\pi$
$73$	76611.5			1.68262			0.841312	+	0.540550i	$0.181784\pi$
$74$	7926.51	−	22385.1i	0.168268	−	0.475205i
$75$		0			0
$76$			64277.1i			1.27650i
$77$	−54366.4			−1.04497
$78$		0			0
$79$		−	43134.5i		−	0.777601i	−0.921322	−	0.388800i	$-0.872890\pi$
$79$		−	43134.5i		−	0.777601i	0.921322	−	0.388800i	$-0.127110\pi$
$80$		−	11842.5i		−	0.206880i
$81$		0			0
$82$		−	18535.8i		−	0.304423i
$83$	48677.9			0.775599			0.387799	−	0.921744i	$-0.373235\pi$
$83$	48677.9			0.775599			0.387799	+	0.921744i	$0.373235\pi$
$84$		0			0
$85$	79403.8			1.19205
$86$	24487.1			0.357019
$87$		0			0
$88$		−	89935.0i		−	1.23800i
$89$		−	46571.1i		−	0.623220i	−0.950210	−	0.311610i	$-0.899132\pi$
$89$		−	46571.1i		−	0.623220i	0.950210	−	0.311610i	$-0.100868\pi$
$90$		0			0
$91$		−	65338.5i		−	0.827114i
$92$		−	73320.0i		−	0.903135i
$93$		0			0
$94$		−	24235.1i		−	0.282894i
$95$	103067.			1.17168
$96$		0			0
$97$			135193.i			1.45889i	0.684038	+	0.729446i	$0.260222\pi$
$97$			135193.i			1.45889i	−0.684038	+	0.729446i	$0.739778\pi$
$98$			21477.7i			0.225903i
$99$		0			0
$100$	39627.7			0.396277
$101$	200856.			1.95921			0.979605	−	0.200933i	$-0.0643972\pi$
$101$	200856.			1.95921			0.979605	+	0.200933i	$0.0643972\pi$
$102$		0			0
$103$			62474.9i			0.580247i	0.956989	+	0.290123i	$0.0936963\pi$
$103$			62474.9i			0.580247i	−0.956989	+	0.290123i	$0.906304\pi$
$104$	108086.			0.979906
$105$		0			0
$106$			64807.6i			0.560223i
$107$	39477.9			0.333345			0.166673	−	0.986012i	$-0.446698\pi$
$107$	39477.9			0.333345			0.166673	+	0.986012i	$0.446698\pi$
$108$		0			0
$109$		−	117545.i		−	0.947628i	−0.880625	−	0.473814i	$-0.842877\pi$
$109$		−	117545.i		−	0.947628i	0.880625	−	0.473814i	$-0.157123\pi$
$110$	−61608.6			−0.485467
$111$		0			0
$112$	−29801.5			−0.224488
$113$			158580.i			1.16830i	0.811647	+	0.584148i	$0.198571\pi$
$113$			158580.i			1.16830i	−0.811647	+	0.584148i	$0.801429\pi$
$114$		0			0
$115$	−117567.			−0.828972
$116$			70801.9i			0.488540i
$117$		0			0
$118$	−67365.7			−0.445383
$119$		−	199819.i		−	1.29351i
$120$		0			0
$121$	157606.			0.978608
$122$	41225.6			0.250765
$123$		0			0
$124$		−	124096.i		−	0.724777i
$125$		−	183140.i		−	1.04835i
$126$		0			0
$127$	77966.9			0.428944			0.214472	−	0.976730i	$-0.431197\pi$
$127$	77966.9			0.428944			0.214472	+	0.976730i	$0.431197\pi$
$128$		−	170981.i		−	0.922409i
$129$		0			0
$130$		−	74042.3i		−	0.384257i
$131$		−	160530.i		−	0.817294i	−0.912693	−	0.408647i	$-0.866001\pi$
$131$		−	160530.i		−	0.817294i	0.912693	−	0.408647i	$-0.133999\pi$
$132$		0			0
$133$		−	259367.i		−	1.27141i
$134$			14203.0i			0.0683310i
$135$		0			0
$136$	330548.			1.53246
$137$	−51981.2			−0.236616			−0.118308	−	0.992977i	$-0.537747\pi$
$137$	−51981.2			−0.236616			−0.118308	+	0.992977i	$0.537747\pi$
$138$		0			0
$139$	−402175.			−1.76554			−0.882771	−	0.469803i	$-0.844325\pi$
$139$	−402175.			−1.76554			−0.882771	+	0.469803i	$0.844325\pi$
$140$			87973.7i			0.379344i
$141$		0			0
$142$		−	21801.0i		−	0.0907309i
$143$			382968.i			1.56611i
$144$		0			0
$145$	113529.			0.448422
$146$		−	218474.i		−	0.848239i
$147$		0			0
$148$	187355.	+	66341.8i	0.703089	+	0.248962i
$149$	−33622.9			−0.124071			−0.0620353	−	0.998074i	$-0.519759\pi$
$149$	−33622.9			−0.124071			−0.0620353	+	0.998074i	$0.519759\pi$
$150$		0			0
$151$	46126.5			0.164630			0.0823148	−	0.996606i	$-0.473769\pi$
$151$	46126.5			0.164630			0.0823148	+	0.996606i	$0.473769\pi$
$152$	429055.			1.50627
$153$		0			0
$154$			155037.i			0.526786i
$155$	−198985.			−0.665261
$156$		0			0
$157$	−189721.			−0.614281			−0.307140	−	0.951664i	$-0.599372\pi$
$157$	−189721.			−0.614281			−0.307140	+	0.951664i	$0.599372\pi$
$158$	−123007.			−0.392002
$159$		0			0
$160$	−228886.			−0.706838
$161$			295856.i			0.899529i
$162$		0			0
$163$		−	89298.7i		−	0.263255i	−0.991299	−	0.131627i	$-0.957980\pi$
$163$		−	89298.7i		−	0.263255i	0.991299	−	0.131627i	$-0.0420202\pi$
$164$	155137.			0.450409
$165$		0			0
$166$		−	138816.i		−	0.390992i
$167$		−	84828.3i		−	0.235369i	−0.993051	−	0.117685i	$-0.962453\pi$
$167$		−	84828.3i		−	0.235369i	0.993051	−	0.117685i	$-0.0375472\pi$
$168$		0			0
$169$	−88964.7			−0.239608
$170$		−	226437.i		−	0.600932i
$171$		0			0
$172$			204947.i			0.528227i
$173$	−352138.			−0.894536			−0.447268	−	0.894400i	$-0.647603\pi$
$173$	−352138.			−0.894536			−0.447268	+	0.894400i	$0.647603\pi$
$174$		0			0
$175$	−159903.			−0.394695
$176$	174675.			0.425059
$177$		0			0
$178$	−132808.			−0.314176
$179$		−	167857.i		−	0.391568i	−0.980647	−	0.195784i	$-0.937275\pi$
$179$		−	167857.i		−	0.391568i	0.980647	−	0.195784i	$-0.0627252\pi$
$180$		0			0
$181$	416443.			0.944842			0.472421	−	0.881373i	$-0.343380\pi$
$181$	416443.			0.944842			0.472421	+	0.881373i	$0.343380\pi$
$182$	−186327.			−0.416962
$183$		0			0
$184$	−489416.			−1.06570
$185$	106377.	−	300419.i	0.228518	−	0.645354i
$186$		0			0
$187$			1.17120e6i			2.44921i
$188$	202838.			0.418557
$189$		0			0
$190$		−	293917.i		−	0.590664i
$191$			713843.i			1.41586i	0.706284	+	0.707929i	$0.250370\pi$
$191$			713843.i			1.41586i	−0.706284	+	0.707929i	$0.749630\pi$
$192$		0			0
$193$			133580.i			0.258136i	0.991636	+	0.129068i	$0.0411986\pi$
$193$			133580.i			0.258136i	−0.991636	+	0.129068i	$0.958801\pi$
$194$	385531.			0.735452
$195$		0			0
$196$	−179760.			−0.334235
$197$	−637649.			−1.17062			−0.585310	−	0.810809i	$-0.699027\pi$
$197$	−637649.			−1.17062			−0.585310	+	0.810809i	$0.699027\pi$
$198$		0			0
$199$			363643.i			0.650942i	0.945552	+	0.325471i	$0.105523\pi$
$199$			363643.i			0.650942i	−0.945552	+	0.325471i	$0.894477\pi$
$200$		−	264518.i		−	0.467606i
$201$		0			0
$202$		−	572784.i		−	0.987671i
$203$		−	285695.i		−	0.486589i
$204$		0			0
$205$		−	248759.i		−	0.413423i
$206$	178161.			0.292512
$207$		0			0
$208$			209928.i			0.336443i
$209$			1.52022e6i			2.40736i
$210$		0			0
$211$	1.19996e6			1.85550			0.927752	−	0.373197i	$-0.121738\pi$
$211$	1.19996e6			1.85550			0.927752	+	0.373197i	$0.121738\pi$
$212$	−542413.			−0.828878
$213$		0			0
$214$		−	112580.i		−	0.168045i
$215$	328628.			0.484851
$216$		0			0
$217$			500745.i			0.721883i
$218$	−335205.			−0.477715
$219$		0			0
$220$		−	515639.i		−	0.718272i
$221$	−1.40757e6			−1.93860
$222$		0			0
$223$	−54778.0			−0.0737640			−0.0368820	−	0.999320i	$-0.511743\pi$
$223$	−54778.0			−0.0737640			−0.0368820	+	0.999320i	$0.511743\pi$
$224$			575990.i			0.766999i
$225$		0			0
$226$	452225.			0.588957
$227$			802415.i			1.03356i	0.856119	+	0.516778i	$0.172869\pi$
$227$			802415.i			1.03356i	−0.856119	+	0.516778i	$0.827131\pi$
$228$		0			0
$229$	710586.			0.895423			0.447711	−	0.894178i	$-0.352239\pi$
$229$	710586.			0.895423			0.447711	+	0.894178i	$0.352239\pi$
$230$			335267.i			0.417899i
$231$		0			0
$232$	472608.			0.576476
$233$	−307185.			−0.370689			−0.185345	−	0.982674i	$-0.559340\pi$
$233$	−307185.			−0.370689			−0.185345	+	0.982674i	$0.559340\pi$
$234$		0			0
$235$		−	325246.i		−	0.384186i
$236$		−	563824.i		−	0.658967i
$237$		0			0
$238$	−569827.			−0.652079
$239$		−	1.34121e6i		−	1.51881i	−0.650621	−	0.759403i	$-0.725491\pi$
$239$		−	1.34121e6i		−	1.51881i	0.650621	−	0.759403i	$-0.274509\pi$
$240$		0			0
$241$			1.02266e6i			1.13420i	0.823650	+	0.567099i	$0.191934\pi$
$241$			1.02266e6i			1.13420i	−0.823650	+	0.567099i	$0.808066\pi$
$242$		−	449447.i		−	0.493333i
$243$		0			0
$244$			345042.i			0.371020i
$245$			288241.i			0.306789i
$246$		0			0
$247$	−1.82703e6			−1.90548
$248$	−828352.			−0.855235
$249$		0			0
$250$	−522263.			−0.528493
$251$			25154.5i			0.0252018i	0.999921	+	0.0126009i	$0.00401110\pi$
$251$			25154.5i			0.0252018i	−0.999921	+	0.0126009i	$0.995989\pi$
$252$		0			0
$253$		−	1.73410e6i		−	1.70322i
$254$		−	222339.i		−	0.216238i
$255$		0			0
$256$	−716489.			−0.683298
$257$		−	815697.i		−	0.770364i	−0.922841	−	0.385182i	$-0.874139\pi$
$257$		−	815697.i		−	0.770364i	0.922841	−	0.385182i	$-0.125861\pi$
$258$		0			0
$259$	−756001.	−	267698.i	−0.700282	−	0.247968i
$260$	619705.			0.568527
$261$		0			0
$262$	−457786.			−0.412011
$263$	−323462.			−0.288359			−0.144180	−	0.989552i	$-0.546054\pi$
$263$	−323462.			−0.288359			−0.144180	+	0.989552i	$0.546054\pi$
$264$		0			0
$265$			869747.i			0.760813i
$266$	−739640.			−0.640938
$267$		0			0
$268$	−118873.			−0.101099
$269$	−866044.			−0.729725			−0.364863	−	0.931061i	$-0.618884\pi$
$269$	−866044.			−0.729725			−0.364863	+	0.931061i	$0.618884\pi$
$270$		0			0
$271$	120109.			0.0993463			0.0496732	−	0.998766i	$-0.484182\pi$
$271$	120109.			0.0993463			0.0496732	+	0.998766i	$0.484182\pi$
$272$			642004.i			0.526157i
$273$		0			0
$274$			148236.i			0.119282i
$275$	937238.			0.747340
$276$		0			0
$277$		−	1.46087e6i		−	1.14397i	−0.820265	−	0.571983i	$-0.806174\pi$
$277$		−	1.46087e6i		−	1.14397i	0.820265	−	0.571983i	$-0.193826\pi$
$278$			1.14689e6i			0.890039i
$279$		0			0
$280$	587231.			0.447625
$281$			755744.i			0.570964i	0.958384	+	0.285482i	$0.0921537\pi$
$281$			755744.i			0.570964i	−0.958384	+	0.285482i	$0.907846\pi$
$282$		0			0
$283$			2.59322e6i			1.92474i	0.271735	+	0.962372i	$0.412402\pi$
$283$			2.59322e6i			1.92474i	−0.271735	+	0.962372i	$0.587598\pi$
$284$	182466.			0.134241
$285$		0			0
$286$	1.09211e6			0.789502
$287$	−626000.			−0.448610
$288$		0			0
$289$	−2.88477e6			−2.03174
$290$		−	323753.i		−	0.226057i
$291$		0			0
$292$	1.82854e6			1.25501
$293$	433664.			0.295111			0.147555	−	0.989054i	$-0.452860\pi$
$293$	433664.			0.295111			0.147555	+	0.989054i	$0.452860\pi$
$294$		0			0
$295$	−904078.			−0.604854
$296$	442836.	−	1.25061e6i	0.293774	−	0.829644i
$297$		0			0
$298$			95882.8i			0.0625461i
$299$	2.08407e6			1.34814
$300$		0			0
$301$		−	826988.i		−	0.526118i
$302$		−	131540.i		−	0.0829925i
$303$		0			0
$304$			833326.i			0.517167i
$305$	553267.			0.340553
$306$		0			0
$307$	1.00344e6			0.607641			0.303820	−	0.952729i	$-0.401738\pi$
$307$	1.00344e6			0.607641			0.303820	+	0.952729i	$0.401738\pi$
$308$	−1.29760e6			−0.779407
$309$		0			0
$310$			567449.i			0.335369i
$311$			1.51587e6i			0.888714i	0.895850	+	0.444357i	$0.146568\pi$
$311$			1.51587e6i			0.888714i	−0.895850	+	0.444357i	$0.853432\pi$
$312$		0			0
$313$			141947.i			0.0818963i	0.999161	+	0.0409482i	$0.0130378\pi$
$313$			141947.i			0.0818963i	−0.999161	+	0.0409482i	$0.986962\pi$
$314$			541031.i			0.309669i
$315$		0			0
$316$		−	1.02952e6i		−	0.579986i
$317$	−1.48491e6			−0.829948			−0.414974	−	0.909833i	$-0.636209\pi$
$317$	−1.48491e6			−0.829948			−0.414974	+	0.909833i	$0.636209\pi$
$318$		0			0
$319$			1.67454e6i			0.921338i
$320$			273759.i			0.149449i
$321$		0			0
$322$	843696.			0.453468
$323$	−5.58745e6			−2.97994
$324$		0			0
$325$			1.12639e6i			0.591535i
$326$	−254655.			−0.132711
$327$		0			0
$328$		−	1.03555e6i		−	0.531481i
$329$	−818477.			−0.416886
$330$		0			0
$331$		−	2.66337e6i		−	1.33617i	−0.744085	−	0.668085i	$-0.767114\pi$
$331$		−	2.66337e6i		−	1.33617i	0.744085	−	0.668085i	$-0.232886\pi$
$332$	1.16183e6			0.578493
$333$		0			0
$334$	−241906.			−0.118654
$335$			190611.i			0.0927973i
$336$		0			0
$337$	−2.02951e6			−0.973458			−0.486729	−	0.873553i	$-0.661810\pi$
$337$	−2.02951e6			−0.973458			−0.486729	+	0.873553i	$0.661810\pi$
$338$			253702.i			0.120790i
$339$		0			0
$340$	1.89519e6			0.889109
$341$		−	2.93501e6i		−	1.36686i
$342$		0			0
$343$	2.34403e6			1.07579
$344$	1.36804e6			0.623307
$345$		0			0
$346$			1.00420e6i			0.450951i
$347$			3.16483e6i			1.41100i	0.708710	+	0.705500i	$0.249277\pi$
$347$			3.16483e6i			1.41100i	−0.708710	+	0.705500i	$0.750723\pi$
$348$		0			0
$349$	−1.43841e6			−0.632147			−0.316073	−	0.948735i	$-0.602365\pi$
$349$	−1.43841e6			−0.632147			−0.316073	+	0.948735i	$0.602365\pi$
$350$			455998.i			0.198972i
$351$		0			0
$352$		−	3.37604e6i		−	1.45228i
$353$			677495.i			0.289381i	0.989477	+	0.144690i	$0.0462186\pi$
$353$			677495.i			0.289381i	−0.989477	+	0.144690i	$0.953781\pi$
$354$		0			0
$355$		−	292579.i		−	0.123218i
$356$		−	1.11155e6i		−	0.464839i
$357$		0			0
$358$	−478681.			−0.197396
$359$	1.97222e6			0.807642			0.403821	−	0.914838i	$-0.367682\pi$
$359$	1.97222e6			0.807642			0.403821	+	0.914838i	$0.367682\pi$
$360$		0			0
$361$	−4.77645e6			−1.92902
$362$		−	1.18758e6i		−	0.476311i
$363$		0			0
$364$		−	1.55948e6i		−	0.616917i
$365$		−	2.93202e6i		−	1.15195i
$366$		0			0
$367$	−2.97537e6			−1.15312			−0.576562	−	0.817053i	$-0.695606\pi$
$367$	−2.97537e6			−1.15312			−0.576562	+	0.817053i	$0.695606\pi$
$368$		−	950563.i		−	0.365899i
$369$		0			0
$370$	−856709.	−	303358.i	−0.325334	−	0.115200i
$371$	2.18871e6			0.825569
$372$		0			0
$373$	−2.19422e6			−0.816597			−0.408299	−	0.912848i	$-0.633878\pi$
$373$	−2.19422e6			−0.816597			−0.408299	+	0.912848i	$0.633878\pi$
$374$	3.33992e6			1.23469
$375$		0			0
$376$		−	1.35396e6i		−	0.493896i
$377$	−2.01249e6			−0.729258
$378$		0			0
$379$	2.44142e6			0.873061			0.436530	−	0.899689i	$-0.356207\pi$
$379$	2.44142e6			0.873061			0.436530	+	0.899689i	$0.356207\pi$
$380$	2.45997e6			0.873918
$381$		0			0
$382$	2.03568e6			0.713757
$383$		−	685919.i		−	0.238933i	−0.992838	−	0.119466i	$-0.961882\pi$
$383$		−	685919.i		−	0.238933i	0.992838	−	0.119466i	$-0.0381184\pi$
$384$		0			0
$385$			2.08067e6i			0.715405i
$386$	380933.			0.130131
$387$		0			0
$388$			3.22674e6i			1.08814i
$389$		−	4.07598e6i		−	1.36571i	−0.730554	−	0.682855i	$-0.760738\pi$
$389$		−	4.07598e6i		−	1.36571i	0.730554	−	0.682855i	$-0.239262\pi$
$390$		0			0
$391$	6.37352e6			2.10832
$392$			1.19991e6i			0.394397i
$393$		0			0
$394$			1.81839e6i			0.590129i
$395$	−1.65081e6			−0.532360
$396$		0			0
$397$	862797.			0.274746			0.137373	−	0.990519i	$-0.456134\pi$
$397$	862797.			0.274746			0.137373	+	0.990519i	$0.456134\pi$
$398$	1.03701e6			0.328151
$399$		0			0
$400$	513757.			0.160549
$401$			121613.i			0.0377676i	0.999822	+	0.0188838i	$0.00601125\pi$
$401$			121613.i			0.0377676i	−0.999822	+	0.0188838i	$0.993989\pi$
$402$		0			0
$403$	3.52735e6			1.08190
$404$	4.79397e6			1.46131
$405$		0			0
$406$	−814720.			−0.245298
$407$	4.43114e6	+	1.56905e6i	1.32596	+	0.469517i
$408$		0			0
$409$		−	5.72992e6i		−	1.69372i	−0.531819	−	0.846858i	$-0.678491\pi$
$409$		−	5.72992e6i		−	1.69372i	0.531819	−	0.846858i	$-0.321509\pi$
$410$	−709390.			−0.208413
$411$		0			0
$412$			1.49113e6i			0.432786i
$413$			2.27510e6i			0.656336i
$414$		0			0
$415$		−	1.86297e6i		−	0.530989i
$416$	4.05739e6			1.14951
$417$		0			0
$418$	4.33524e6			1.21359
$419$	−5.11602e6			−1.42363			−0.711816	−	0.702366i	$-0.752127\pi$
$419$	−5.11602e6			−1.42363			−0.711816	+	0.702366i	$0.752127\pi$
$420$		0			0
$421$			972912.i			0.267527i	0.991013	+	0.133764i	$0.0427063\pi$
$421$			972912.i			0.267527i	−0.991013	+	0.133764i	$0.957294\pi$
$422$		−	3.42195e6i		−	0.935391i
$423$		0			0
$424$			3.62065e6i			0.978075i
$425$			3.44474e6i			0.925090i
$426$		0			0
$427$		−	1.39229e6i		−	0.369539i
$428$	942247.			0.248631
$429$		0			0
$430$		−	937152.i		−	0.244421i
$431$		−	2.25886e6i		−	0.585728i	−0.956154	−	0.292864i	$-0.905392\pi$
$431$		−	2.25886e6i		−	0.585728i	0.956154	−	0.292864i	$-0.0946084\pi$
$432$		0			0
$433$	4.76746e6			1.22199			0.610994	−	0.791635i	$-0.290770\pi$
$433$	4.76746e6			1.22199			0.610994	+	0.791635i	$0.290770\pi$
$434$	1.42798e6			0.363913
$435$		0			0
$436$		−	2.80553e6i		−	0.706804i
$437$	8.27288e6			2.07230
$438$		0			0
$439$		−	6.47953e6i		−	1.60466i	−0.596882	−	0.802329i	$-0.703594\pi$
$439$		−	6.47953e6i		−	1.60466i	0.596882	−	0.802329i	$-0.296406\pi$
$440$	−3.44193e6			−0.847560
$441$		0			0
$442$			4.01397e6i			0.977280i
$443$	−7.13256e6			−1.72678			−0.863389	−	0.504540i	$-0.831662\pi$
$443$	−7.13256e6			−1.72678			−0.863389	+	0.504540i	$0.831662\pi$
$444$		0			0
$445$	−1.78234e6			−0.426668
$446$			156211.i			0.0371856i
$447$		0			0
$448$	688911.			0.162169
$449$		−	6.31912e6i		−	1.47925i	−0.673021	−	0.739624i	$-0.735003\pi$
$449$		−	6.31912e6i		−	1.47925i	0.673021	−	0.739624i	$-0.264997\pi$
$450$		0			0
$451$	3.66916e6			0.849426
$452$			3.78495e6i			0.871392i
$453$		0			0
$454$	2.28826e6			0.521033
$455$	−2.50059e6			−0.566257
$456$		0			0
$457$			7.64055e6i			1.71133i	0.517529	+	0.855666i	$0.326852\pi$
$457$			7.64055e6i			1.71133i	−0.517529	+	0.855666i	$0.673148\pi$
$458$		−	2.02639e6i		−	0.451398i
$459$		0			0
$460$	−2.80605e6			−0.618303
$461$			3.92010e6i			0.859102i	0.903043	+	0.429551i	$0.141328\pi$
$461$			3.92010e6i			0.859102i	−0.903043	+	0.429551i	$0.858672\pi$
$462$		0			0
$463$		−	6.57986e6i		−	1.42647i	−0.700923	−	0.713237i	$-0.747228\pi$
$463$		−	6.57986e6i		−	1.42647i	0.700923	−	0.713237i	$-0.252772\pi$
$464$			917917.i			0.197929i
$465$		0			0
$466$			876004.i			0.186871i
$467$			3.88545e6i			0.824421i	0.911089	+	0.412211i	$0.135243\pi$
$467$			3.88545e6i			0.824421i	−0.911089	+	0.412211i	$0.864757\pi$
$468$		0			0
$469$	479670.			0.100696
$470$	−927508.			−0.193675
$471$		0			0
$472$	−3.76357e6			−0.777579
$473$			4.84722e6i			0.996184i
$474$		0			0
$475$			4.47130e6i			0.909284i
$476$		−	4.76922e6i		−	0.964784i
$477$		0			0
$478$	−3.82475e6			−0.765656
$479$		−	752587.i		−	0.149871i	−0.997188	−	0.0749355i	$-0.976125\pi$
$479$		−	752587.i		−	0.149871i	0.997188	−	0.0749355i	$-0.0238751\pi$
$480$		0			0
$481$	−1.88572e6	+	5.32543e6i	−0.371633	+	1.04952i
$482$	2.91634e6			0.571768
$483$		0			0
$484$	3.76169e6			0.729911
$485$	5.17400e6			0.998784
$486$		0			0
$487$			1.01221e7i			1.93396i	0.254846	+	0.966982i	$0.417975\pi$
$487$			1.01221e7i			1.93396i	−0.254846	+	0.966982i	$0.582025\pi$
$488$	2.30318e6			0.437803
$489$		0			0
$490$	821980.			0.154657
$491$	−3.12950e6			−0.585829			−0.292915	−	0.956139i	$-0.594625\pi$
$491$	−3.12950e6			−0.585829			−0.292915	+	0.956139i	$0.594625\pi$
$492$		0			0
$493$	−6.15463e6			−1.14047
$494$			5.21017e6i			0.960582i
$495$		0			0
$496$		−	1.60886e6i		−	0.293639i
$497$	−736273.			−0.133705
$498$		0			0
$499$			5032.18i			0.000904700i	1.00000		0.000452350i	$0.000143987\pi$
$499$			5032.18i			0.000904700i	−1.00000		0.000452350i	$0.999856\pi$
$500$		−	4.37113e6i		−	0.781932i
$501$		0			0
$502$	71733.5			0.0127046
$503$		−	1.88460e6i		−	0.332123i	−0.986115	−	0.166062i	$-0.946895\pi$
$503$		−	1.88460e6i		−	0.332123i	0.986115	−	0.166062i	$-0.0531050\pi$
$504$		0			0
$505$		−	7.68702e6i		−	1.34131i
$506$	−4.94515e6			−0.858624
$507$		0			0
$508$	1.86089e6			0.319935
$509$	6.71398e6			1.14864			0.574322	−	0.818629i	$-0.305266\pi$
$509$	6.71398e6			1.14864			0.574322	+	0.818629i	$0.305266\pi$
$510$		0			0
$511$	−7.37841e6			−1.25000
$512$		−	3.42818e6i		−	0.577948i
$513$		0			0
$514$	−2.32613e6			−0.388353
$515$	2.39100e6			0.397247
$516$		0			0
$517$	4.79733e6			0.789357
$518$	−763397.	+	2.15590e6i	−0.125005	+	0.353024i
$519$		0			0
$520$		−	4.13658e6i		−	0.670861i
$521$	3.33825e6			0.538796			0.269398	−	0.963029i	$-0.413175\pi$
$521$	3.33825e6			0.538796			0.269398	+	0.963029i	$0.413175\pi$
$522$		0			0
$523$			5.47242e6i			0.874832i	0.899259	+	0.437416i	$0.144106\pi$
$523$			5.47242e6i			0.874832i	−0.899259	+	0.437416i	$0.855894\pi$
$524$		−	3.83149e6i		−	0.609592i
$525$		0			0
$526$			922421.i			0.145367i
$527$	1.07874e7			1.69196
$528$		0			0
$529$	−3.00041e6			−0.466167
$530$	2.48027e6			0.383539
$531$		0			0
$532$		−	6.19049e6i		−	0.948300i
$533$			4.40967e6i			0.672338i
$534$		0			0
$535$		−	1.51087e6i		−	0.228214i
$536$			793489.i			0.119297i
$537$		0			0
$538$			2.46971e6i			0.367867i
$539$	−4.25151e6			−0.630335
$540$		0			0
$541$			850109.i			0.124877i	0.998049	+	0.0624383i	$0.0198877\pi$
$541$			850109.i			0.124877i	−0.998049	+	0.0624383i	$0.980112\pi$
$542$		−	342516.i		−	0.0500821i
$543$		0			0
$544$	1.24084e7			1.79770
$545$	−4.49860e6			−0.648763
$546$		0			0
$547$		−	4.42050e6i		−	0.631688i	−0.948811	−	0.315844i	$-0.897712\pi$
$547$		−	4.42050e6i		−	0.631688i	0.948811	−	0.315844i	$-0.102288\pi$
$548$	−1.24067e6			−0.176484
$549$		0			0
$550$		−	2.67274e6i		−	0.376747i
$551$	−7.98876e6			−1.12099
$552$		0			0
$553$			4.15425e6i			0.577670i
$554$	−4.16599e6			−0.576692
$555$		0			0
$556$	−9.59900e6			−1.31686
$557$		−	5.79127e6i		−	0.790926i	−0.918482	−	0.395463i	$-0.870584\pi$
$557$		−	5.79127e6i		−	0.790926i	0.918482	−	0.395463i	$-0.129416\pi$
$558$		0			0
$559$	−5.82547e6			−0.788500
$560$			1.14054e6i			0.153689i
$561$		0			0
$562$	2.15517e6			0.287833
$563$		−	5.05335e6i		−	0.671906i	−0.941879	−	0.335953i	$-0.890942\pi$
$563$		−	5.05335e6i		−	0.671906i	0.941879	−	0.335953i	$-0.109058\pi$
$564$		0			0
$565$	6.06907e6			0.799836
$566$	7.39512e6			0.970296
$567$		0			0
$568$		−	1.21797e6i		−	0.158404i
$569$			3.57229e6i			0.462558i	0.972888	+	0.231279i	$0.0742910\pi$
$569$			3.57229e6i			0.462558i	−0.972888	+	0.231279i	$0.925709\pi$
$570$		0			0
$571$	1.29233e7			1.65876			0.829380	−	0.558684i	$-0.188694\pi$
$571$	1.29233e7			1.65876			0.829380	+	0.558684i	$0.188694\pi$
$572$			9.14057e6i			1.16811i
$573$		0			0
$574$			1.78517e6i			0.226152i
$575$		−	5.10035e6i		−	0.643324i
$576$		0			0
$577$		−	1.61007e6i		−	0.201329i	−0.994920	−	0.100664i	$-0.967903\pi$
$577$		−	1.61007e6i		−	0.201329i	0.994920	−	0.100664i	$-0.0320968\pi$
$578$			8.22656e6i			1.02423i
$579$		0			0
$580$	2.70968e6			0.334463
$581$	−4.68814e6			−0.576183
$582$		0			0
$583$	−1.28287e7			−1.56318
$584$		−	1.22057e7i		−	1.48091i
$585$		0			0
$586$		−	1.23669e6i		−	0.148770i
$587$		−	4.70903e6i		−	0.564075i	−0.959403	−	0.282037i	$-0.908990\pi$
$587$		−	4.70903e6i		−	0.564075i	0.959403	−	0.282037i	$-0.0910102\pi$
$588$		0			0
$589$	1.40021e7			1.66305
$590$			2.57817e6i			0.304917i
$591$		0			0
$592$	2.42898e6	+	860094.i	0.284852	+	0.100865i
$593$	−6.00154e6			−0.700851			−0.350426	−	0.936591i	$-0.613963\pi$
$593$	−6.00154e6			−0.700851			−0.350426	+	0.936591i	$0.613963\pi$
$594$		0			0
$595$	−7.64733e6			−0.885559
$596$	−802501.			−0.0925401
$597$		0			0
$598$		−	5.94317e6i		−	0.679618i
$599$	−1.17352e7			−1.33636			−0.668179	−	0.744000i	$-0.732926\pi$
$599$	−1.17352e7			−1.33636			−0.668179	+	0.744000i	$0.732926\pi$
$600$		0			0
$601$	−9.96280e6			−1.12511			−0.562555	−	0.826760i	$-0.690182\pi$
$601$	−9.96280e6			−1.12511			−0.562555	+	0.826760i	$0.690182\pi$
$602$	−2.35833e6			−0.265225
$603$		0			0
$604$	1.10093e6			0.122792
$605$		−	6.03178e6i		−	0.669973i
$606$		0			0
$607$		−	1.17448e6i		−	0.129382i	−0.997905	−	0.0646908i	$-0.979394\pi$
$607$		−	1.17448e6i		−	0.129382i	0.997905	−	0.0646908i	$-0.0206061\pi$
$608$	1.61062e7			1.76699
$609$		0			0
$610$		−	1.57776e6i		−	0.171679i
$611$			5.76552e6i			0.624792i
$612$		0			0
$613$	−1.45455e7			−1.56343			−0.781715	−	0.623636i	$-0.785655\pi$
$613$	−1.45455e7			−1.56343			−0.781715	+	0.623636i	$0.785655\pi$
$614$		−	2.86153e6i		−	0.306322i
$615$		0			0
$616$			8.66158e6i			0.919699i
$617$	−8.19052e6			−0.866161			−0.433081	−	0.901355i	$-0.642573\pi$
$617$	−8.19052e6			−0.866161			−0.433081	+	0.901355i	$0.642573\pi$
$618$		0			0
$619$	595920.			0.0625117			0.0312558	−	0.999511i	$-0.490049\pi$
$619$	595920.			0.0625117			0.0312558	+	0.999511i	$0.490049\pi$
$620$	−4.74933e6			−0.496195
$621$		0			0
$622$	4.32284e6			0.448016
$623$			4.48524e6i			0.462983i
$624$		0			0
$625$	−1.82055e6			−0.186424
$626$	404791.			0.0412853
$627$		0			0
$628$	−4.52822e6			−0.458171
$629$	−5.76692e6	+	1.62863e7i	−0.581189	+	1.64133i
$630$		0			0
$631$			2.68979e6i			0.268934i	0.990918	+	0.134467i	$0.0429322\pi$
$631$			2.68979e6i			0.268934i	−0.990918	+	0.134467i	$0.957068\pi$
$632$	−6.87213e6			−0.684382
$633$		0			0
$634$			4.23453e6i			0.418390i
$635$		−	2.98390e6i		−	0.293663i
$636$		0			0
$637$		−	5.10954e6i		−	0.498923i
$638$	4.77531e6			0.464462
$639$		0			0
$640$	−6.54368e6			−0.631498
$641$	−1.40540e6			−0.135100			−0.0675498	−	0.997716i	$-0.521518\pi$
$641$	−1.40540e6			−0.135100			−0.0675498	+	0.997716i	$0.521518\pi$
$642$		0			0
$643$		−	1.06168e7i		−	1.01267i	−0.862337	−	0.506335i	$-0.831000\pi$
$643$		−	1.06168e7i		−	1.01267i	0.862337	−	0.506335i	$-0.169000\pi$
$644$			7.06140e6i			0.670928i
$645$		0			0
$646$			1.59338e7i			1.50224i
$647$		−	2.90444e6i		−	0.272773i	−0.990656	−	0.136387i	$-0.956451\pi$
$647$		−	2.90444e6i		−	0.272773i	0.990656	−	0.136387i	$-0.0435489\pi$
$648$		0			0
$649$		−	1.33350e7i		−	1.24275i
$650$	3.21214e6			0.298203
$651$		0			0
$652$		−	2.13136e6i		−	0.196353i
$653$		−	6.64094e6i		−	0.609462i	−0.952438	−	0.304731i	$-0.901433\pi$
$653$		−	6.64094e6i		−	0.609462i	0.952438	−	0.304731i	$-0.0985666\pi$
$654$		0			0
$655$	−6.14370e6			−0.559534
$656$	2.01129e6			0.182480
$657$		0			0
$658$			2.33406e6i			0.210159i
$659$	−1.05401e6			−0.0945431			−0.0472716	−	0.998882i	$-0.515053\pi$
$659$	−1.05401e6			−0.0945431			−0.0472716	+	0.998882i	$0.515053\pi$
$660$		0			0
$661$		−	6.02242e6i		−	0.536126i	−0.963401	−	0.268063i	$-0.913616\pi$
$661$		−	6.02242e6i		−	0.536126i	0.963401	−	0.268063i	$-0.0863836\pi$
$662$	−7.59518e6			−0.673586
$663$		0			0
$664$		−	7.75531e6i		−	0.682620i
$665$	−9.92630e6			−0.870429
$666$		0			0
$667$	9.11266e6			0.793105
$668$		−	2.02466e6i		−	0.175554i
$669$		0			0
$670$	543567.			0.0467807
$671$			8.16061e6i			0.699707i
$672$		0			0
$673$	6.71709e6			0.571667			0.285834	−	0.958279i	$-0.407730\pi$
$673$	6.71709e6			0.571667			0.285834	+	0.958279i	$0.407730\pi$
$674$			5.78759e6i			0.490736i
$675$		0			0
$676$	−2.12338e6			−0.178715
$677$	1.24184e7			1.04134			0.520670	−	0.853758i	$-0.325682\pi$
$677$	1.24184e7			1.04134			0.520670	+	0.853758i	$0.325682\pi$
$678$		0			0
$679$		−	1.30203e7i		−	1.08379i
$680$		−	1.26505e7i		−	1.04915i
$681$		0			0
$682$	−8.36981e6			−0.689056
$683$		−	1.47559e6i		−	0.121036i	−0.998167	−	0.0605180i	$-0.980725\pi$
$683$		−	1.47559e6i		−	0.121036i	0.998167	−	0.0605180i	$-0.0192753\pi$
$684$		0			0
$685$			1.98939e6i			0.161992i
$686$		−	6.68449e6i		−	0.542323i
$687$		0			0
$688$			2.65706e6i			0.214008i
$689$		−	1.54177e7i		−	1.23729i
$690$		0			0
$691$	−7.94552e6			−0.633034			−0.316517	−	0.948587i	$-0.602513\pi$
$691$	−7.94552e6			−0.633034			−0.316517	+	0.948587i	$0.602513\pi$
$692$	−8.40473e6			−0.667204
$693$		0			0
$694$	9.02520e6			0.711308
$695$			1.53918e7i			1.20872i
$696$		0			0
$697$			1.34857e7i			1.05146i
$698$			4.10192e6i			0.318676i
$699$		0			0
$700$	−3.81652e6			−0.294390
$701$			1.46257e7i			1.12414i	0.827088	+	0.562072i	$0.189996\pi$
$701$			1.46257e7i			1.12414i	−0.827088	+	0.562072i	$0.810004\pi$
$702$		0			0
$703$	−7.48551e6	+	2.11397e7i	−0.571259	+	1.61328i
$704$	−4.03791e6			−0.307061
$705$		0			0
$706$	1.93202e6			0.145882
$707$	−1.93443e7			−1.45547
$708$		0			0
$709$			2.36304e7i			1.76545i	0.469892	+	0.882724i	$0.344293\pi$
$709$			2.36304e7i			1.76545i	−0.469892	+	0.882724i	$0.655707\pi$
$710$	−834352.			−0.0621160
$711$		0			0
$712$	−7.41966e6			−0.548509
$713$	−1.59720e7			−1.17662
$714$		0			0
$715$	1.46567e7			1.07219
$716$		−	4.00637e6i		−	0.292057i
$717$		0			0
$718$		−	5.62421e6i		−	0.407146i
$719$	5.85300e6			0.422237			0.211118	−	0.977460i	$-0.432289\pi$
$719$	5.85300e6			0.422237			0.211118	+	0.977460i	$0.432289\pi$
$720$		0			0
$721$		−	6.01692e6i		−	0.431058i
$722$			1.36211e7i			0.972453i
$723$		0			0
$724$	9.93955e6			0.704726
$725$			4.92518e6i			0.347998i
$726$		0			0
$727$		−	636687.i		−	0.0446776i	−0.999750	−	0.0223388i	$-0.992889\pi$
$727$		−	636687.i		−	0.0446776i	0.999750	−	0.0223388i	$-0.00711126\pi$
$728$	−1.04097e7			−0.727960
$729$		0			0
$730$	−8.36129e6			−0.580719
$731$	−1.78155e7			−1.23312
$732$		0			0
$733$	1.67411e7			1.15087			0.575433	−	0.817849i	$-0.304833\pi$
$733$	1.67411e7			1.15087			0.575433	+	0.817849i	$0.304833\pi$
$734$			8.48491e6i			0.581309i
$735$		0			0
$736$	−1.83721e7			−1.25015
$737$	−2.81148e6			−0.190663
$738$		0			0
$739$	1.36468e7			0.919217			0.459609	−	0.888122i	$-0.347990\pi$
$739$	1.36468e7			0.919217			0.459609	+	0.888122i	$0.347990\pi$
$740$	2.53899e6	−	7.17031e6i	0.170444	−	0.481348i
$741$		0			0
$742$		−	6.24158e6i		−	0.416183i
$743$	1.59940e6			0.106288			0.0531440	−	0.998587i	$-0.483076\pi$
$743$	1.59940e6			0.106288			0.0531440	+	0.998587i	$0.483076\pi$
$744$		0			0
$745$			1.28679e6i			0.0849410i
$746$			6.25729e6i			0.411660i
$747$		0			0
$748$			2.79538e7i			1.82678i
$749$	−3.80209e6			−0.247638
$750$		0			0
$751$	−1.94616e6			−0.125915			−0.0629575	−	0.998016i	$-0.520053\pi$
$751$	−1.94616e6			−0.125915			−0.0629575	+	0.998016i	$0.520053\pi$
$752$	2.62971e6			0.169576
$753$		0			0
$754$			5.73906e6i			0.367631i
$755$		−	1.76532e6i		−	0.112708i
$756$		0			0
$757$		−	2.01297e7i		−	1.27673i	−0.769735	−	0.638364i	$-0.779611\pi$
$757$		−	2.01297e7i		−	1.27673i	0.769735	−	0.638364i	$-0.220389\pi$
$758$		−	6.96223e6i		−	0.440125i
$759$		0			0
$760$		−	1.64205e7i		−	1.03122i
$761$	−7.09820e6			−0.444310			−0.222155	−	0.975011i	$-0.571309\pi$
$761$	−7.09820e6			−0.444310			−0.222155	+	0.975011i	$0.571309\pi$
$762$		0			0
$763$			1.13207e7i			0.703982i
$764$			1.70378e7i			1.05604i
$765$		0			0
$766$	−1.95604e6			−0.120450
$767$	1.60263e7			0.983659
$768$		0			0
$769$		−	6.44116e6i		−	0.392779i	−0.980526	−	0.196389i	$-0.937078\pi$
$769$		−	6.44116e6i		−	0.392779i	0.980526	−	0.196389i	$-0.0629217\pi$
$770$	5.93348e6			0.360647
$771$		0			0
$772$			3.18826e6i			0.192535i
$773$	9.51628e6			0.572820			0.286410	−	0.958107i	$-0.407538\pi$
$773$	9.51628e6			0.572820			0.286410	+	0.958107i	$0.407538\pi$
$774$		0			0
$775$		−	8.63249e6i		−	0.516276i
$776$	2.15387e7			1.28400
$777$		0			0
$778$	−1.16235e7			−0.688477
$779$			1.75045e7i			1.03349i
$780$		0			0
$781$	4.31550e6			0.253165
$782$		−	1.81755e7i		−	1.06284i
$783$		0			0
$784$	−2.33051e6			−0.135413
$785$			7.26088e6i			0.420548i
$786$		0			0
$787$	−2.72531e7			−1.56848			−0.784241	−	0.620457i	$-0.786947\pi$
$787$	−2.72531e7			−1.56848			−0.784241	+	0.620457i	$0.786947\pi$
$788$	−1.52192e7			−0.873126
$789$		0			0
$790$			4.70765e6i			0.268371i
$791$		−	1.52728e7i		−	0.867913i
$792$		0			0
$793$	−9.80757e6			−0.553833
$794$		−	2.46045e6i		−	0.138504i
$795$		0			0
$796$			8.67932e6i			0.485516i
$797$		−	2.81998e7i		−	1.57253i	−0.617888	−	0.786266i	$-0.712012\pi$
$797$		−	2.81998e7i		−	1.57253i	0.617888	−	0.786266i	$-0.287988\pi$
$798$		0			0
$799$			1.76322e7i			0.977100i
$800$		−	9.92967e6i		−	0.548542i
$801$		0			0
$802$	346806.			0.0190393
$803$	4.32470e7			2.36683
$804$		0			0
$805$	1.13228e7			0.615834
$806$		−	1.00590e7i		−	0.545402i
$807$		0			0
$808$		−	3.20001e7i		−	1.72434i
$809$			1.93886e7i			1.04154i	0.853698	+	0.520768i	$0.174354\pi$
$809$			1.93886e7i			1.04154i	−0.853698	+	0.520768i	$0.825646\pi$
$810$		0			0
$811$	4.62132e6			0.246725			0.123363	−	0.992362i	$-0.460632\pi$
$811$	4.62132e6			0.246725			0.123363	+	0.992362i	$0.460632\pi$
$812$		−	6.81889e6i		−	0.362930i
$813$		0			0
$814$	4.47449e6	−	1.26363e7i	0.236692	−	0.668437i
$815$	−3.41758e6			−0.180229
$816$		0			0
$817$	−2.31247e7			−1.21205
$818$	−1.63401e7			−0.853831
$819$		0			0
$820$		−	5.93731e6i		−	0.308358i
$821$	2.53220e7			1.31111			0.655557	−	0.755145i	$-0.272434\pi$
$821$	2.53220e7			1.31111			0.655557	+	0.755145i	$0.272434\pi$
$822$		0			0
$823$	3.21181e7			1.65291			0.826457	−	0.563001i	$-0.190353\pi$
$823$	3.21181e7			1.65291			0.826457	+	0.563001i	$0.190353\pi$
$824$	9.95343e6			0.510687
$825$		0			0
$826$	6.48795e6			0.330870
$827$		−	1.20543e7i		−	0.612883i	−0.951889	−	0.306442i	$-0.900862\pi$
$827$		−	1.20543e7i		−	0.612883i	0.951889	−	0.306442i	$-0.0991384\pi$
$828$		0			0
$829$			1.95634e7i			0.988684i	0.869267	+	0.494342i	$0.164591\pi$
$829$			1.95634e7i			0.988684i	−0.869267	+	0.494342i	$0.835409\pi$
$830$	−5.31266e6			−0.267680
$831$		0			0
$832$		−	4.85283e6i		−	0.243045i
$833$		−	1.56261e7i		−	0.780256i
$834$		0			0
$835$	−3.24649e6			−0.161138
$836$			3.62842e7i			1.79557i
$837$		0			0
$838$			1.45894e7i			0.717676i
$839$	1.14371e7			0.560931			0.280466	−	0.959864i	$-0.409511\pi$
$839$	1.14371e7			0.560931			0.280466	+	0.959864i	$0.409511\pi$
$840$		0			0
$841$	1.17115e7			0.570980
$842$	2.77447e6			0.134865
$843$		0			0
$844$	2.86404e7			1.38396
$845$			3.40479e6i			0.164040i
$846$		0			0
$847$	−1.51789e7			−0.726996
$848$	−7.03217e6			−0.335815
$849$		0			0
$850$	9.82341e6			0.466353
$851$	8.53861e6	−	2.41138e7i	0.404169	−	1.14141i
$852$		0			0
$853$			1.88357e7i			0.886359i	0.896433	+	0.443180i	$0.146150\pi$
$853$			1.88357e7i			0.886359i	−0.896433	+	0.443180i	$0.853850\pi$
$854$	−3.97041e6			−0.186291
$855$		0			0
$856$		−	6.28957e6i		−	0.293384i
$857$			1.22399e7i			0.569280i	0.958635	+	0.284640i	$0.0918740\pi$
$857$			1.22399e7i			0.569280i	−0.958635	+	0.284640i	$0.908126\pi$
$858$		0			0
$859$		−	2.16729e7i		−	1.00216i	−0.865402	−	0.501078i	$-0.832937\pi$
$859$		−	2.16729e7i		−	1.00216i	0.865402	−	0.501078i	$-0.167063\pi$
$860$	7.84359e6			0.361634
$861$		0			0
$862$	−6.44162e6			−0.295275
$863$	4.04170e7			1.84730			0.923649	−	0.383240i	$-0.125192\pi$
$863$	4.04170e7			1.84730			0.923649	+	0.383240i	$0.125192\pi$
$864$		0			0
$865$			1.34768e7i			0.612415i
$866$		−	1.35954e7i		−	0.616025i
$867$		0			0
$868$			1.19516e7i			0.538428i
$869$		−	2.43493e7i		−	1.09380i
$870$		0			0
$871$		−	3.37889e6i		−	0.150914i
$872$	−1.87271e7			−0.834027
$873$		0			0
$874$		−	2.35919e7i		−	1.04468i
$875$			1.76381e7i			0.778810i
$876$		0			0
$877$	−1.77282e7			−0.778333			−0.389166	−	0.921167i	$-0.627237\pi$
$877$	−1.77282e7			−0.778333			−0.389166	+	0.921167i	$0.627237\pi$
$878$	−1.84778e7			−0.808935
$879$		0			0
$880$		−	6.68505e6i		−	0.291003i
$881$	2.08301e7			0.904172			0.452086	−	0.891974i	$-0.350680\pi$
$881$	2.08301e7			0.904172			0.452086	+	0.891974i	$0.350680\pi$
$882$		0			0
$883$			1.01542e7i			0.438270i	0.975695	+	0.219135i	$0.0703236\pi$
$883$			1.01542e7i			0.438270i	−0.975695	+	0.219135i	$0.929676\pi$
$884$	−3.35954e7			−1.44594
$885$		0			0
$886$			2.03400e7i			0.870497i
$887$	3.19961e7			1.36549			0.682744	−	0.730657i	$-0.260786\pi$
$887$	3.19961e7			1.36549			0.682744	+	0.730657i	$0.260786\pi$
$888$		0			0
$889$	−7.50894e6			−0.318658
$890$			5.08272e6i			0.215090i
$891$		0			0
$892$	−1.30743e6			−0.0550181
$893$			2.28867e7i			0.960406i
$894$		0			0
$895$	−6.42411e6			−0.268075
$896$			1.64671e7i			0.685247i
$897$		0			0
$898$	−1.80203e7			−0.745713
$899$	1.54235e7			0.636476
$900$		0			0
$901$		−	4.71506e7i		−	1.93498i
$902$		−	1.04634e7i		−	0.428210i
$903$		0			0
$904$	2.52648e7			1.02824
$905$		−	1.59378e7i		−	0.646856i
$906$		0			0
$907$		−	1.27218e7i		−	0.513486i	−0.966480	−	0.256743i	$-0.917351\pi$
$907$		−	1.27218e7i		−	0.513486i	0.966480	−	0.256743i	$-0.0826494\pi$
$908$			1.91518e7i			0.770895i
$909$		0			0
$910$			7.13097e6i			0.285460i
$911$		−	1.63465e7i		−	0.652571i	−0.945271	−	0.326285i	$-0.894203\pi$
$911$		−	1.63465e7i		−	0.652571i	0.945271	−	0.326285i	$-0.105797\pi$
$912$		0			0
$913$	2.74786e7			1.09098
$914$	2.17887e7			0.862711
$915$		0			0
$916$	1.69601e7			0.667866
$917$			1.54606e7i			0.607158i
$918$		0			0
$919$			3.39032e7i			1.32419i	0.749418	+	0.662097i	$0.230333\pi$
$919$			3.39032e7i			1.32419i	−0.749418	+	0.662097i	$0.769667\pi$
$920$			1.87306e7i			0.729596i
$921$		0			0
$922$	1.11790e7			0.433088
$923$			5.18645e6i			0.200385i
$924$		0			0
$925$	1.30329e7	+	4.61492e6i	0.500827	+	0.177341i
$926$	−1.87639e7			−0.719110
$927$		0			0
$928$	1.77411e7			0.676255
$929$	−4.54616e7			−1.72824			−0.864122	−	0.503282i	$-0.832126\pi$
$929$	−4.54616e7			−1.72824			−0.864122	+	0.503282i	$0.832126\pi$
$930$		0			0
$931$		−	2.02828e7i		−	0.766925i
$932$	−7.33180e6			−0.276485
$933$		0			0
$934$	1.10802e7			0.415605
$935$	4.48232e7			1.67677
$936$		0			0
$937$	−1.11969e6			−0.0416629			−0.0208314	−	0.999783i	$-0.506631\pi$
$937$	−1.11969e6			−0.0416629			−0.0208314	+	0.999783i	$0.506631\pi$
$938$		−	1.36788e6i		−	0.0507623i
$939$		0			0
$940$		−	7.76287e6i		−	0.286551i
$941$	−2.25114e7			−0.828760			−0.414380	−	0.910104i	$-0.636001\pi$
$941$	−2.25114e7			−0.828760			−0.414380	+	0.910104i	$0.636001\pi$
$942$		0			0
$943$		−	1.99672e7i		−	0.731202i
$944$		−	7.30975e6i		−	0.266976i
$945$		0			0
$946$	1.38229e7			0.502193
$947$		−	3.50838e7i		−	1.27125i	−0.771997	−	0.635626i	$-0.780742\pi$
$947$		−	3.50838e7i		−	1.27125i	0.771997	−	0.635626i	$-0.219258\pi$
$948$		0			0
$949$			5.19750e7i			1.87339i
$950$	1.27509e7			0.458385
$951$		0			0
$952$	−3.18349e7			−1.13844
$953$	−8.82460e6			−0.314748			−0.157374	−	0.987539i	$-0.550303\pi$
$953$	−8.82460e6			−0.314748			−0.157374	+	0.987539i	$0.550303\pi$
$954$		0			0
$955$	2.73197e7			0.969322
$956$		−	3.20116e7i		−	1.13283i
$957$		0			0
$958$	−2.14616e6			−0.0755525
$959$	5.00628e6			0.175779
$960$		0			0
$961$	1.59608e6			0.0557501
$962$	1.51866e7	+	5.37753e6i	0.529082	+	0.187346i
$963$		0			0
$964$			2.44086e7i			0.845960i
$965$	5.11229e6			0.176725
$966$		0			0
$967$		−	2.91030e7i		−	1.00086i	−0.865778	−	0.500428i	$-0.833176\pi$
$967$		−	2.91030e7i		−	1.00086i	0.865778	−	0.500428i	$-0.166824\pi$
$968$		−	2.51096e7i		−	0.861293i
$969$		0			0
$970$		−	1.47548e7i		−	0.503504i
$971$	1.31598e7			0.447921			0.223961	−	0.974598i	$-0.428101\pi$
$971$	1.31598e7			0.447921			0.223961	+	0.974598i	$0.428101\pi$
$972$		0			0
$973$	3.87332e7			1.31160
$974$	2.88653e7			0.974943
$975$		0			0
$976$			4.47333e6i			0.150316i
$977$			3.87316e7i			1.29816i	0.760720	+	0.649080i	$0.224846\pi$
$977$			3.87316e7i			1.29816i	−0.760720	+	0.649080i	$0.775154\pi$
$978$		0			0
$979$		−	2.62893e7i		−	0.876641i
$980$			6.87964e6i			0.228824i
$981$		0			0
$982$			8.92444e6i			0.295326i
$983$	1.63230e7			0.538788			0.269394	−	0.963030i	$-0.413177\pi$
$983$	1.63230e7			0.538788			0.269394	+	0.963030i	$0.413177\pi$
$984$		0			0
$985$			2.44037e7i			0.801428i
$986$			1.75512e7i			0.574931i
$987$		0			0
$988$	−4.36071e7			−1.42123
$989$	2.63780e7			0.857534
$990$		0			0
$991$		−	1.49634e7i		−	0.484001i	−0.970276	−	0.242000i	$-0.922196\pi$
$991$		−	1.49634e7i		−	0.484001i	0.970276	−	0.242000i	$-0.0778035\pi$
$992$	−3.10953e7			−1.00326
$993$		0			0
$994$			2.09964e6i			0.0674029i
$995$	1.39171e7			0.445647
$996$		0			0
$997$			2.63578e7i			0.839791i	0.907572	+	0.419896i	$0.137933\pi$
$997$			2.63578e7i			0.839791i	−0.907572	+	0.419896i	$0.862067\pi$
$998$	14350.3			0.000456074
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	333.6.c.d.73.7		16
3.2	odd	2		37.6.b.a.36.10	yes	16
12.11	even	2		592.6.g.c.369.12		16
37.36	even	2	inner	333.6.c.d.73.10		16
111.110	odd	2		37.6.b.a.36.7	✓	16
444.443	even	2		592.6.g.c.369.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.b.a.36.7	✓	16	111.110	odd	2
37.6.b.a.36.10	yes	16	3.2	odd	2
333.6.c.d.73.7		16	1.1	even	1	trivial
333.6.c.d.73.10		16	37.36	even	2	inner
592.6.g.c.369.11		16	444.443	even	2
592.6.g.c.369.12		16	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:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	333.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.85171i q^{2} +23.8677 q^{4} -38.2713i q^{5} -96.3094 q^{7} -159.319i q^{8} +O(q^{10})\)	\(q-2.85171i q^{2} +23.8677 q^{4} -38.2713i q^{5} -96.3094 q^{7} -159.319i q^{8} -109.139 q^{10} +564.497 q^{11} +678.423i q^{13} +274.647i q^{14} +309.435 q^{16} +2074.76i q^{17} +2693.06i q^{19} -913.449i q^{20} -1609.78i q^{22} -3071.93i q^{23} +1660.31 q^{25} +1934.67 q^{26} -2298.69 q^{28} +2966.43i q^{29} -5199.33i q^{31} -5980.62i q^{32} +5916.63 q^{34} +3685.89i q^{35} +(7849.71 + 2779.56i) q^{37} +7679.83 q^{38} -6097.34 q^{40} +6499.88 q^{41} +8586.79i q^{43} +13473.3 q^{44} -8760.27 q^{46} +8498.42 q^{47} -7531.50 q^{49} -4734.72i q^{50} +16192.4i q^{52} -22725.8 q^{53} -21604.0i q^{55} +15343.9i q^{56} +8459.41 q^{58} -23622.9i q^{59} +14456.4i q^{61} -14827.0 q^{62} -7153.11 q^{64} +25964.1 q^{65} -4980.51 q^{67} +49519.8i q^{68} +10511.1 q^{70} +7644.87 q^{71} +76611.5 q^{73} +(7926.51 - 22385.1i) q^{74} +64277.1i q^{76} -54366.4 q^{77} -43134.5i q^{79} -11842.5i q^{80} -18535.8i q^{82} +48677.9 q^{83} +79403.8 q^{85} +24487.1 q^{86} -89935.0i q^{88} -46571.1i q^{89} -65338.5i q^{91} -73320.0i q^{92} -24235.1i q^{94} +103067. q^{95} +135193. i q^{97} +21477.7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 268 q^{4} + 190 q^{7}+O(q^{10}) \) 16 * q - 268 * q^4 + 190 * q^7	\( 16 q - 268 q^{4} + 190 q^{7} - 74 q^{10} + 1110 q^{11} + 2900 q^{16} - 12052 q^{25} - 4902 q^{26} - 16824 q^{28} + 20556 q^{34} - 11400 q^{37} - 12108 q^{38} + 16966 q^{40} - 3918 q^{41} - 125394 q^{44} + 17470 q^{46} - 3822 q^{47} - 32618 q^{49} + 24126 q^{53} - 164718 q^{58} + 81426 q^{62} + 158076 q^{64} - 98976 q^{65} + 23560 q^{67} - 222404 q^{70} + 50046 q^{71} - 196274 q^{73} - 141216 q^{74} + 239574 q^{77} + 215814 q^{83} - 346472 q^{85} - 197640 q^{86} + 132504 q^{95}+O(q^{100}) \) 16 * q - 268 * q^4 + 190 * q^7 - 74 * q^10 + 1110 * q^11 + 2900 * q^16 - 12052 * q^25 - 4902 * q^26 - 16824 * q^28 + 20556 * q^34 - 11400 * q^37 - 12108 * q^38 + 16966 * q^40 - 3918 * q^41 - 125394 * q^44 + 17470 * q^46 - 3822 * q^47 - 32618 * q^49 + 24126 * q^53 - 164718 * q^58 + 81426 * q^62 + 158076 * q^64 - 98976 * q^65 + 23560 * q^67 - 222404 * q^70 + 50046 * q^71 - 196274 * q^73 - 141216 * q^74 + 239574 * q^77 + 215814 * q^83 - 346472 * q^85 - 197640 * q^86 + 132504 * q^95

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