LMFDB - Embedded newform 150.3.d.d.101.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,3,Mod(101,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("150.101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	150.d (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:	$4.08720396540$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 16x^{2} + 81 $ x^4 - 16*x^2 + 81
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.1
Root			$2.91548 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	150.101
Dual form			150.3.d.d.101.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +(-2.91548 - 0.707107i) q^{3} -2.00000 q^{4} +(-1.00000 + 4.12311i) q^{6} -5.83095 q^{7} +2.82843i q^{8} +(8.00000 + 4.12311i) q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +(-2.91548 - 0.707107i) q^{3} -2.00000 q^{4} +(-1.00000 + 4.12311i) q^{6} -5.83095 q^{7} +2.82843i q^{8} +(8.00000 + 4.12311i) q^{9} +16.4924i q^{11} +(5.83095 + 1.41421i) q^{12} +8.24621i q^{14} +4.00000 q^{16} +11.3137i q^{17} +(5.83095 - 11.3137i) q^{18} -12.0000 q^{19} +(17.0000 + 4.12311i) q^{21} +23.3238 q^{22} +24.0416i q^{23} +(2.00000 - 8.24621i) q^{24} +(-20.4083 - 17.6777i) q^{27} +11.6619 q^{28} -32.0000 q^{31} -5.65685i q^{32} +(11.6619 - 48.0833i) q^{33} +16.0000 q^{34} +(-16.0000 - 8.24621i) q^{36} -23.3238 q^{37} +16.9706i q^{38} -57.7235i q^{41} +(5.83095 - 24.0416i) q^{42} -40.8167 q^{43} -32.9848i q^{44} +34.0000 q^{46} +35.3553i q^{47} +(-11.6619 - 2.82843i) q^{48} -15.0000 q^{49} +(8.00000 - 32.9848i) q^{51} +67.8823i q^{53} +(-25.0000 + 28.8617i) q^{54} -16.4924i q^{56} +(34.9857 + 8.48528i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +45.2548i q^{62} +(-46.6476 - 24.0416i) q^{63} -8.00000 q^{64} +(-68.0000 - 16.4924i) q^{66} +5.83095 q^{67} -22.6274i q^{68} +(17.0000 - 70.0928i) q^{69} +(-11.6619 + 22.6274i) q^{72} +116.619 q^{73} +32.9848i q^{74} +24.0000 q^{76} -96.1665i q^{77} +72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} -81.6333 q^{82} -43.8406i q^{83} +(-34.0000 - 8.24621i) q^{84} +57.7235i q^{86} -46.6476 q^{88} +65.9697i q^{89} -48.0833i q^{92} +(93.2952 + 22.6274i) q^{93} +50.0000 q^{94} +(-4.00000 + 16.4924i) q^{96} +163.267 q^{97} +21.2132i q^{98} +(-68.0000 + 131.939i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{6} + 32 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^6 + 32 * q^9	$ 4 q - 8 q^{4} - 4 q^{6} + 32 q^{9} + 16 q^{16} - 48 q^{19} + 68 q^{21} + 8 q^{24} - 128 q^{31} + 64 q^{34} - 64 q^{36} + 136 q^{46} - 60 q^{49} + 32 q^{51} - 100 q^{54} - 64 q^{61} - 32 q^{64} - 272 q^{66} + 68 q^{69} + 96 q^{76} + 288 q^{79} + 188 q^{81} - 136 q^{84} + 200 q^{94} - 16 q^{96} - 272 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^6 + 32 * q^9 + 16 * q^16 - 48 * q^19 + 68 * q^21 + 8 * q^24 - 128 * q^31 + 64 * q^34 - 64 * q^36 + 136 * q^46 - 60 * q^49 + 32 * q^51 - 100 * q^54 - 64 * q^61 - 32 * q^64 - 272 * q^66 + 68 * q^69 + 96 * q^76 + 288 * q^79 + 188 * q^81 - 136 * q^84 + 200 * q^94 - 16 * q^96 - 272 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$		−	1.41421i		−	0.707107i
$2$		−	1.41421i		−	0.707107i
$3$	−2.91548	−	0.707107i	−0.971825	−	0.235702i
$3$	−2.91548	−	0.707107i	−0.971825	−	0.235702i
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$	−1.00000	+	4.12311i	−0.166667	+	0.687184i
$7$	−5.83095			−0.832993			−0.416497	−	0.909137i	$-0.636742\pi$
$7$	−5.83095			−0.832993			−0.416497	+	0.909137i	$0.636742\pi$
$8$			2.82843i			0.353553i
$9$	8.00000	+	4.12311i	0.888889	+	0.458123i
$10$		0			0
$11$			16.4924i			1.49931i	0.661828	+	0.749656i	$0.269781\pi$
$11$			16.4924i			1.49931i	−0.661828	+	0.749656i	$0.730219\pi$
$12$	5.83095	+	1.41421i	0.485913	+	0.117851i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$			8.24621i			0.589015i
$15$		0			0
$16$	4.00000			0.250000
$17$			11.3137i			0.665512i	0.943013	+	0.332756i	$0.107979\pi$
$17$			11.3137i			0.665512i	−0.943013	+	0.332756i	$0.892021\pi$
$18$	5.83095	−	11.3137i	0.323942	−	0.628539i
$19$	−12.0000			−0.631579			−0.315789	−	0.948829i	$-0.602269\pi$
$19$	−12.0000			−0.631579			−0.315789	+	0.948829i	$0.602269\pi$
$20$		0			0
$21$	17.0000	+	4.12311i	0.809524	+	0.196338i
$22$	23.3238			1.06017
$23$			24.0416i			1.04529i	0.852551	+	0.522644i	$0.175054\pi$
$23$			24.0416i			1.04529i	−0.852551	+	0.522644i	$0.824946\pi$
$24$	2.00000	−	8.24621i	0.0833333	−	0.343592i
$25$		0			0
$26$		0			0
$27$	−20.4083	−	17.6777i	−0.755864	−	0.654729i
$28$	11.6619			0.416497
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−32.0000			−1.03226			−0.516129	−	0.856511i	$-0.672628\pi$
$31$	−32.0000			−1.03226			−0.516129	+	0.856511i	$0.672628\pi$
$32$		−	5.65685i		−	0.176777i
$33$	11.6619	−	48.0833i	0.353391	−	1.45707i
$34$	16.0000			0.470588
$35$		0			0
$36$	−16.0000	−	8.24621i	−0.444444	−	0.229061i
$37$	−23.3238			−0.630373			−0.315187	−	0.949030i	$-0.602067\pi$
$37$	−23.3238			−0.630373			−0.315187	+	0.949030i	$0.602067\pi$
$38$			16.9706i			0.446594i
$39$		0			0
$40$		0			0
$41$		−	57.7235i		−	1.40789i	−0.710255	−	0.703945i	$-0.751420\pi$
$41$		−	57.7235i		−	1.40789i	0.710255	−	0.703945i	$-0.248580\pi$
$42$	5.83095	−	24.0416i	0.138832	−	0.572420i
$43$	−40.8167			−0.949225			−0.474612	−	0.880195i	$-0.657412\pi$
$43$	−40.8167			−0.949225			−0.474612	+	0.880195i	$0.657412\pi$
$44$		−	32.9848i		−	0.749656i
$45$		0			0
$46$	34.0000			0.739130
$47$			35.3553i			0.752241i	0.926571	+	0.376121i	$0.122742\pi$
$47$			35.3553i			0.752241i	−0.926571	+	0.376121i	$0.877258\pi$
$48$	−11.6619	−	2.82843i	−0.242956	−	0.0589256i
$49$	−15.0000			−0.306122
$50$		0			0
$51$	8.00000	−	32.9848i	0.156863	−	0.646762i
$52$		0			0
$53$			67.8823i			1.28080i	0.768043	+	0.640399i	$0.221231\pi$
$53$			67.8823i			1.28080i	−0.768043	+	0.640399i	$0.778769\pi$
$54$	−25.0000	+	28.8617i	−0.462963	+	0.534477i
$55$		0			0
$56$		−	16.4924i		−	0.294508i
$57$	34.9857	+	8.48528i	0.613784	+	0.148865i
$58$		0			0
$59$		−	16.4924i		−	0.279533i	−0.990185	−	0.139766i	$-0.955365\pi$
$59$		−	16.4924i		−	0.279533i	0.990185	−	0.139766i	$-0.0446351\pi$
$60$		0			0
$61$	−16.0000			−0.262295			−0.131148	−	0.991363i	$-0.541866\pi$
$61$	−16.0000			−0.262295			−0.131148	+	0.991363i	$0.541866\pi$
$62$			45.2548i			0.729917i
$63$	−46.6476	−	24.0416i	−0.740438	−	0.381613i
$64$	−8.00000			−0.125000
$65$		0			0
$66$	−68.0000	−	16.4924i	−1.03030	−	0.249885i
$67$	5.83095			0.0870291			0.0435146	−	0.999053i	$-0.486145\pi$
$67$	5.83095			0.0870291			0.0435146	+	0.999053i	$0.486145\pi$
$68$		−	22.6274i		−	0.332756i
$69$	17.0000	−	70.0928i	0.246377	−	1.01584i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−11.6619	+	22.6274i	−0.161971	+	0.314270i
$73$	116.619			1.59752			0.798761	−	0.601649i	$-0.205489\pi$
$73$	116.619			1.59752			0.798761	+	0.601649i	$0.205489\pi$
$74$			32.9848i			0.445741i
$75$		0			0
$76$	24.0000			0.315789
$77$		−	96.1665i		−	1.24892i
$78$		0			0
$79$	72.0000			0.911392			0.455696	−	0.890135i	$-0.349390\pi$
$79$	72.0000			0.911392			0.455696	+	0.890135i	$0.349390\pi$
$80$		0			0
$81$	47.0000	+	65.9697i	0.580247	+	0.814441i
$82$	−81.6333			−0.995528
$83$		−	43.8406i		−	0.528200i	−0.964495	−	0.264100i	$-0.914925\pi$
$83$		−	43.8406i		−	0.528200i	0.964495	−	0.264100i	$-0.0850749\pi$
$84$	−34.0000	−	8.24621i	−0.404762	−	0.0981692i
$85$		0			0
$86$			57.7235i			0.671203i
$87$		0			0
$88$	−46.6476			−0.530087
$89$			65.9697i			0.741232i	0.928786	+	0.370616i	$0.120853\pi$
$89$			65.9697i			0.741232i	−0.928786	+	0.370616i	$0.879147\pi$
$90$		0			0
$91$		0			0
$92$		−	48.0833i		−	0.522644i
$93$	93.2952	+	22.6274i	1.00317	+	0.243306i
$94$	50.0000			0.531915
$95$		0			0
$96$	−4.00000	+	16.4924i	−0.0416667	+	0.171796i
$97$	163.267			1.68316			0.841581	−	0.540131i	$-0.181625\pi$
$97$	163.267			1.68316			0.841581	+	0.540131i	$0.181625\pi$
$98$			21.2132i			0.216461i
$99$	−68.0000	+	131.939i	−0.686869	+	1.33272i
$100$		0			0
$101$			131.939i			1.30633i	0.757215	+	0.653165i	$0.226559\pi$
$101$			131.939i			1.30633i	−0.757215	+	0.653165i	$0.773441\pi$
$102$	−46.6476	−	11.3137i	−0.457330	−	0.110919i
$103$	−99.1262			−0.962390			−0.481195	−	0.876614i	$-0.659797\pi$
$103$	−99.1262			−0.962390			−0.481195	+	0.876614i	$0.659797\pi$
$104$		0			0
$105$		0			0
$106$	96.0000			0.905660
$107$			55.1543i			0.515461i	0.966217	+	0.257731i	$0.0829747\pi$
$107$			55.1543i			0.515461i	−0.966217	+	0.257731i	$0.917025\pi$
$108$	40.8167	+	35.3553i	0.377932	+	0.327364i
$109$	−80.0000			−0.733945			−0.366972	−	0.930232i	$-0.619606\pi$
$109$	−80.0000			−0.733945			−0.366972	+	0.930232i	$0.619606\pi$
$110$		0			0
$111$	68.0000	+	16.4924i	0.612613	+	0.148580i
$112$	−23.3238			−0.208248
$113$		−	152.735i		−	1.35164i	−0.737068	−	0.675819i	$-0.763790\pi$
$113$		−	152.735i		−	1.35164i	0.737068	−	0.675819i	$-0.236210\pi$
$114$	12.0000	−	49.4773i	0.105263	−	0.434011i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−23.3238			−0.197659
$119$		−	65.9697i		−	0.554367i
$120$		0			0
$121$	−151.000			−1.24793
$122$			22.6274i			0.185471i
$123$	−40.8167	+	168.291i	−0.331843	+	1.36822i
$124$	64.0000			0.516129
$125$		0			0
$126$	−34.0000	+	65.9697i	−0.269841	+	0.523569i
$127$	−40.8167			−0.321391			−0.160696	−	0.987004i	$-0.551374\pi$
$127$	−40.8167			−0.321391			−0.160696	+	0.987004i	$0.551374\pi$
$128$			11.3137i			0.0883883i
$129$	119.000	+	28.8617i	0.922481	+	0.223734i
$130$		0			0
$131$			49.4773i			0.377689i	0.982007	+	0.188845i	$0.0604742\pi$
$131$			49.4773i			0.377689i	−0.982007	+	0.188845i	$0.939526\pi$
$132$	−23.3238	+	96.1665i	−0.176696	+	0.728534i
$133$	69.9714			0.526101
$134$		−	8.24621i		−	0.0615389i
$135$		0			0
$136$	−32.0000			−0.235294
$137$			50.9117i			0.371618i	0.982586	+	0.185809i	$0.0594906\pi$
$137$			50.9117i			0.371618i	−0.982586	+	0.185809i	$0.940509\pi$
$138$	−99.1262	−	24.0416i	−0.718306	−	0.174215i
$139$	−44.0000			−0.316547			−0.158273	−	0.987395i	$-0.550593\pi$
$139$	−44.0000			−0.316547			−0.158273	+	0.987395i	$0.550593\pi$
$140$		0			0
$141$	25.0000	−	103.078i	0.177305	−	0.731047i
$142$		0			0
$143$		0			0
$144$	32.0000	+	16.4924i	0.222222	+	0.114531i
$145$		0			0
$146$		−	164.924i		−	1.12962i
$147$	43.7321	+	10.6066i	0.297498	+	0.0721538i
$148$	46.6476			0.315187
$149$			8.24621i			0.0553437i	0.999617	+	0.0276718i	$0.00880935\pi$
$149$			8.24621i			0.0553437i	−0.999617	+	0.0276718i	$0.991191\pi$
$150$		0			0
$151$	136.000			0.900662			0.450331	−	0.892862i	$-0.351306\pi$
$151$	136.000			0.900662			0.450331	+	0.892862i	$0.351306\pi$
$152$		−	33.9411i		−	0.223297i
$153$	−46.6476	+	90.5097i	−0.304886	+	0.591566i
$154$	−136.000			−0.883117
$155$		0			0
$156$		0			0
$157$	−116.619			−0.742796			−0.371398	−	0.928474i	$-0.621122\pi$
$157$	−116.619			−0.742796			−0.371398	+	0.928474i	$0.621122\pi$
$158$		−	101.823i		−	0.644452i
$159$	48.0000	−	197.909i	0.301887	−	1.24471i
$160$		0			0
$161$		−	140.186i		−	0.870718i
$162$	93.2952	−	66.4680i	0.575896	−	0.410297i
$163$	−99.1262			−0.608136			−0.304068	−	0.952650i	$-0.598345\pi$
$163$	−99.1262			−0.608136			−0.304068	+	0.952650i	$0.598345\pi$
$164$			115.447i			0.703945i
$165$		0			0
$166$	−62.0000			−0.373494
$167$		−	292.742i		−	1.75295i	−0.481450	−	0.876474i	$-0.659890\pi$
$167$		−	292.742i		−	1.75295i	0.481450	−	0.876474i	$-0.340110\pi$
$168$	−11.6619	+	48.0833i	−0.0694161	+	0.286210i
$169$	−169.000			−1.00000
$170$		0			0
$171$	−96.0000	−	49.4773i	−0.561404	−	0.289341i
$172$	81.6333			0.474612
$173$		−	164.049i		−	0.948259i	−0.880455	−	0.474129i	$-0.842763\pi$
$173$		−	164.049i		−	0.948259i	0.880455	−	0.474129i	$-0.157237\pi$
$174$		0			0
$175$		0			0
$176$			65.9697i			0.374828i
$177$	−11.6619	+	48.0833i	−0.0658865	+	0.271657i
$178$	93.2952			0.524131
$179$		−	16.4924i		−	0.0921364i	−0.998938	−	0.0460682i	$-0.985331\pi$
$179$		−	16.4924i		−	0.0921364i	0.998938	−	0.0460682i	$-0.0146692\pi$
$180$		0			0
$181$	−82.0000			−0.453039			−0.226519	−	0.974007i	$-0.572735\pi$
$181$	−82.0000			−0.453039			−0.226519	+	0.974007i	$0.572735\pi$
$182$		0			0
$183$	46.6476	+	11.3137i	0.254905	+	0.0618235i
$184$	−68.0000			−0.369565
$185$		0			0
$186$	32.0000	−	131.939i	0.172043	−	0.709352i
$187$	−186.590			−0.997810
$188$		−	70.7107i		−	0.376121i
$189$	119.000	+	103.078i	0.629630	+	0.545384i
$190$		0			0
$191$		−	296.864i		−	1.55426i	−0.629340	−	0.777130i	$-0.716675\pi$
$191$		−	296.864i		−	1.55426i	0.629340	−	0.777130i	$-0.283325\pi$
$192$	23.3238	+	5.65685i	0.121478	+	0.0294628i
$193$	116.619			0.604244			0.302122	−	0.953269i	$-0.402305\pi$
$193$	116.619			0.604244			0.302122	+	0.953269i	$0.402305\pi$
$194$		−	230.894i		−	1.19017i
$195$		0			0
$196$	30.0000			0.153061
$197$			192.333i			0.976310i	0.872757	+	0.488155i	$0.162330\pi$
$197$			192.333i			0.976310i	−0.872757	+	0.488155i	$0.837670\pi$
$198$	186.590	+	96.1665i	0.942376	+	0.485690i
$199$	312.000			1.56784			0.783920	−	0.620862i	$-0.213217\pi$
$199$	312.000			1.56784			0.783920	+	0.620862i	$0.213217\pi$
$200$		0			0
$201$	−17.0000	−	4.12311i	−0.0845771	−	0.0205130i
$202$	186.590			0.923715
$203$		0			0
$204$	−16.0000	+	65.9697i	−0.0784314	+	0.323381i
$205$		0			0
$206$			140.186i			0.680513i
$207$	−99.1262	+	192.333i	−0.478870	+	0.929145i
$208$		0			0
$209$		−	197.909i		−	0.946933i
$210$		0			0
$211$	−12.0000			−0.0568720			−0.0284360	−	0.999596i	$-0.509053\pi$
$211$	−12.0000			−0.0568720			−0.0284360	+	0.999596i	$0.509053\pi$
$212$		−	135.765i		−	0.640399i
$213$		0			0
$214$	78.0000			0.364486
$215$		0			0
$216$	50.0000	−	57.7235i	0.231481	−	0.267238i
$217$	186.590			0.859864
$218$			113.137i			0.518977i
$219$	−340.000	−	82.4621i	−1.55251	−	0.376539i
$220$		0			0
$221$		0			0
$222$	23.3238	−	96.1665i	0.105062	−	0.433183i
$223$	−40.8167			−0.183034			−0.0915172	−	0.995803i	$-0.529172\pi$
$223$	−40.8167			−0.183034			−0.0915172	+	0.995803i	$0.529172\pi$
$224$			32.9848i			0.147254i
$225$		0			0
$226$	−216.000			−0.955752
$227$			159.806i			0.703992i	0.936002	+	0.351996i	$0.114497\pi$
$227$			159.806i			0.703992i	−0.936002	+	0.351996i	$0.885503\pi$
$228$	−69.9714	−	16.9706i	−0.306892	−	0.0744323i
$229$	−82.0000			−0.358079			−0.179039	−	0.983842i	$-0.557299\pi$
$229$	−82.0000			−0.358079			−0.179039	+	0.983842i	$0.557299\pi$
$230$		0			0
$231$	−68.0000	+	280.371i	−0.294372	+	1.21373i
$232$		0			0
$233$			192.333i			0.825464i	0.910853	+	0.412732i	$0.135425\pi$
$233$			192.333i			0.825464i	−0.910853	+	0.412732i	$0.864575\pi$
$234$		0			0
$235$		0			0
$236$			32.9848i			0.139766i
$237$	−209.914	−	50.9117i	−0.885714	−	0.214817i
$238$	−93.2952			−0.391997
$239$			461.788i			1.93217i	0.258231	+	0.966083i	$0.416861\pi$
$239$			461.788i			1.93217i	−0.258231	+	0.966083i	$0.583139\pi$
$240$		0			0
$241$	304.000			1.26141			0.630705	−	0.776022i	$-0.282766\pi$
$241$	304.000			1.26141			0.630705	+	0.776022i	$0.282766\pi$
$242$			213.546i			0.882423i
$243$	−90.3798	−	225.567i	−0.371933	−	0.928260i
$244$	32.0000			0.131148
$245$		0			0
$246$	238.000	+	57.7235i	0.967480	+	0.234648i
$247$		0			0
$248$		−	90.5097i		−	0.364958i
$249$	−31.0000	+	127.816i	−0.124498	+	0.513318i
$250$		0			0
$251$			346.341i			1.37984i	0.723884	+	0.689922i	$0.242355\pi$
$251$			346.341i			1.37984i	−0.723884	+	0.689922i	$0.757645\pi$
$252$	93.2952	+	48.0833i	0.370219	+	0.190807i
$253$	−396.505			−1.56721
$254$			57.7235i			0.227258i
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	390.323i		−	1.51877i	−0.650644	−	0.759383i	$-0.725501\pi$
$257$		−	390.323i		−	1.51877i	0.650644	−	0.759383i	$-0.274499\pi$
$258$	40.8167	−	168.291i	0.158204	−	0.652292i
$259$	136.000			0.525097
$260$		0			0
$261$		0			0
$262$	69.9714			0.267066
$263$			295.571i			1.12384i	0.827191	+	0.561921i	$0.189938\pi$
$263$			295.571i			1.12384i	−0.827191	+	0.561921i	$0.810062\pi$
$264$	136.000	+	32.9848i	0.515152	+	0.124943i
$265$		0			0
$266$		−	98.9545i		−	0.372010i
$267$	46.6476	−	192.333i	0.174710	−	0.720348i
$268$	−11.6619			−0.0435146
$269$		−	74.2159i		−	0.275896i	−0.990440	−	0.137948i	$-0.955949\pi$
$269$		−	74.2159i		−	0.275896i	0.990440	−	0.137948i	$-0.0440506\pi$
$270$		0			0
$271$	40.0000			0.147601			0.0738007	−	0.997273i	$-0.476487\pi$
$271$	40.0000			0.147601			0.0738007	+	0.997273i	$0.476487\pi$
$272$			45.2548i			0.166378i
$273$		0			0
$274$	72.0000			0.262774
$275$		0			0
$276$	−34.0000	+	140.186i	−0.123188	+	0.507919i
$277$	−443.152			−1.59983			−0.799914	−	0.600115i	$-0.795122\pi$
$277$	−443.152			−1.59983			−0.799914	+	0.600115i	$0.795122\pi$
$278$			62.2254i			0.223832i
$279$	−256.000	−	131.939i	−0.917563	−	0.472901i
$280$		0			0
$281$			519.511i			1.84879i	0.381431	+	0.924397i	$0.375431\pi$
$281$			519.511i			1.84879i	−0.381431	+	0.924397i	$0.624569\pi$
$282$	−145.774	−	35.3553i	−0.516928	−	0.125374i
$283$	320.702			1.13322			0.566612	−	0.823985i	$-0.308254\pi$
$283$	320.702			1.13322			0.566612	+	0.823985i	$0.308254\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			336.583i			1.17276i
$288$	23.3238	−	45.2548i	0.0809854	−	0.157135i
$289$	161.000			0.557093
$290$		0			0
$291$	−476.000	−	115.447i	−1.63574	−	0.396725i
$292$	−233.238			−0.798761
$293$			84.8528i			0.289600i	0.989461	+	0.144800i	$0.0462539\pi$
$293$			84.8528i			0.289600i	−0.989461	+	0.144800i	$0.953746\pi$
$294$	15.0000	−	61.8466i	0.0510204	−	0.210363i
$295$		0			0
$296$		−	65.9697i		−	0.222871i
$297$	291.548	−	336.583i	0.981642	−	1.13328i
$298$	11.6619			0.0391339
$299$		0			0
$300$		0			0
$301$	238.000			0.790698
$302$		−	192.333i		−	0.636864i
$303$	93.2952	−	384.666i	0.307905	−	1.26953i
$304$	−48.0000			−0.157895
$305$		0			0
$306$	128.000	+	65.9697i	0.418301	+	0.215587i
$307$	367.350			1.19658			0.598290	−	0.801280i	$-0.295847\pi$
$307$	367.350			1.19658			0.598290	+	0.801280i	$0.295847\pi$
$308$			192.333i			0.624458i
$309$	289.000	+	70.0928i	0.935275	+	0.226838i
$310$		0			0
$311$		−	98.9545i		−	0.318182i	−0.987264	−	0.159091i	$-0.949144\pi$
$311$		−	98.9545i		−	0.318182i	0.987264	−	0.159091i	$-0.0508563\pi$
$312$		0			0
$313$	186.590			0.596136			0.298068	−	0.954545i	$-0.403658\pi$
$313$	186.590			0.596136			0.298068	+	0.954545i	$0.403658\pi$
$314$			164.924i			0.525236i
$315$		0			0
$316$	−144.000			−0.455696
$317$			520.431i			1.64174i	0.571117	+	0.820868i	$0.306510\pi$
$317$			520.431i			1.64174i	−0.571117	+	0.820868i	$0.693490\pi$
$318$	−279.886	−	67.8823i	−0.880144	−	0.213466i
$319$		0			0
$320$		0			0
$321$	39.0000	−	160.801i	0.121495	−	0.500938i
$322$	−198.252			−0.615691
$323$		−	135.765i		−	0.420324i
$324$	−94.0000	−	131.939i	−0.290123	−	0.407220i
$325$		0			0
$326$			140.186i			0.430017i
$327$	233.238	+	56.5685i	0.713266	+	0.172992i
$328$	163.267			0.497764
$329$		−	206.155i		−	0.626612i
$330$		0			0
$331$	292.000			0.882175			0.441088	−	0.897464i	$-0.354593\pi$
$331$	292.000			0.882175			0.441088	+	0.897464i	$0.354593\pi$
$332$			87.6812i			0.264100i
$333$	−186.590	−	96.1665i	−0.560332	−	0.288788i
$334$	−414.000			−1.23952
$335$		0			0
$336$	68.0000	+	16.4924i	0.202381	+	0.0490846i
$337$	326.533			0.968942			0.484471	−	0.874807i	$-0.339012\pi$
$337$	326.533			0.968942			0.484471	+	0.874807i	$0.339012\pi$
$338$			239.002i			0.707107i
$339$	−108.000	+	445.295i	−0.318584	+	1.31356i
$340$		0			0
$341$		−	527.758i		−	1.54768i
$342$	−69.9714	+	135.765i	−0.204595	+	0.396972i
$343$	373.181			1.08799
$344$		−	115.447i		−	0.335602i
$345$		0			0
$346$	−232.000			−0.670520
$347$		−	394.566i		−	1.13708i	−0.822657	−	0.568538i	$-0.807509\pi$
$347$		−	394.566i		−	1.13708i	0.822657	−	0.568538i	$-0.192491\pi$
$348$		0			0
$349$	−254.000			−0.727794			−0.363897	−	0.931439i	$-0.618554\pi$
$349$	−254.000			−0.727794			−0.363897	+	0.931439i	$0.618554\pi$
$350$		0			0
$351$		0			0
$352$	93.2952			0.265043
$353$			345.068i			0.977530i	0.872415	+	0.488765i	$0.162552\pi$
$353$			345.068i			0.977530i	−0.872415	+	0.488765i	$0.837448\pi$
$354$	68.0000	+	16.4924i	0.192090	+	0.0465888i
$355$		0			0
$356$		−	131.939i		−	0.370616i
$357$	−46.6476	+	192.333i	−0.130666	+	0.538748i
$358$	−23.3238			−0.0651503
$359$		−	395.818i		−	1.10256i	−0.834321	−	0.551279i	$-0.814140\pi$
$359$		−	395.818i		−	1.10256i	0.834321	−	0.551279i	$-0.185860\pi$
$360$		0			0
$361$	−217.000			−0.601108
$362$			115.966i			0.320347i
$363$	440.237	+	106.773i	1.21277	+	0.294141i
$364$		0			0
$365$		0			0
$366$	16.0000	−	65.9697i	0.0437158	−	0.180245i
$367$	413.998			1.12806			0.564029	−	0.825755i	$-0.309250\pi$
$367$	413.998			1.12806			0.564029	+	0.825755i	$0.309250\pi$
$368$			96.1665i			0.261322i
$369$	238.000	−	461.788i	0.644986	−	1.25146i
$370$		0			0
$371$		−	395.818i		−	1.06690i
$372$	−186.590	−	45.2548i	−0.501587	−	0.121653i
$373$	−629.743			−1.68832			−0.844159	−	0.536092i	$-0.819900\pi$
$373$	−629.743			−1.68832			−0.844159	+	0.536092i	$0.819900\pi$
$374$			263.879i			0.705558i
$375$		0			0
$376$	−100.000			−0.265957
$377$		0			0
$378$	145.774	−	168.291i	0.385645	−	0.445215i
$379$	572.000			1.50923			0.754617	−	0.656165i	$-0.227822\pi$
$379$	572.000			1.50923			0.754617	+	0.656165i	$0.227822\pi$
$380$		0			0
$381$	119.000	+	28.8617i	0.312336	+	0.0757526i
$382$	−419.829			−1.09903
$383$		−	193.747i		−	0.505868i	−0.967484	−	0.252934i	$-0.918605\pi$
$383$		−	193.747i		−	0.505868i	0.967484	−	0.252934i	$-0.0813955\pi$
$384$	8.00000	−	32.9848i	0.0208333	−	0.0858980i
$385$		0			0
$386$		−	164.924i		−	0.427265i
$387$	−326.533	−	168.291i	−0.843755	−	0.434862i
$388$	−326.533			−0.841581
$389$			387.572i			0.996329i	0.867083	+	0.498164i	$0.165992\pi$
$389$			387.572i			0.996329i	−0.867083	+	0.498164i	$0.834008\pi$
$390$		0			0
$391$	−272.000			−0.695652
$392$		−	42.4264i		−	0.108231i
$393$	34.9857	−	144.250i	0.0890222	−	0.367048i
$394$	272.000			0.690355
$395$		0			0
$396$	136.000	−	263.879i	0.343434	−	0.666361i
$397$	−513.124			−1.29250			−0.646252	−	0.763124i	$-0.723664\pi$
$397$	−513.124			−1.29250			−0.646252	+	0.763124i	$0.723664\pi$
$398$		−	441.235i		−	1.10863i
$399$	−204.000	−	49.4773i	−0.511278	−	0.124003i
$400$		0			0
$401$		−	65.9697i		−	0.164513i	−0.996611	−	0.0822565i	$-0.973787\pi$
$401$		−	65.9697i		−	0.164513i	0.996611	−	0.0822565i	$-0.0262127\pi$
$402$	−5.83095	+	24.0416i	−0.0145049	+	0.0598051i
$403$		0			0
$404$		−	263.879i		−	0.653165i
$405$		0			0
$406$		0			0
$407$		−	384.666i		−	0.945126i
$408$	93.2952	+	22.6274i	0.228665	+	0.0554594i
$409$	−640.000			−1.56479			−0.782396	−	0.622781i	$-0.786003\pi$
$409$	−640.000			−1.56479			−0.782396	+	0.622781i	$0.786003\pi$
$410$		0			0
$411$	36.0000	−	148.432i	0.0875912	−	0.361148i
$412$	198.252			0.481195
$413$			96.1665i			0.232849i
$414$	272.000	+	140.186i	0.657005	+	0.338613i
$415$		0			0
$416$		0			0
$417$	128.281	+	31.1127i	0.307628	+	0.0746108i
$418$	−279.886			−0.669583
$419$		−	577.235i		−	1.37765i	−0.724928	−	0.688824i	$-0.758127\pi$
$419$		−	577.235i		−	1.37765i	0.724928	−	0.688824i	$-0.241873\pi$
$420$		0			0
$421$	−656.000			−1.55819			−0.779097	−	0.626903i	$-0.784322\pi$
$421$	−656.000			−1.55819			−0.779097	+	0.626903i	$0.784322\pi$
$422$			16.9706i			0.0402146i
$423$	−145.774	+	282.843i	−0.344619	+	0.668659i
$424$	−192.000			−0.452830
$425$		0			0
$426$		0			0
$427$	93.2952			0.218490
$428$		−	110.309i		−	0.257731i
$429$		0			0
$430$		0			0
$431$			362.833i			0.841841i	0.907098	+	0.420920i	$0.138293\pi$
$431$			362.833i			0.841841i	−0.907098	+	0.420920i	$0.861707\pi$
$432$	−81.6333	−	70.7107i	−0.188966	−	0.163682i
$433$	−163.267			−0.377059			−0.188530	−	0.982068i	$-0.560372\pi$
$433$	−163.267			−0.377059			−0.188530	+	0.982068i	$0.560372\pi$
$434$		−	263.879i		−	0.608016i
$435$		0			0
$436$	160.000			0.366972
$437$		−	288.500i		−	0.660182i
$438$	−116.619	+	480.833i	−0.266254	+	1.09779i
$439$	−432.000			−0.984055			−0.492027	−	0.870580i	$-0.663744\pi$
$439$	−432.000			−0.984055			−0.492027	+	0.870580i	$0.663744\pi$
$440$		0			0
$441$	−120.000	−	61.8466i	−0.272109	−	0.140242i
$442$		0			0
$443$			123.037i			0.277735i	0.990311	+	0.138867i	$0.0443462\pi$
$443$			123.037i			0.277735i	−0.990311	+	0.138867i	$0.955654\pi$
$444$	−136.000	−	32.9848i	−0.306306	−	0.0742902i
$445$		0			0
$446$			57.7235i			0.129425i
$447$	5.83095	−	24.0416i	0.0130446	−	0.0537844i
$448$	46.6476			0.104124
$449$			865.852i			1.92840i	0.265174	+	0.964201i	$0.414571\pi$
$449$			865.852i			1.92840i	−0.265174	+	0.964201i	$0.585429\pi$
$450$		0			0
$451$	952.000			2.11086
$452$			305.470i			0.675819i
$453$	−396.505	−	96.1665i	−0.875286	−	0.212288i
$454$	226.000			0.497797
$455$		0			0
$456$	−24.0000	+	98.9545i	−0.0526316	+	0.217006i
$457$	466.476			1.02074			0.510368	−	0.859956i	$-0.329509\pi$
$457$	466.476			1.02074			0.510368	+	0.859956i	$0.329509\pi$
$458$			115.966i			0.253200i
$459$	200.000	−	230.894i	0.435730	−	0.503037i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$	396.505	+	96.1665i	0.858235	+	0.208153i
$463$	612.250			1.32235			0.661177	−	0.750230i	$-0.270057\pi$
$463$	612.250			1.32235			0.661177	+	0.750230i	$0.270057\pi$
$464$		0			0
$465$		0			0
$466$	272.000			0.583691
$467$			767.918i			1.64436i	0.569225	+	0.822182i	$0.307243\pi$
$467$			767.918i			1.64436i	−0.569225	+	0.822182i	$0.692757\pi$
$468$		0			0
$469$	−34.0000			−0.0724947
$470$		0			0
$471$	340.000	+	82.4621i	0.721868	+	0.175079i
$472$	46.6476			0.0988297
$473$		−	673.166i		−	1.42318i
$474$	−72.0000	+	296.864i	−0.151899	+	0.626295i
$475$		0			0
$476$			131.939i			0.277184i
$477$	−279.886	+	543.058i	−0.586762	+	1.13849i
$478$	653.067			1.36625
$479$			560.742i			1.17065i	0.810798	+	0.585326i	$0.199033\pi$
$479$			560.742i			1.17065i	−0.810798	+	0.585326i	$0.800967\pi$
$480$		0			0
$481$		0			0
$482$		−	429.921i		−	0.891952i
$483$	−99.1262	+	408.708i	−0.205230	+	0.846186i
$484$	302.000			0.623967
$485$		0			0
$486$	−319.000	+	127.816i	−0.656379	+	0.262996i
$487$	−647.236			−1.32903			−0.664513	−	0.747277i	$-0.731361\pi$
$487$	−647.236			−1.32903			−0.664513	+	0.747277i	$0.731361\pi$
$488$		−	45.2548i		−	0.0927353i
$489$	289.000	+	70.0928i	0.591002	+	0.143339i
$490$		0			0
$491$			346.341i			0.705379i	0.935740	+	0.352689i	$0.114733\pi$
$491$			346.341i			0.705379i	−0.935740	+	0.352689i	$0.885267\pi$
$492$	81.6333	−	336.583i	0.165921	−	0.684111i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−128.000			−0.258065
$497$		0			0
$498$	180.760	+	43.8406i	0.362971	+	0.0880334i
$499$	660.000			1.32265			0.661323	−	0.750102i	$-0.269995\pi$
$499$	660.000			1.32265			0.661323	+	0.750102i	$0.269995\pi$
$500$		0			0
$501$	−207.000	+	853.483i	−0.413174	+	1.70356i
$502$	489.800			0.975697
$503$		−	182.434i		−	0.362691i	−0.983419	−	0.181345i	$-0.941955\pi$
$503$		−	182.434i		−	0.362691i	0.983419	−	0.181345i	$-0.0580452\pi$
$504$	68.0000	−	131.939i	0.134921	−	0.261784i
$505$		0			0
$506$			560.742i			1.10819i
$507$	492.715	+	119.501i	0.971825	+	0.235702i
$508$	81.6333			0.160696
$509$		−	395.818i		−	0.777639i	−0.921314	−	0.388819i	$-0.872883\pi$
$509$		−	395.818i		−	0.777639i	0.921314	−	0.388819i	$-0.127117\pi$
$510$		0			0
$511$	−680.000			−1.33072
$512$		−	22.6274i		−	0.0441942i
$513$	244.900	+	212.132i	0.477388	+	0.413513i
$514$	−552.000			−1.07393
$515$		0			0
$516$	−238.000	−	57.7235i	−0.461240	−	0.111867i
$517$	−583.095			−1.12784
$518$		−	192.333i		−	0.371299i
$519$	−116.000	+	478.280i	−0.223507	+	0.921542i
$520$		0			0
$521$			131.939i			0.253243i	0.991951	+	0.126621i	$0.0404133\pi$
$521$			131.939i			0.253243i	−0.991951	+	0.126621i	$0.959587\pi$
$522$		0			0
$523$	145.774			0.278726			0.139363	−	0.990241i	$-0.455494\pi$
$523$	145.774			0.278726			0.139363	+	0.990241i	$0.455494\pi$
$524$		−	98.9545i		−	0.188845i
$525$		0			0
$526$	418.000			0.794677
$527$		−	362.039i		−	0.686980i
$528$	46.6476	−	192.333i	0.0883478	−	0.364267i
$529$	−49.0000			−0.0926276
$530$		0			0
$531$	68.0000	−	131.939i	0.128060	−	0.248473i
$532$	−139.943			−0.263050
$533$		0			0
$534$	−272.000	−	65.9697i	−0.509363	−	0.123539i
$535$		0			0
$536$			16.4924i			0.0307694i
$537$	−11.6619	+	48.0833i	−0.0217168	+	0.0895405i
$538$	−104.957			−0.195088
$539$		−	247.386i		−	0.458973i
$540$		0			0
$541$	418.000			0.772643			0.386322	−	0.922364i	$-0.373745\pi$
$541$	418.000			0.772643			0.386322	+	0.922364i	$0.373745\pi$
$542$		−	56.5685i		−	0.104370i
$543$	239.069	+	57.9828i	0.440274	+	0.106782i
$544$	64.0000			0.117647
$545$		0			0
$546$		0			0
$547$	−285.717			−0.522334			−0.261167	−	0.965294i	$-0.584107\pi$
$547$	−285.717			−0.522334			−0.261167	+	0.965294i	$0.584107\pi$
$548$		−	101.823i		−	0.185809i
$549$	−128.000	−	65.9697i	−0.233151	−	0.120163i
$550$		0			0
$551$		0			0
$552$	198.252	+	48.0833i	0.359153	+	0.0871074i
$553$	−419.829			−0.759184
$554$			626.712i			1.13125i
$555$		0			0
$556$	88.0000			0.158273
$557$		−	424.264i		−	0.761695i	−0.924638	−	0.380847i	$-0.875632\pi$
$557$		−	424.264i		−	0.761695i	0.924638	−	0.380847i	$-0.124368\pi$
$558$	−186.590	+	362.039i	−0.334392	+	0.648815i
$559$		0			0
$560$		0			0
$561$	544.000	+	131.939i	0.969697	+	0.235186i
$562$	734.700			1.30730
$563$		−	813.173i		−	1.44436i	−0.691707	−	0.722178i	$-0.743141\pi$
$563$		−	813.173i		−	1.44436i	0.691707	−	0.722178i	$-0.256859\pi$
$564$	−50.0000	+	206.155i	−0.0886525	+	0.365524i
$565$		0			0
$566$		−	453.542i		−	0.801310i
$567$	−274.055	−	384.666i	−0.483342	−	0.678423i
$568$		0			0
$569$		−	453.542i		−	0.797085i	−0.917150	−	0.398543i	$-0.869516\pi$
$569$		−	453.542i		−	0.797085i	0.917150	−	0.398543i	$-0.130484\pi$
$570$		0			0
$571$	220.000			0.385289			0.192644	−	0.981269i	$-0.438294\pi$
$571$	220.000			0.385289			0.192644	+	0.981269i	$0.438294\pi$
$572$		0			0
$573$	−209.914	+	865.499i	−0.366343	+	1.51047i
$574$	476.000			0.829268
$575$		0			0
$576$	−64.0000	−	32.9848i	−0.111111	−	0.0572654i
$577$	46.6476			0.0808451			0.0404225	−	0.999183i	$-0.487130\pi$
$577$	46.6476			0.0808451			0.0404225	+	0.999183i	$0.487130\pi$
$578$		−	227.688i		−	0.393925i
$579$	−340.000	−	82.4621i	−0.587219	−	0.142422i
$580$		0			0
$581$			255.633i			0.439987i
$582$	−163.267	+	673.166i	−0.280527	+	1.15664i
$583$	−1119.54			−1.92031
$584$			329.848i			0.564809i
$585$		0			0
$586$	120.000			0.204778
$587$		−	55.1543i		−	0.0939597i	−0.998896	−	0.0469798i	$-0.985040\pi$
$587$		−	55.1543i		−	0.0939597i	0.998896	−	0.0469798i	$-0.0149597\pi$
$588$	−87.4643	−	21.2132i	−0.148749	−	0.0360769i
$589$	384.000			0.651952
$590$		0			0
$591$	136.000	−	560.742i	0.230118	−	0.948803i
$592$	−93.2952			−0.157593
$593$			390.323i			0.658217i	0.944292	+	0.329109i	$0.106748\pi$
$593$			390.323i			0.658217i	−0.944292	+	0.329109i	$0.893252\pi$
$594$	−476.000	−	412.311i	−0.801347	−	0.694126i
$595$		0			0
$596$		−	16.4924i		−	0.0276718i
$597$	−909.628	−	220.617i	−1.52367	−	0.369543i
$598$		0			0
$599$			98.9545i			0.165200i	0.996583	+	0.0825998i	$0.0263223\pi$
$599$			98.9545i			0.165200i	−0.996583	+	0.0825998i	$0.973678\pi$
$600$		0			0
$601$	−880.000			−1.46423			−0.732113	−	0.681183i	$-0.761466\pi$
$601$	−880.000			−1.46423			−0.732113	+	0.681183i	$0.761466\pi$
$602$		−	336.583i		−	0.559108i
$603$	46.6476	+	24.0416i	0.0773592	+	0.0398700i
$604$	−272.000			−0.450331
$605$		0			0
$606$	−544.000	−	131.939i	−0.897690	−	0.217722i
$607$	425.659			0.701251			0.350626	−	0.936516i	$-0.385969\pi$
$607$	425.659			0.701251			0.350626	+	0.936516i	$0.385969\pi$
$608$			67.8823i			0.111648i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	93.2952	−	181.019i	0.152443	−	0.295783i
$613$	606.419			0.989264			0.494632	−	0.869102i	$-0.335303\pi$
$613$	606.419			0.989264			0.494632	+	0.869102i	$0.335303\pi$
$614$		−	519.511i		−	0.846110i
$615$		0			0
$616$	272.000			0.441558
$617$			113.137i			0.183366i	0.995788	+	0.0916832i	$0.0292247\pi$
$617$			113.137i			0.183366i	−0.995788	+	0.0916832i	$0.970775\pi$
$618$	99.1262	−	408.708i	0.160398	−	0.661339i
$619$	−52.0000			−0.0840065			−0.0420032	−	0.999117i	$-0.513374\pi$
$619$	−52.0000			−0.0840065			−0.0420032	+	0.999117i	$0.513374\pi$
$620$		0			0
$621$	425.000	−	490.650i	0.684380	−	0.790096i
$622$	−139.943			−0.224988
$623$		−	384.666i		−	0.617442i
$624$		0			0
$625$		0			0
$626$		−	263.879i		−	0.421532i
$627$	−139.943	+	576.999i	−0.223194	+	0.920254i
$628$	233.238			0.371398
$629$		−	263.879i		−	0.419521i
$630$		0			0
$631$	−544.000			−0.862124			−0.431062	−	0.902322i	$-0.641861\pi$
$631$	−544.000			−0.862124			−0.431062	+	0.902322i	$0.641861\pi$
$632$			203.647i			0.322226i
$633$	34.9857	+	8.48528i	0.0552697	+	0.0134049i
$634$	736.000			1.16088
$635$		0			0
$636$	−96.0000	+	395.818i	−0.150943	+	0.622356i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	849.360i		−	1.32505i	−0.749038	−	0.662527i	$-0.769484\pi$
$641$		−	849.360i		−	1.32505i	0.749038	−	0.662527i	$-0.230516\pi$
$642$	−227.407	−	55.1543i	−0.354217	−	0.0859102i
$643$	−367.350			−0.571306			−0.285653	−	0.958333i	$-0.592211\pi$
$643$	−367.350			−0.571306			−0.285653	+	0.958333i	$0.592211\pi$
$644$			280.371i			0.435359i
$645$		0			0
$646$	−192.000			−0.297214
$647$			971.565i			1.50165i	0.660504	+	0.750823i	$0.270343\pi$
$647$			971.565i			1.50165i	−0.660504	+	0.750823i	$0.729657\pi$
$648$	−186.590	+	132.936i	−0.287948	+	0.205148i
$649$	272.000			0.419106
$650$		0			0
$651$	−544.000	−	131.939i	−0.835637	−	0.202672i
$652$	198.252			0.304068
$653$			350.725i			0.537098i	0.963266	+	0.268549i	$0.0865441\pi$
$653$			350.725i			0.537098i	−0.963266	+	0.268549i	$0.913456\pi$
$654$	80.0000	−	329.848i	0.122324	−	0.504355i
$655$		0			0
$656$		−	230.894i		−	0.351972i
$657$	932.952	+	480.833i	1.42002	+	0.731861i
$658$	−291.548			−0.443081
$659$			577.235i			0.875925i	0.898993	+	0.437963i	$0.144300\pi$
$659$			577.235i			0.875925i	−0.898993	+	0.437963i	$0.855700\pi$
$660$		0			0
$661$	80.0000			0.121029			0.0605144	−	0.998167i	$-0.480726\pi$
$661$	80.0000			0.121029			0.0605144	+	0.998167i	$0.480726\pi$
$662$		−	412.950i		−	0.623792i
$663$		0			0
$664$	124.000			0.186747
$665$		0			0
$666$	−136.000	+	263.879i	−0.204204	+	0.396214i
$667$		0			0
$668$			585.484i			0.876474i
$669$	119.000	+	28.8617i	0.177877	+	0.0431416i
$670$		0			0
$671$		−	263.879i		−	0.393262i
$672$	23.3238	−	96.1665i	0.0347080	−	0.143105i
$673$	489.800			0.727786			0.363893	−	0.931441i	$-0.381447\pi$
$673$	489.800			0.727786			0.363893	+	0.931441i	$0.381447\pi$
$674$		−	461.788i		−	0.685145i
$675$		0			0
$676$	338.000			0.500000
$677$		−	192.333i		−	0.284096i	−0.989860	−	0.142048i	$-0.954631\pi$
$677$		−	192.333i		−	0.284096i	0.989860	−	0.142048i	$-0.0453687\pi$
$678$	629.743	+	152.735i	0.928824	+	0.225273i
$679$	−952.000			−1.40206
$680$		0			0
$681$	113.000	−	465.911i	0.165932	−	0.684157i
$682$	−746.362			−1.09437
$683$			236.174i			0.345789i	0.984940	+	0.172894i	$0.0553119\pi$
$683$			236.174i			0.345789i	−0.984940	+	0.172894i	$0.944688\pi$
$684$	192.000	+	98.9545i	0.280702	+	0.144670i
$685$		0			0
$686$		−	527.758i		−	0.769326i
$687$	239.069	+	57.9828i	0.347990	+	0.0843999i
$688$	−163.267			−0.237306
$689$		0			0
$690$		0			0
$691$	−548.000			−0.793054			−0.396527	−	0.918023i	$-0.629785\pi$
$691$	−548.000			−0.793054			−0.396527	+	0.918023i	$0.629785\pi$
$692$			328.098i			0.474129i
$693$	396.505	−	769.332i	0.572157	−	1.11015i
$694$	−558.000			−0.804035
$695$		0			0
$696$		0			0
$697$	653.067			0.936968
$698$			359.210i			0.514628i
$699$	136.000	−	560.742i	0.194564	−	0.802207i
$700$		0			0
$701$		−	57.7235i		−	0.0823445i	−0.999152	−	0.0411722i	$-0.986891\pi$
$701$		−	57.7235i		−	0.0823445i	0.999152	−	0.0411722i	$-0.0131092\pi$
$702$		0			0
$703$	279.886			0.398130
$704$		−	131.939i		−	0.187414i
$705$		0			0
$706$	488.000			0.691218
$707$		−	769.332i		−	1.08816i
$708$	23.3238	−	96.1665i	0.0329432	−	0.135828i
$709$	−1230.00			−1.73484			−0.867419	−	0.497579i	$-0.834223\pi$
$709$	−1230.00			−1.73484			−0.867419	+	0.497579i	$0.834223\pi$
$710$		0			0
$711$	576.000	+	296.864i	0.810127	+	0.417530i
$712$	−186.590			−0.262065
$713$		−	769.332i		−	1.07901i
$714$	272.000	+	65.9697i	0.380952	+	0.0923945i
$715$		0			0
$716$			32.9848i			0.0460682i
$717$	326.533	−	1346.33i	0.455416	−	1.87773i
$718$	−559.771			−0.779626
$719$		−	626.712i		−	0.871644i	−0.900033	−	0.435822i	$-0.856458\pi$
$719$		−	626.712i		−	0.871644i	0.900033	−	0.435822i	$-0.143542\pi$
$720$		0			0
$721$	578.000			0.801664
$722$			306.884i			0.425048i
$723$	−886.305	−	214.960i	−1.22587	−	0.297317i
$724$	164.000			0.226519
$725$		0			0
$726$	151.000	−	622.589i	0.207989	−	0.857561i
$727$	−367.350			−0.505296			−0.252648	−	0.967558i	$-0.581301\pi$
$727$	−367.350			−0.505296			−0.252648	+	0.967558i	$0.581301\pi$
$728$		0			0
$729$	104.000	+	721.543i	0.142661	+	0.989772i
$730$		0			0
$731$		−	461.788i		−	0.631721i
$732$	−93.2952	−	22.6274i	−0.127453	−	0.0309118i
$733$	1002.92			1.36825			0.684123	−	0.729367i	$-0.260185\pi$
$733$	1002.92			1.36825			0.684123	+	0.729367i	$0.260185\pi$
$734$		−	585.481i		−	0.797658i
$735$		0			0
$736$	136.000			0.184783
$737$			96.1665i			0.130484i
$738$	−653.067	−	336.583i	−0.884914	−	0.456074i
$739$	340.000			0.460081			0.230041	−	0.973181i	$-0.426114\pi$
$739$	340.000			0.460081			0.230041	+	0.973181i	$0.426114\pi$
$740$		0			0
$741$		0			0
$742$	−559.771			−0.754409
$743$			1243.09i			1.67307i	0.547911	+	0.836537i	$0.315423\pi$
$743$			1243.09i			1.67307i	−0.547911	+	0.836537i	$0.684577\pi$
$744$	−64.0000	+	263.879i	−0.0860215	+	0.354676i
$745$		0			0
$746$			890.591i			1.19382i
$747$	180.760	−	350.725i	0.241981	−	0.469511i
$748$	373.181			0.498905
$749$		−	321.602i		−	0.429375i
$750$		0			0
$751$	−520.000			−0.692410			−0.346205	−	0.938159i	$-0.612530\pi$
$751$	−520.000			−0.692410			−0.346205	+	0.938159i	$0.612530\pi$
$752$			141.421i			0.188060i
$753$	244.900	−	1009.75i	0.325232	−	1.34097i
$754$		0			0
$755$		0			0
$756$	−238.000	−	206.155i	−0.314815	−	0.272692i
$757$	816.333			1.07838			0.539190	−	0.842184i	$-0.318731\pi$
$757$	816.333			1.07838			0.539190	+	0.842184i	$0.318731\pi$
$758$		−	808.930i		−	1.06719i
$759$	1156.00	+	280.371i	1.52306	+	0.369395i
$760$		0			0
$761$			395.818i			0.520129i	0.965591	+	0.260064i	$0.0837438\pi$
$761$			395.818i			0.520129i	−0.965591	+	0.260064i	$0.916256\pi$
$762$	40.8167	−	168.291i	0.0535652	−	0.220855i
$763$	466.476			0.611371
$764$			593.727i			0.777130i
$765$		0			0
$766$	−274.000			−0.357702
$767$		0			0
$768$	−46.6476	−	11.3137i	−0.0607391	−	0.0147314i
$769$	306.000			0.397919			0.198960	−	0.980008i	$-0.436244\pi$
$769$	306.000			0.397919			0.198960	+	0.980008i	$0.436244\pi$
$770$		0			0
$771$	−276.000	+	1137.98i	−0.357977	+	1.47598i
$772$	−233.238			−0.302122
$773$			305.470i			0.395175i	0.980285	+	0.197587i	$0.0633106\pi$
$773$			305.470i			0.395175i	−0.980285	+	0.197587i	$0.936689\pi$
$774$	−238.000	+	461.788i	−0.307494	+	0.596625i
$775$		0			0
$776$			461.788i			0.595087i
$777$	−396.505	−	96.1665i	−0.510302	−	0.123766i
$778$	548.109			0.704511
$779$			692.682i			0.889194i
$780$		0			0
$781$		0			0
$782$			384.666i			0.491900i
$783$		0			0
$784$	−60.0000			−0.0765306
$785$		0			0
$786$	−204.000	−	49.4773i	−0.259542	−	0.0629482i
$787$	−565.602			−0.718681			−0.359341	−	0.933206i	$-0.616998\pi$
$787$	−565.602			−0.718681

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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(-2.91548 - 0.707107i) q^{3} -2.00000 q^{4} +(-1.00000 + 4.12311i) q^{6} -5.83095 q^{7} +2.82843i q^{8} +(8.00000 + 4.12311i) q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +(-2.91548 - 0.707107i) q^{3} -2.00000 q^{4} +(-1.00000 + 4.12311i) q^{6} -5.83095 q^{7} +2.82843i q^{8} +(8.00000 + 4.12311i) q^{9} +16.4924i q^{11} +(5.83095 + 1.41421i) q^{12} +8.24621i q^{14} +4.00000 q^{16} +11.3137i q^{17} +(5.83095 - 11.3137i) q^{18} -12.0000 q^{19} +(17.0000 + 4.12311i) q^{21} +23.3238 q^{22} +24.0416i q^{23} +(2.00000 - 8.24621i) q^{24} +(-20.4083 - 17.6777i) q^{27} +11.6619 q^{28} -32.0000 q^{31} -5.65685i q^{32} +(11.6619 - 48.0833i) q^{33} +16.0000 q^{34} +(-16.0000 - 8.24621i) q^{36} -23.3238 q^{37} +16.9706i q^{38} -57.7235i q^{41} +(5.83095 - 24.0416i) q^{42} -40.8167 q^{43} -32.9848i q^{44} +34.0000 q^{46} +35.3553i q^{47} +(-11.6619 - 2.82843i) q^{48} -15.0000 q^{49} +(8.00000 - 32.9848i) q^{51} +67.8823i q^{53} +(-25.0000 + 28.8617i) q^{54} -16.4924i q^{56} +(34.9857 + 8.48528i) q^{57} -16.4924i q^{59} -16.0000 q^{61} +45.2548i q^{62} +(-46.6476 - 24.0416i) q^{63} -8.00000 q^{64} +(-68.0000 - 16.4924i) q^{66} +5.83095 q^{67} -22.6274i q^{68} +(17.0000 - 70.0928i) q^{69} +(-11.6619 + 22.6274i) q^{72} +116.619 q^{73} +32.9848i q^{74} +24.0000 q^{76} -96.1665i q^{77} +72.0000 q^{79} +(47.0000 + 65.9697i) q^{81} -81.6333 q^{82} -43.8406i q^{83} +(-34.0000 - 8.24621i) q^{84} +57.7235i q^{86} -46.6476 q^{88} +65.9697i q^{89} -48.0833i q^{92} +(93.2952 + 22.6274i) q^{93} +50.0000 q^{94} +(-4.00000 + 16.4924i) q^{96} +163.267 q^{97} +21.2132i q^{98} +(-68.0000 + 131.939i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{6} + 32 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^6 + 32 * q^9	\( 4 q - 8 q^{4} - 4 q^{6} + 32 q^{9} + 16 q^{16} - 48 q^{19} + 68 q^{21} + 8 q^{24} - 128 q^{31} + 64 q^{34} - 64 q^{36} + 136 q^{46} - 60 q^{49} + 32 q^{51} - 100 q^{54} - 64 q^{61} - 32 q^{64} - 272 q^{66} + 68 q^{69} + 96 q^{76} + 288 q^{79} + 188 q^{81} - 136 q^{84} + 200 q^{94} - 16 q^{96} - 272 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^6 + 32 * q^9 + 16 * q^16 - 48 * q^19 + 68 * q^21 + 8 * q^24 - 128 * q^31 + 64 * q^34 - 64 * q^36 + 136 * q^46 - 60 * q^49 + 32 * q^51 - 100 * q^54 - 64 * q^61 - 32 * q^64 - 272 * q^66 + 68 * q^69 + 96 * q^76 + 288 * q^79 + 188 * q^81 - 136 * q^84 + 200 * q^94 - 16 * q^96 - 272 * q^99

Introduction

L-functions

Modular forms

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