LMFDB - Embedded newform 270.3.b.c.269.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [270,3,Mod(269,270)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("270.269"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$270 = 2 \cdot 3^{3} \cdot 5$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	270.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.35696713773$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			269.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	270.269
Dual form			270.3.b.c.269.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(-3.53553 - 3.53553i) q^{5} +5.00000i q^{7} -2.82843 q^{8} +(5.00000 + 5.00000i) q^{10} +1.41421i q^{11} +9.00000i q^{13} -7.07107i q^{14} +4.00000 q^{16} +11.3137 q^{17} +21.0000 q^{19} +(-7.07107 - 7.07107i) q^{20} -2.00000i q^{22} +1.41421 q^{23} +25.0000i q^{25} -12.7279i q^{26} +10.0000i q^{28} +38.1838i q^{29} +40.0000 q^{31} -5.65685 q^{32} -16.0000 q^{34} +(17.6777 - 17.6777i) q^{35} -25.0000i q^{37} -29.6985 q^{38} +(10.0000 + 10.0000i) q^{40} +52.3259i q^{41} +64.0000i q^{43} +2.82843i q^{44} -2.00000 q^{46} +22.6274 q^{47} +24.0000 q^{49} -35.3553i q^{50} +18.0000i q^{52} +72.1249 q^{53} +(5.00000 - 5.00000i) q^{55} -14.1421i q^{56} -54.0000i q^{58} +90.5097i q^{59} -97.0000 q^{61} -56.5685 q^{62} +8.00000 q^{64} +(31.8198 - 31.8198i) q^{65} -131.000i q^{67} +22.6274 q^{68} +(-25.0000 + 25.0000i) q^{70} -89.0955i q^{71} -17.0000i q^{73} +35.3553i q^{74} +42.0000 q^{76} -7.07107 q^{77} -117.000 q^{79} +(-14.1421 - 14.1421i) q^{80} -74.0000i q^{82} -57.9828 q^{83} +(-40.0000 - 40.0000i) q^{85} -90.5097i q^{86} -4.00000i q^{88} +147.078i q^{89} -45.0000 q^{91} +2.82843 q^{92} -32.0000 q^{94} +(-74.2462 - 74.2462i) q^{95} -41.0000i q^{97} -33.9411 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{4} + 20 q^{10} + 16 q^{16} + 84 q^{19} + 160 q^{31} - 64 q^{34} + 40 q^{40} - 8 q^{46} + 96 q^{49} + 20 q^{55} - 388 q^{61} + 32 q^{64} - 100 q^{70} + 168 q^{76} - 468 q^{79} - 160 q^{85} - 180 q^{91}+ \cdots - 128 q^{94}+O(q^{100})$ 4 * q + 8 * q^4 + 20 * q^10 + 16 * q^16 + 84 * q^19 + 160 * q^31 - 64 * q^34 + 40 * q^40 - 8 * q^46 + 96 * q^49 + 20 * q^55 - 388 * q^61 + 32 * q^64 - 100 * q^70 + 168 * q^76 - 468 * q^79 - 160 * q^85 - 180 * q^91 - 128 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$ .

$n$	$191$	$217$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$6$		0			0
$7$			5.00000i			0.714286i	0.934050	+	0.357143i	$0.116249\pi$
$7$			5.00000i			0.714286i	−0.934050	+	0.357143i	$0.883751\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	5.00000	+	5.00000i	0.500000	+	0.500000i
$11$			1.41421i			0.128565i	0.997932	+	0.0642824i	$0.0204759\pi$
$11$			1.41421i			0.128565i	−0.997932	+	0.0642824i	$0.979524\pi$
$12$		0			0
$13$			9.00000i			0.692308i	0.938178	+	0.346154i	$0.112512\pi$
$13$			9.00000i			0.692308i	−0.938178	+	0.346154i	$0.887488\pi$
$14$		−	7.07107i		−	0.505076i
$15$		0			0
$16$	4.00000			0.250000
$17$	11.3137			0.665512			0.332756	−	0.943013i	$-0.392021\pi$
$17$	11.3137			0.665512			0.332756	+	0.943013i	$0.392021\pi$
$18$		0			0
$19$	21.0000			1.10526			0.552632	−	0.833426i	$-0.313624\pi$
$19$	21.0000			1.10526			0.552632	+	0.833426i	$0.313624\pi$
$20$	−7.07107	−	7.07107i	−0.353553	−	0.353553i
$21$		0			0
$22$		−	2.00000i		−	0.0909091i
$23$	1.41421			0.0614875			0.0307438	−	0.999527i	$-0.490212\pi$
$23$	1.41421			0.0614875			0.0307438	+	0.999527i	$0.490212\pi$
$24$		0			0
$25$			25.0000i			1.00000i
$26$		−	12.7279i		−	0.489535i
$27$		0			0
$28$			10.0000i			0.357143i
$29$			38.1838i			1.31668i	0.752720	+	0.658341i	$0.228741\pi$
$29$			38.1838i			1.31668i	−0.752720	+	0.658341i	$0.771259\pi$
$30$		0			0
$31$	40.0000			1.29032			0.645161	−	0.764046i	$-0.276790\pi$
$31$	40.0000			1.29032			0.645161	+	0.764046i	$0.276790\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	−16.0000			−0.470588
$35$	17.6777	−	17.6777i	0.505076	−	0.505076i
$36$		0			0
$37$		−	25.0000i		−	0.675676i	−0.941204	−	0.337838i	$-0.890304\pi$
$37$		−	25.0000i		−	0.675676i	0.941204	−	0.337838i	$-0.109696\pi$
$38$	−29.6985			−0.781539
$39$		0			0
$40$	10.0000	+	10.0000i	0.250000	+	0.250000i
$41$			52.3259i			1.27624i	0.769936	+	0.638121i	$0.220288\pi$
$41$			52.3259i			1.27624i	−0.769936	+	0.638121i	$0.779712\pi$
$42$		0			0
$43$			64.0000i			1.48837i	0.667972	+	0.744186i	$0.267162\pi$
$43$			64.0000i			1.48837i	−0.667972	+	0.744186i	$0.732838\pi$
$44$			2.82843i			0.0642824i
$45$		0			0
$46$	−2.00000			−0.0434783
$47$	22.6274			0.481434			0.240717	−	0.970595i	$-0.422617\pi$
$47$	22.6274			0.481434			0.240717	+	0.970595i	$0.422617\pi$
$48$		0			0
$49$	24.0000			0.489796
$50$		−	35.3553i		−	0.707107i
$51$		0			0
$52$			18.0000i			0.346154i
$53$	72.1249			1.36085			0.680424	−	0.732819i	$-0.261796\pi$
$53$	72.1249			1.36085			0.680424	+	0.732819i	$0.261796\pi$
$54$		0			0
$55$	5.00000	−	5.00000i	0.0909091	−	0.0909091i
$56$		−	14.1421i		−	0.252538i
$57$		0			0
$58$		−	54.0000i		−	0.931034i
$59$			90.5097i			1.53406i	0.641610	+	0.767031i	$0.278267\pi$
$59$			90.5097i			1.53406i	−0.641610	+	0.767031i	$0.721733\pi$
$60$		0			0
$61$	−97.0000			−1.59016			−0.795082	−	0.606502i	$-0.792572\pi$
$61$	−97.0000			−1.59016			−0.795082	+	0.606502i	$0.792572\pi$
$62$	−56.5685			−0.912396
$63$		0			0
$64$	8.00000			0.125000
$65$	31.8198	−	31.8198i	0.489535	−	0.489535i
$66$		0			0
$67$		−	131.000i		−	1.95522i	−0.210416	−	0.977612i	$-0.567482\pi$
$67$		−	131.000i		−	1.95522i	0.210416	−	0.977612i	$-0.432518\pi$
$68$	22.6274			0.332756
$69$		0			0
$70$	−25.0000	+	25.0000i	−0.357143	+	0.357143i
$71$		−	89.0955i		−	1.25487i	−0.778671	−	0.627433i	$-0.784106\pi$
$71$		−	89.0955i		−	1.25487i	0.778671	−	0.627433i	$-0.215894\pi$
$72$		0			0
$73$		−	17.0000i		−	0.232877i	−0.993198	−	0.116438i	$-0.962852\pi$
$73$		−	17.0000i		−	0.232877i	0.993198	−	0.116438i	$-0.0371477\pi$
$74$			35.3553i			0.477775i
$75$		0			0
$76$	42.0000			0.552632
$77$	−7.07107			−0.0918320
$78$		0			0
$79$	−117.000			−1.48101			−0.740506	−	0.672049i	$-0.765414\pi$
$79$	−117.000			−1.48101			−0.740506	+	0.672049i	$0.765414\pi$
$80$	−14.1421	−	14.1421i	−0.176777	−	0.176777i
$81$		0			0
$82$		−	74.0000i		−	0.902439i
$83$	−57.9828			−0.698587			−0.349294	−	0.937013i	$-0.613578\pi$
$83$	−57.9828			−0.698587			−0.349294	+	0.937013i	$0.613578\pi$
$84$		0			0
$85$	−40.0000	−	40.0000i	−0.470588	−	0.470588i
$86$		−	90.5097i		−	1.05244i
$87$		0			0
$88$		−	4.00000i		−	0.0454545i
$89$			147.078i			1.65256i	0.563257	+	0.826282i	$0.309548\pi$
$89$			147.078i			1.65256i	−0.563257	+	0.826282i	$0.690452\pi$
$90$		0			0
$91$	−45.0000			−0.494505
$92$	2.82843			0.0307438
$93$		0			0
$94$	−32.0000			−0.340426
$95$	−74.2462	−	74.2462i	−0.781539	−	0.781539i
$96$		0			0
$97$		−	41.0000i		−	0.422680i	−0.977413	−	0.211340i	$-0.932217\pi$
$97$		−	41.0000i		−	0.422680i	0.977413	−	0.211340i	$-0.0677828\pi$
$98$	−33.9411			−0.346338
$99$		0			0
$100$			50.0000i			0.500000i
$101$		−	90.5097i		−	0.896135i	−0.894000	−	0.448068i	$-0.852112\pi$
$101$		−	90.5097i		−	0.896135i	0.894000	−	0.448068i	$-0.147888\pi$
$102$		0			0
$103$		−	13.0000i		−	0.126214i	−0.998007	−	0.0631068i	$-0.979899\pi$
$103$		−	13.0000i		−	0.126214i	0.998007	−	0.0631068i	$-0.0201009\pi$
$104$		−	25.4558i		−	0.244768i
$105$		0			0
$106$	−102.000			−0.962264
$107$	−123.037			−1.14987			−0.574937	−	0.818197i	$-0.694974\pi$
$107$	−123.037			−1.14987			−0.574937	+	0.818197i	$0.694974\pi$
$108$		0			0
$109$	8.00000			0.0733945			0.0366972	−	0.999326i	$-0.488316\pi$
$109$	8.00000			0.0733945			0.0366972	+	0.999326i	$0.488316\pi$
$110$	−7.07107	+	7.07107i	−0.0642824	+	0.0642824i
$111$		0			0
$112$			20.0000i			0.178571i
$113$	−38.1838			−0.337909			−0.168955	−	0.985624i	$-0.554039\pi$
$113$	−38.1838			−0.337909			−0.168955	+	0.985624i	$0.554039\pi$
$114$		0			0
$115$	−5.00000	−	5.00000i	−0.0434783	−	0.0434783i
$116$			76.3675i			0.658341i
$117$		0			0
$118$		−	128.000i		−	1.08475i
$119$			56.5685i			0.475366i
$120$		0			0
$121$	119.000			0.983471
$122$	137.179			1.12442
$123$		0			0
$124$	80.0000			0.645161
$125$	88.3883	−	88.3883i	0.707107	−	0.707107i
$126$		0			0
$127$			8.00000i			0.0629921i	0.999504	+	0.0314961i	$0.0100272\pi$
$127$			8.00000i			0.0629921i	−0.999504	+	0.0314961i	$0.989973\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	−45.0000	+	45.0000i	−0.346154	+	0.346154i
$131$		−	135.765i		−	1.03637i	−0.855268	−	0.518185i	$-0.826608\pi$
$131$		−	135.765i		−	1.03637i	0.855268	−	0.518185i	$-0.173392\pi$
$132$		0			0
$133$			105.000i			0.789474i
$134$			185.262i			1.38255i
$135$		0			0
$136$	−32.0000			−0.235294
$137$	267.286			1.95100			0.975498	−	0.220010i	$-0.0706089\pi$
$137$	267.286			1.95100			0.975498	+	0.220010i	$0.0706089\pi$
$138$		0			0
$139$	−37.0000			−0.266187			−0.133094	−	0.991103i	$-0.542491\pi$
$139$	−37.0000			−0.266187			−0.133094	+	0.991103i	$0.542491\pi$
$140$	35.3553	−	35.3553i	0.252538	−	0.252538i
$141$		0			0
$142$			126.000i			0.887324i
$143$	−12.7279			−0.0890064
$144$		0			0
$145$	135.000	−	135.000i	0.931034	−	0.931034i
$146$			24.0416i			0.164669i
$147$		0			0
$148$		−	50.0000i		−	0.337838i
$149$			260.215i			1.74641i	0.487352	+	0.873206i	$0.337963\pi$
$149$			260.215i			1.74641i	−0.487352	+	0.873206i	$0.662037\pi$
$150$		0			0
$151$	109.000			0.721854			0.360927	−	0.932594i	$-0.382460\pi$
$151$	109.000			0.721854			0.360927	+	0.932594i	$0.382460\pi$
$152$	−59.3970			−0.390770
$153$		0			0
$154$	10.0000			0.0649351
$155$	−141.421	−	141.421i	−0.912396	−	0.912396i
$156$		0			0
$157$			118.000i			0.751592i	0.926702	+	0.375796i	$0.122631\pi$
$157$			118.000i			0.751592i	−0.926702	+	0.375796i	$0.877369\pi$
$158$	165.463			1.04723
$159$		0			0
$160$	20.0000	+	20.0000i	0.125000	+	0.125000i
$161$			7.07107i			0.0439197i
$162$		0			0
$163$			203.000i			1.24540i	0.782461	+	0.622699i	$0.213964\pi$
$163$			203.000i			1.24540i	−0.782461	+	0.622699i	$0.786036\pi$
$164$			104.652i			0.638121i
$165$		0			0
$166$	82.0000			0.493976
$167$	−101.823			−0.609721			−0.304860	−	0.952397i	$-0.598610\pi$
$167$	−101.823			−0.609721			−0.304860	+	0.952397i	$0.598610\pi$
$168$		0			0
$169$	88.0000			0.520710
$170$	56.5685	+	56.5685i	0.332756	+	0.332756i
$171$		0			0
$172$			128.000i			0.744186i
$173$	11.3137			0.0653972			0.0326986	−	0.999465i	$-0.489590\pi$
$173$	11.3137			0.0653972			0.0326986	+	0.999465i	$0.489590\pi$
$174$		0			0
$175$	−125.000			−0.714286
$176$			5.65685i			0.0321412i
$177$		0			0
$178$		−	208.000i		−	1.16854i
$179$		−	125.865i		−	0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$179$		−	125.865i		−	0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$180$		0			0
$181$	−127.000			−0.701657			−0.350829	−	0.936440i	$-0.614100\pi$
$181$	−127.000			−0.701657			−0.350829	+	0.936440i	$0.614100\pi$
$182$	63.6396			0.349668
$183$		0			0
$184$	−4.00000			−0.0217391
$185$	−88.3883	+	88.3883i	−0.477775	+	0.477775i
$186$		0			0
$187$			16.0000i			0.0855615i
$188$	45.2548			0.240717
$189$		0			0
$190$	105.000	+	105.000i	0.552632	+	0.552632i
$191$		−	101.823i		−	0.533107i	−0.963820	−	0.266553i	$-0.914115\pi$
$191$		−	101.823i		−	0.533107i	0.963820	−	0.266553i	$-0.0858849\pi$
$192$		0			0
$193$			271.000i			1.40415i	0.712105	+	0.702073i	$0.247742\pi$
$193$			271.000i			1.40415i	−0.712105	+	0.702073i	$0.752258\pi$
$194$			57.9828i			0.298880i
$195$		0			0
$196$	48.0000			0.244898
$197$	−316.784			−1.60804			−0.804020	−	0.594602i	$-0.797309\pi$
$197$	−316.784			−1.60804			−0.804020	+	0.594602i	$0.797309\pi$
$198$		0			0
$199$	147.000			0.738693			0.369347	−	0.929292i	$-0.379581\pi$
$199$	147.000			0.738693			0.369347	+	0.929292i	$0.379581\pi$
$200$		−	70.7107i		−	0.353553i
$201$		0			0
$202$			128.000i			0.633663i
$203$	−190.919			−0.940487
$204$		0			0
$205$	185.000	−	185.000i	0.902439	−	0.902439i
$206$			18.3848i			0.0892465i
$207$		0			0
$208$			36.0000i			0.173077i
$209$			29.6985i			0.142098i
$210$		0			0
$211$	141.000			0.668246			0.334123	−	0.942529i	$-0.391560\pi$
$211$	141.000			0.668246			0.334123	+	0.942529i	$0.391560\pi$
$212$	144.250			0.680424
$213$		0			0
$214$	174.000			0.813084
$215$	226.274	−	226.274i	1.05244	−	1.05244i
$216$		0			0
$217$			200.000i			0.921659i
$218$	−11.3137			−0.0518977
$219$		0			0
$220$	10.0000	−	10.0000i	0.0454545	−	0.0454545i
$221$			101.823i			0.460739i
$222$		0			0
$223$		−	8.00000i		−	0.0358744i	−0.999839	−	0.0179372i	$-0.994290\pi$
$223$		−	8.00000i		−	0.0358744i	0.999839	−	0.0179372i	$-0.00570990\pi$
$224$		−	28.2843i		−	0.126269i
$225$		0			0
$226$	54.0000			0.238938
$227$	−69.2965			−0.305271			−0.152635	−	0.988283i	$-0.548776\pi$
$227$	−69.2965			−0.305271			−0.152635	+	0.988283i	$0.548776\pi$
$228$		0			0
$229$	−8.00000			−0.0349345			−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000			−0.0349345			−0.0174672	+	0.999847i	$0.505560\pi$
$230$	7.07107	+	7.07107i	0.0307438	+	0.0307438i
$231$		0			0
$232$		−	108.000i		−	0.465517i
$233$	316.784			1.35959			0.679794	−	0.733403i	$-0.262069\pi$
$233$	316.784			1.35959			0.679794	+	0.733403i	$0.262069\pi$
$234$		0			0
$235$	−80.0000	−	80.0000i	−0.340426	−	0.340426i
$236$			181.019i			0.767031i
$237$		0			0
$238$		−	80.0000i		−	0.336134i
$239$		−	205.061i		−	0.857996i	−0.903306	−	0.428998i	$-0.858867\pi$
$239$		−	205.061i		−	0.857996i	0.903306	−	0.428998i	$-0.141133\pi$
$240$		0			0
$241$	79.0000			0.327801			0.163900	−	0.986477i	$-0.447592\pi$
$241$	79.0000			0.327801			0.163900	+	0.986477i	$0.447592\pi$
$242$	−168.291			−0.695419
$243$		0			0
$244$	−194.000			−0.795082
$245$	−84.8528	−	84.8528i	−0.346338	−	0.346338i
$246$		0			0
$247$			189.000i			0.765182i
$248$	−113.137			−0.456198
$249$		0			0
$250$	−125.000	+	125.000i	−0.500000	+	0.500000i
$251$		−	46.6690i		−	0.185932i	−0.995669	−	0.0929662i	$-0.970365\pi$
$251$		−	46.6690i		−	0.185932i	0.995669	−	0.0929662i	$-0.0296349\pi$
$252$		0			0
$253$			2.00000i			0.00790514i
$254$		−	11.3137i		−	0.0445422i
$255$		0			0
$256$	16.0000			0.0625000
$257$	169.706			0.660333			0.330167	−	0.943923i	$-0.392895\pi$
$257$	169.706			0.660333			0.330167	+	0.943923i	$0.392895\pi$
$258$		0			0
$259$	125.000			0.482625
$260$	63.6396	−	63.6396i	0.244768	−	0.244768i
$261$		0			0
$262$			192.000i			0.732824i
$263$	−282.843			−1.07545			−0.537724	−	0.843121i	$-0.680716\pi$
$263$	−282.843			−1.07545			−0.537724	+	0.843121i	$0.680716\pi$
$264$		0			0
$265$	−255.000	−	255.000i	−0.962264	−	0.962264i
$266$		−	148.492i		−	0.558242i
$267$		0			0
$268$		−	262.000i		−	0.977612i
$269$		−	101.823i		−	0.378526i	−0.981926	−	0.189263i	$-0.939390\pi$
$269$		−	101.823i		−	0.378526i	0.981926	−	0.189263i	$-0.0606098\pi$
$270$		0			0
$271$	−221.000			−0.815498			−0.407749	−	0.913094i	$-0.633686\pi$
$271$	−221.000			−0.815498			−0.407749	+	0.913094i	$0.633686\pi$
$272$	45.2548			0.166378
$273$		0			0
$274$	−378.000			−1.37956
$275$	−35.3553			−0.128565
$276$		0			0
$277$		−	88.0000i		−	0.317690i	−0.987304	−	0.158845i	$-0.949223\pi$
$277$		−	88.0000i		−	0.317690i	0.987304	−	0.158845i	$-0.0507769\pi$
$278$	52.3259			0.188223
$279$		0			0
$280$	−50.0000	+	50.0000i	−0.178571	+	0.178571i
$281$			203.647i			0.724722i	0.932038	+	0.362361i	$0.118029\pi$
$281$			203.647i			0.724722i	−0.932038	+	0.362361i	$0.881971\pi$
$282$		0			0
$283$			32.0000i			0.113074i	0.998400	+	0.0565371i	$0.0180059\pi$
$283$			32.0000i			0.113074i	−0.998400	+	0.0565371i	$0.981994\pi$
$284$		−	178.191i		−	0.627433i
$285$		0			0
$286$	18.0000			0.0629371
$287$	−261.630			−0.911601
$288$		0			0
$289$	−161.000			−0.557093
$290$	−190.919	+	190.919i	−0.658341	+	0.658341i
$291$		0			0
$292$		−	34.0000i		−	0.116438i
$293$	−173.948			−0.593680			−0.296840	−	0.954927i	$-0.595933\pi$
$293$	−173.948			−0.593680			−0.296840	+	0.954927i	$0.595933\pi$
$294$		0			0
$295$	320.000	−	320.000i	1.08475	−	1.08475i
$296$			70.7107i			0.238887i
$297$		0			0
$298$		−	368.000i		−	1.23490i
$299$			12.7279i			0.0425683i
$300$		0			0
$301$	−320.000			−1.06312
$302$	−154.149			−0.510428
$303$		0			0
$304$	84.0000			0.276316
$305$	342.947	+	342.947i	1.12442	+	1.12442i
$306$		0			0
$307$		−	486.000i		−	1.58306i	−0.611129	−	0.791531i	$-0.709284\pi$
$307$		−	486.000i		−	1.58306i	0.611129	−	0.791531i	$-0.290716\pi$
$308$	−14.1421			−0.0459160
$309$		0			0
$310$	200.000	+	200.000i	0.645161	+	0.645161i
$311$			67.8823i			0.218271i	0.994027	+	0.109135i	$0.0348082\pi$
$311$			67.8823i			0.218271i	−0.994027	+	0.109135i	$0.965192\pi$
$312$		0			0
$313$		−	281.000i		−	0.897764i	−0.893591	−	0.448882i	$-0.851822\pi$
$313$		−	281.000i		−	0.897764i	0.893591	−	0.448882i	$-0.148178\pi$
$314$		−	166.877i		−	0.531456i
$315$		0			0
$316$	−234.000			−0.740506
$317$	520.431			1.64174			0.820868	−	0.571117i	$-0.193490\pi$
$317$	520.431			1.64174			0.820868	+	0.571117i	$0.193490\pi$
$318$		0			0
$319$	−54.0000			−0.169279
$320$	−28.2843	−	28.2843i	−0.0883883	−	0.0883883i
$321$		0			0
$322$		−	10.0000i		−	0.0310559i
$323$	237.588			0.735566
$324$		0			0
$325$	−225.000			−0.692308
$326$		−	287.085i		−	0.880630i
$327$		0			0
$328$		−	148.000i		−	0.451220i
$329$			113.137i			0.343882i
$330$		0			0
$331$	−59.0000			−0.178248			−0.0891239	−	0.996021i	$-0.528407\pi$
$331$	−59.0000			−0.178248			−0.0891239	+	0.996021i	$0.528407\pi$
$332$	−115.966			−0.349294
$333$		0			0
$334$	144.000			0.431138
$335$	−463.155	+	463.155i	−1.38255	+	1.38255i
$336$		0			0
$337$		−	55.0000i		−	0.163205i	−0.996665	−	0.0816024i	$-0.973996\pi$
$337$		−	55.0000i		−	0.163205i	0.996665	−	0.0816024i	$-0.0260038\pi$
$338$	−124.451			−0.368198
$339$		0			0
$340$	−80.0000	−	80.0000i	−0.235294	−	0.235294i
$341$			56.5685i			0.165890i
$342$		0			0
$343$			365.000i			1.06414i
$344$		−	181.019i		−	0.526219i
$345$		0			0
$346$	−16.0000			−0.0462428
$347$	598.212			1.72395			0.861977	−	0.506947i	$-0.169226\pi$
$347$	598.212			1.72395			0.861977	+	0.506947i	$0.169226\pi$
$348$		0			0
$349$	−439.000			−1.25788			−0.628940	−	0.777454i	$-0.716511\pi$
$349$	−439.000			−1.25788			−0.628940	+	0.777454i	$0.716511\pi$
$350$	176.777			0.505076
$351$		0			0
$352$		−	8.00000i		−	0.0227273i
$353$	520.431			1.47431			0.737154	−	0.675725i	$-0.236169\pi$
$353$	520.431			1.47431			0.737154	+	0.675725i	$0.236169\pi$
$354$		0			0
$355$	−315.000	+	315.000i	−0.887324	+	0.887324i
$356$			294.156i			0.826282i
$357$		0			0
$358$			178.000i			0.497207i
$359$		−	55.1543i		−	0.153633i	−0.997045	−	0.0768166i	$-0.975524\pi$
$359$		−	55.1543i		−	0.153633i	0.997045	−	0.0768166i	$-0.0244756\pi$
$360$		0			0
$361$	80.0000			0.221607
$362$	179.605			0.496147
$363$		0			0
$364$	−90.0000			−0.247253
$365$	−60.1041	+	60.1041i	−0.164669	+	0.164669i
$366$		0			0
$367$		−	589.000i		−	1.60490i	−0.596716	−	0.802452i	$-0.703528\pi$
$367$		−	589.000i		−	1.60490i	0.596716	−	0.802452i	$-0.296472\pi$
$368$	5.65685			0.0153719
$369$		0			0
$370$	125.000	−	125.000i	0.337838	−	0.337838i
$371$			360.624i			0.972034i
$372$		0			0
$373$			9.00000i			0.0241287i	0.999927	+	0.0120643i	$0.00384029\pi$
$373$			9.00000i			0.0241287i	−0.999927	+	0.0120643i	$0.996160\pi$
$374$		−	22.6274i		−	0.0605011i
$375$		0			0
$376$	−64.0000			−0.170213
$377$	−343.654			−0.911549
$378$		0			0
$379$	157.000			0.414248			0.207124	−	0.978315i	$-0.433590\pi$
$379$	157.000			0.414248			0.207124	+	0.978315i	$0.433590\pi$
$380$	−148.492	−	148.492i	−0.390770	−	0.390770i
$381$		0			0
$382$			144.000i			0.376963i
$383$	282.843			0.738493			0.369246	−	0.929332i	$-0.379616\pi$
$383$	282.843			0.738493			0.369246	+	0.929332i	$0.379616\pi$
$384$		0			0
$385$	25.0000	+	25.0000i	0.0649351	+	0.0649351i
$386$		−	383.252i		−	0.992881i
$387$		0			0
$388$		−	82.0000i		−	0.211340i
$389$		−	599.627i		−	1.54146i	−0.637164	−	0.770728i	$-0.719893\pi$
$389$		−	599.627i		−	1.54146i	0.637164	−	0.770728i	$-0.280107\pi$
$390$		0			0
$391$	16.0000			0.0409207
$392$	−67.8823			−0.173169
$393$		0			0
$394$	448.000			1.13706
$395$	413.657	+	413.657i	1.04723	+	1.04723i
$396$		0			0
$397$			296.000i			0.745592i	0.927913	+	0.372796i	$0.121601\pi$
$397$			296.000i			0.745592i	−0.927913	+	0.372796i	$0.878399\pi$
$398$	−207.889			−0.522335
$399$		0			0
$400$			100.000i			0.250000i
$401$			388.909i			0.969847i	0.874557	+	0.484924i	$0.161153\pi$
$401$			388.909i			0.969847i	−0.874557	+	0.484924i	$0.838847\pi$
$402$		0			0
$403$			360.000i			0.893300i
$404$		−	181.019i		−	0.448068i
$405$		0			0
$406$	270.000			0.665025
$407$	35.3553			0.0868682
$408$		0			0
$409$	145.000			0.354523			0.177262	−	0.984164i	$-0.443276\pi$
$409$	145.000			0.354523			0.177262	+	0.984164i	$0.443276\pi$
$410$	−261.630	+	261.630i	−0.638121	+	0.638121i
$411$		0			0
$412$		−	26.0000i		−	0.0631068i
$413$	−452.548			−1.09576
$414$		0			0
$415$	205.000	+	205.000i	0.493976	+	0.493976i
$416$		−	50.9117i		−	0.122384i
$417$		0			0
$418$		−	42.0000i		−	0.100478i
$419$		−	701.450i		−	1.67410i	−0.547123	−	0.837052i	$-0.684277\pi$
$419$		−	701.450i		−	1.67410i	0.547123	−	0.837052i	$-0.315723\pi$
$420$		0			0
$421$	505.000			1.19952			0.599762	−	0.800178i	$-0.295262\pi$
$421$	505.000			1.19952			0.599762	+	0.800178i	$0.295262\pi$
$422$	−199.404			−0.472522
$423$		0			0
$424$	−204.000			−0.481132
$425$			282.843i			0.665512i
$426$		0			0
$427$		−	485.000i		−	1.13583i
$428$	−246.073			−0.574937
$429$		0			0
$430$	−320.000	+	320.000i	−0.744186	+	0.744186i
$431$			43.8406i			0.101718i	0.998706	+	0.0508592i	$0.0161960\pi$
$431$			43.8406i			0.101718i	−0.998706	+	0.0508592i	$0.983804\pi$
$432$		0			0
$433$		−	32.0000i		−	0.0739030i	−0.999317	−	0.0369515i	$-0.988235\pi$
$433$		−	32.0000i		−	0.0739030i	0.999317	−	0.0369515i	$-0.0117647\pi$
$434$		−	282.843i		−	0.651711i
$435$		0			0
$436$	16.0000			0.0366972
$437$	29.6985			0.0679599
$438$		0			0
$439$	−504.000			−1.14806			−0.574032	−	0.818833i	$-0.694621\pi$
$439$	−504.000			−1.14806			−0.574032	+	0.818833i	$0.694621\pi$
$440$	−14.1421	+	14.1421i	−0.0321412	+	0.0321412i
$441$		0			0
$442$		−	144.000i		−	0.325792i
$443$	−237.588			−0.536316			−0.268158	−	0.963375i	$-0.586415\pi$
$443$	−237.588			−0.536316			−0.268158	+	0.963375i	$0.586415\pi$
$444$		0			0
$445$	520.000	−	520.000i	1.16854	−	1.16854i
$446$			11.3137i			0.0253671i
$447$		0			0
$448$			40.0000i			0.0892857i
$449$		−	67.8823i		−	0.151185i	−0.997139	−	0.0755927i	$-0.975915\pi$
$449$		−	67.8823i		−	0.151185i	0.997139	−	0.0755927i	$-0.0240849\pi$
$450$		0			0
$451$	−74.0000			−0.164080
$452$	−76.3675			−0.168955
$453$		0			0
$454$	98.0000			0.215859
$455$	159.099	+	159.099i	0.349668	+	0.349668i
$456$		0			0
$457$			752.000i			1.64551i	0.568393	+	0.822757i	$0.307565\pi$
$457$			752.000i			1.64551i	−0.568393	+	0.822757i	$0.692435\pi$
$458$	11.3137			0.0247024
$459$		0			0
$460$	−10.0000	−	10.0000i	−0.0217391	−	0.0217391i
$461$		−	610.940i		−	1.32525i	−0.748951	−	0.662625i	$-0.769442\pi$
$461$		−	610.940i		−	1.32525i	0.748951	−	0.662625i	$-0.230558\pi$
$462$		0			0
$463$			597.000i			1.28942i	0.764429	+	0.644708i	$0.223021\pi$
$463$			597.000i			1.28942i	−0.764429	+	0.644708i	$0.776979\pi$
$464$			152.735i			0.329170i
$465$		0			0
$466$	−448.000			−0.961373
$467$	−848.528			−1.81698			−0.908488	−	0.417910i	$-0.862763\pi$
$467$	−848.528			−1.81698			−0.908488	+	0.417910i	$0.862763\pi$
$468$		0			0
$469$	655.000			1.39659
$470$	113.137	+	113.137i	0.240717	+	0.240717i
$471$		0			0
$472$		−	256.000i		−	0.542373i
$473$	−90.5097			−0.191352
$474$		0			0
$475$			525.000i			1.10526i
$476$			113.137i			0.237683i
$477$		0			0
$478$			290.000i			0.606695i
$479$			168.291i			0.351339i	0.984449	+	0.175670i	$0.0562090\pi$
$479$			168.291i			0.351339i	−0.984449	+	0.175670i	$0.943791\pi$
$480$		0			0
$481$	225.000			0.467775
$482$	−111.723			−0.231790
$483$		0			0
$484$	238.000			0.491736
$485$	−144.957	+	144.957i	−0.298880	+	0.298880i
$486$		0			0
$487$		−	507.000i		−	1.04107i	−0.853841	−	0.520534i	$-0.825733\pi$
$487$		−	507.000i		−	1.04107i	0.853841	−	0.520534i	$-0.174267\pi$
$488$	274.357			0.562208
$489$		0			0
$490$	120.000	+	120.000i	0.244898	+	0.244898i
$491$		−	428.507i		−	0.872722i	−0.899772	−	0.436361i	$-0.856267\pi$
$491$		−	428.507i		−	0.872722i	0.899772	−	0.436361i	$-0.143733\pi$
$492$		0			0
$493$			432.000i			0.876268i
$494$		−	267.286i		−	0.541066i
$495$		0			0
$496$	160.000			0.322581
$497$	445.477			0.896333
$498$		0			0
$499$	870.000			1.74349			0.871743	−	0.489963i	$-0.162990\pi$
$499$	870.000			1.74349			0.871743	+	0.489963i	$0.162990\pi$
$500$	176.777	−	176.777i	0.353553	−	0.353553i
$501$		0			0
$502$			66.0000i			0.131474i
$503$	462.448			0.919379			0.459690	−	0.888080i	$-0.347961\pi$
$503$	462.448			0.919379			0.459690	+	0.888080i	$0.347961\pi$
$504$		0			0
$505$	−320.000	+	320.000i	−0.633663	+	0.633663i
$506$		−	2.82843i		−	0.00558978i
$507$		0			0
$508$			16.0000i			0.0314961i
$509$		−	818.830i		−	1.60870i	−0.594154	−	0.804351i	$-0.702513\pi$
$509$		−	818.830i		−	1.60870i	0.594154	−	0.804351i	$-0.297487\pi$
$510$		0			0
$511$	85.0000			0.166341
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−240.000			−0.466926
$515$	−45.9619	+	45.9619i	−0.0892465	+	0.0892465i
$516$		0			0
$517$			32.0000i			0.0618956i
$518$	−176.777			−0.341268
$519$		0			0
$520$	−90.0000	+	90.0000i	−0.173077	+	0.173077i
$521$			864.084i			1.65851i	0.558869	+	0.829256i	$0.311235\pi$
$521$			864.084i			1.65851i	−0.558869	+	0.829256i	$0.688765\pi$
$522$		0			0
$523$		−	163.000i		−	0.311663i	−0.987784	−	0.155832i	$-0.950194\pi$
$523$		−	163.000i		−	0.311663i	0.987784	−	0.155832i	$-0.0498058\pi$
$524$		−	271.529i		−	0.518185i
$525$		0			0
$526$	400.000			0.760456
$527$	452.548			0.858726
$528$		0			0
$529$	−527.000			−0.996219
$530$	360.624	+	360.624i	0.680424	+	0.680424i
$531$		0			0
$532$			210.000i			0.394737i
$533$	−470.933			−0.883552
$534$		0			0
$535$	435.000	+	435.000i	0.813084	+	0.813084i
$536$			370.524i			0.691276i
$537$		0			0
$538$			144.000i			0.267658i
$539$			33.9411i			0.0629705i
$540$		0			0
$541$	697.000			1.28835			0.644177	−	0.764876i	$-0.277200\pi$
$541$	697.000			1.28835			0.644177	+	0.764876i	$0.277200\pi$
$542$	312.541			0.576644
$543$		0			0
$544$	−64.0000			−0.117647
$545$	−28.2843	−	28.2843i	−0.0518977	−	0.0518977i
$546$		0			0
$547$			389.000i			0.711152i	0.934647	+	0.355576i	$0.115715\pi$
$547$			389.000i			0.711152i	−0.934647	+	0.355576i	$0.884285\pi$
$548$	534.573			0.975498
$549$		0			0
$550$	50.0000			0.0909091
$551$			801.859i			1.45528i
$552$		0			0
$553$		−	585.000i		−	1.05787i
$554$			124.451i			0.224640i
$555$		0			0
$556$	−74.0000			−0.133094
$557$	−1086.12			−1.94994			−0.974969	−	0.222339i	$-0.928631\pi$
$557$	−1086.12			−1.94994			−0.974969	+	0.222339i	$0.928631\pi$
$558$		0			0
$559$	−576.000			−1.03041
$560$	70.7107	−	70.7107i	0.126269	−	0.126269i
$561$		0			0
$562$		−	288.000i		−	0.512456i
$563$	622.254			1.10525			0.552623	−	0.833431i	$-0.313627\pi$
$563$	622.254			1.10525			0.552623	+	0.833431i	$0.313627\pi$
$564$		0			0
$565$	135.000	+	135.000i	0.238938	+	0.238938i
$566$		−	45.2548i		−	0.0799555i
$567$		0			0
$568$			252.000i			0.443662i
$569$		−	463.862i		−	0.815223i	−0.913155	−	0.407612i	$-0.866362\pi$
$569$		−	463.862i		−	0.815223i	0.913155	−	0.407612i	$-0.133638\pi$
$570$		0			0
$571$	−923.000			−1.61646			−0.808231	−	0.588865i	$-0.799575\pi$
$571$	−923.000			−1.61646			−0.808231	+	0.588865i	$0.799575\pi$
$572$	−25.4558			−0.0445032
$573$		0			0
$574$	370.000			0.644599
$575$			35.3553i			0.0614875i
$576$		0			0
$577$		−	247.000i		−	0.428076i	−0.976825	−	0.214038i	$-0.931338\pi$
$577$		−	247.000i		−	0.428076i	0.976825	−	0.214038i	$-0.0686617\pi$
$578$	227.688			0.393925
$579$		0			0
$580$	270.000	−	270.000i	0.465517	−	0.465517i
$581$		−	289.914i		−	0.498991i
$582$		0			0
$583$			102.000i			0.174957i
$584$			48.0833i			0.0823344i
$585$		0			0
$586$	246.000			0.419795
$587$	453.963			0.773360			0.386680	−	0.922214i	$-0.373622\pi$
$587$	453.963			0.773360			0.386680	+	0.922214i	$0.373622\pi$
$588$		0			0
$589$	840.000			1.42615
$590$	−452.548	+	452.548i	−0.767031	+	0.767031i
$591$		0			0
$592$		−	100.000i		−	0.168919i
$593$	4.24264			0.00715454			0.00357727	−	0.999994i	$-0.498861\pi$
$593$	4.24264			0.00715454			0.00357727	+	0.999994i	$0.498861\pi$
$594$		0			0
$595$	200.000	−	200.000i	0.336134	−	0.336134i
$596$			520.431i			0.873206i
$597$		0			0
$598$		−	18.0000i		−	0.0301003i
$599$		−	224.860i		−	0.375392i	−0.982227	−	0.187696i	$-0.939898\pi$
$599$		−	224.860i		−	0.375392i	0.982227	−	0.187696i	$-0.0601020\pi$
$600$		0			0
$601$	−736.000			−1.22463			−0.612313	−	0.790616i	$-0.709761\pi$
$601$	−736.000			−1.22463			−0.612313	+	0.790616i	$0.709761\pi$
$602$	452.548			0.751741
$603$		0			0
$604$	218.000			0.360927
$605$	−420.729	−	420.729i	−0.695419	−	0.695419i
$606$		0			0
$607$		−	437.000i		−	0.719934i	−0.932965	−	0.359967i	$-0.882788\pi$
$607$		−	437.000i		−	0.719934i	0.932965	−	0.359967i	$-0.117212\pi$
$608$	−118.794			−0.195385
$609$		0			0
$610$	−485.000	−	485.000i	−0.795082	−	0.795082i
$611$			203.647i			0.333301i
$612$		0			0
$613$		−	335.000i		−	0.546493i	−0.961944	−	0.273246i	$-0.911903\pi$
$613$		−	335.000i		−	0.546493i	0.961944	−	0.273246i	$-0.0880974\pi$
$614$			687.308i			1.11939i
$615$		0			0
$616$	20.0000			0.0324675
$617$	−255.973			−0.414867			−0.207433	−	0.978249i	$-0.566511\pi$
$617$	−255.973			−0.414867			−0.207433	+	0.978249i	$0.566511\pi$
$618$		0			0
$619$	−965.000			−1.55897			−0.779483	−	0.626423i	$-0.784518\pi$
$619$	−965.000			−1.55897			−0.779483	+	0.626423i	$0.784518\pi$
$620$	−282.843	−	282.843i	−0.456198	−	0.456198i
$621$		0			0
$622$		−	96.0000i		−	0.154341i
$623$	−735.391			−1.18040
$624$		0			0
$625$	−625.000			−1.00000
$626$			397.394i			0.634815i
$627$		0			0
$628$			236.000i			0.375796i
$629$		−	282.843i		−	0.449670i
$630$		0			0
$631$	275.000			0.435816			0.217908	−	0.975969i	$-0.430077\pi$
$631$	275.000			0.435816			0.217908	+	0.975969i	$0.430077\pi$
$632$	330.926			0.523617
$633$		0			0
$634$	−736.000			−1.16088
$635$	28.2843	−	28.2843i	0.0445422	−	0.0445422i
$636$		0			0
$637$			216.000i			0.339089i
$638$	76.3675			0.119698
$639$		0			0
$640$	40.0000	+	40.0000i	0.0625000	+	0.0625000i
$641$		−	486.489i		−	0.758954i	−0.925201	−	0.379477i	$-0.876104\pi$
$641$		−	486.489i		−	0.758954i	0.925201	−	0.379477i	$-0.123896\pi$
$642$		0			0
$643$		−	1152.00i		−	1.79160i	−0.444455	−	0.895801i	$-0.646603\pi$
$643$		−	1152.00i		−	1.79160i	0.444455	−	0.895801i	$-0.353397\pi$
$644$			14.1421i			0.0219598i
$645$		0			0
$646$	−336.000			−0.520124
$647$	691.550			1.06886			0.534428	−	0.845214i	$-0.320527\pi$
$647$	691.550			1.06886			0.534428	+	0.845214i	$0.320527\pi$
$648$		0			0
$649$	−128.000			−0.197227
$650$	318.198			0.489535
$651$		0			0
$652$			406.000i			0.622699i
$653$	−350.725			−0.537098			−0.268549	−	0.963266i	$-0.586544\pi$
$653$	−350.725			−0.537098			−0.268549	+	0.963266i	$0.586544\pi$
$654$		0			0
$655$	−480.000	+	480.000i	−0.732824	+	0.732824i
$656$			209.304i			0.319060i
$657$		0			0
$658$		−	160.000i		−	0.243161i
$659$			497.803i			0.755392i	0.925930	+	0.377696i	$0.123284\pi$
$659$			497.803i			0.755392i	−0.925930	+	0.377696i	$0.876716\pi$
$660$		0			0
$661$	−577.000			−0.872920			−0.436460	−	0.899724i	$-0.643768\pi$
$661$	−577.000			−0.872920			−0.436460	+	0.899724i	$0.643768\pi$
$662$	83.4386			0.126040
$663$		0			0
$664$	164.000			0.246988
$665$	371.231	−	371.231i	0.558242	−	0.558242i
$666$		0			0
$667$			54.0000i			0.0809595i
$668$	−203.647			−0.304860
$669$		0			0
$670$	655.000	−	655.000i	0.977612	−	0.977612i
$671$		−	137.179i		−	0.204439i
$672$		0			0
$673$		−	489.000i		−	0.726597i	−0.931673	−	0.363299i	$-0.881650\pi$
$673$		−	489.000i		−	0.726597i	0.931673	−	0.363299i	$-0.118350\pi$
$674$			77.7817i			0.115403i
$675$		0			0
$676$	176.000			0.260355
$677$	−599.627			−0.885711			−0.442856	−	0.896593i	$-0.646035\pi$
$677$	−599.627			−0.885711			−0.442856	+	0.896593i	$0.646035\pi$
$678$		0			0
$679$	205.000			0.301915
$680$	113.137	+	113.137i	0.166378	+	0.166378i
$681$		0			0
$682$		−	80.0000i		−	0.117302i
$683$	−236.174			−0.345789			−0.172894	−	0.984940i	$-0.555312\pi$
$683$	−236.174			−0.345789			−0.172894	+	0.984940i	$0.555312\pi$
$684$		0			0
$685$	−945.000	−	945.000i	−1.37956	−	1.37956i
$686$		−	516.188i		−	0.752461i
$687$		0			0
$688$			256.000i			0.372093i
$689$			649.124i			0.942125i
$690$		0			0
$691$	640.000			0.926194			0.463097	−	0.886308i	$-0.346738\pi$
$691$	640.000			0.926194			0.463097	+	0.886308i	$0.346738\pi$
$692$	22.6274			0.0326986
$693$		0			0
$694$	−846.000			−1.21902
$695$	130.815	+	130.815i	0.188223	+	0.188223i
$696$		0			0
$697$			592.000i			0.849354i
$698$	620.840			0.889455
$699$		0			0
$700$	−250.000			−0.357143
$701$		−	1093.19i		−	1.55947i	−0.626111	−	0.779734i	$-0.715354\pi$
$701$		−	1093.19i		−	1.55947i	0.626111	−	0.779734i	$-0.284646\pi$
$702$		0			0
$703$		−	525.000i		−	0.746799i
$704$			11.3137i			0.0160706i
$705$		0			0
$706$	−736.000			−1.04249
$707$	452.548			0.640097
$708$		0			0
$709$	−489.000			−0.689704			−0.344852	−	0.938657i	$-0.612071\pi$
$709$	−489.000			−0.689704			−0.344852	+	0.938657i	$0.612071\pi$
$710$	445.477	−	445.477i	0.627433	−	0.627433i
$711$		0			0
$712$		−	416.000i		−	0.584270i
$713$	56.5685			0.0793388
$714$		0			0
$715$	45.0000	+	45.0000i	0.0629371	+	0.0629371i
$716$		−	251.730i		−	0.351578i
$717$		0			0
$718$			78.0000i			0.108635i
$719$		−	620.840i		−	0.863477i	−0.901999	−	0.431738i	$-0.857900\pi$
$719$		−	620.840i		−	0.863477i	0.901999	−	0.431738i	$-0.142100\pi$
$720$		0			0
$721$	65.0000			0.0901526
$722$	−113.137			−0.156700
$723$		0			0
$724$	−254.000			−0.350829
$725$	−954.594			−1.31668
$726$		0			0
$727$		−	1080.00i		−	1.48556i	−0.669537	−	0.742779i	$-0.733508\pi$
$727$		−	1080.00i		−	1.48556i	0.669537	−	0.742779i	$-0.266492\pi$
$728$	127.279			0.174834
$729$		0			0
$730$	85.0000	−	85.0000i	0.116438	−	0.116438i
$731$			724.077i			0.990530i
$732$		0			0
$733$			248.000i			0.338336i	0.985587	+	0.169168i	$0.0541080\pi$
$733$			248.000i			0.338336i	−0.985587	+	0.169168i	$0.945892\pi$
$734$			832.972i			1.13484i
$735$		0			0
$736$	−8.00000			−0.0108696
$737$	185.262			0.251373
$738$		0			0
$739$	848.000			1.14750			0.573748	−	0.819032i	$-0.305489\pi$
$739$	848.000			1.14750			0.573748	+	0.819032i	$0.305489\pi$
$740$	−176.777	+	176.777i	−0.238887	+	0.238887i
$741$		0			0
$742$		−	510.000i		−	0.687332i
$743$	−33.9411			−0.0456812			−0.0228406	−	0.999739i	$-0.507271\pi$
$743$	−33.9411			−0.0456812			−0.0228406	+	0.999739i	$0.507271\pi$
$744$		0			0
$745$	920.000	−	920.000i	1.23490	−	1.23490i
$746$		−	12.7279i		−	0.0170616i
$747$		0			0
$748$			32.0000i			0.0427807i
$749$		−	615.183i		−	0.821339i
$750$		0			0
$751$	−133.000			−0.177097			−0.0885486	−	0.996072i	$-0.528223\pi$
$751$	−133.000			−0.177097			−0.0885486	+	0.996072i	$0.528223\pi$
$752$	90.5097			0.120359
$753$		0			0
$754$	486.000			0.644562
$755$	−385.373	−	385.373i	−0.510428	−	0.510428i
$756$		0			0
$757$			1271.00i			1.67900i	0.543363	+	0.839498i	$0.317151\pi$
$757$			1271.00i			1.67900i	−0.543363	+	0.839498i	$0.682849\pi$
$758$	−222.032			−0.292918
$759$		0			0
$760$	210.000	+	210.000i	0.276316	+	0.276316i
$761$		−	538.815i		−	0.708036i	−0.935239	−	0.354018i	$-0.884815\pi$
$761$		−	538.815i		−	0.708036i	0.935239	−	0.354018i	$-0.115185\pi$
$762$		0			0
$763$			40.0000i			0.0524246i
$764$		−	203.647i		−	0.266553i
$765$		0			0
$766$	−400.000			−0.522193
$767$	−814.587			−1.06204
$768$		0			0
$769$	−81.0000			−0.105332			−0.0526658	−	0.998612i	$-0.516772\pi$
$769$	−81.0000			−0.105332			−0.0526658	+	0.998612i	$0.516772\pi$
$770$	−35.3553	−	35.3553i	−0.0459160	−	0.0459160i
$771$		0			0
$772$			542.000i			0.702073i
$773$	−445.477			−0.576297			−0.288148	−	0.957586i	$-0.593040\pi$
$773$	−445.477			−0.576297			−0.288148	+	0.957586i	$0.593040\pi$
$774$		0			0
$775$			1000.00i			1.29032i
$776$			115.966i			0.149440i
$777$		0			0
$778$			848.000i			1.08997i
$779$			1098.84i			1.41058i
$780$		0			0
$781$	126.000			0.161332
$782$	−22.6274			−0.0289353
$783$		0			0
$784$	96.0000			0.122449
$785$	417.193	−	417.193i	0.531456	−	0.531456i
$786$		0			0
$787$			395.000i			0.501906i	0.967999	+	0.250953i	$0.0807440\pi$
$787$			395.000i			0.501906i	−0.967999	+	0.250953i	$0.919256\pi$
$788$	−633.568			−0.804020
$789$		0			0
$790$	−585.000	−	585.000i	−0.740506	−	0.740506i
$791$		−	190.919i		−	0.241364i
$792$		0			0
$793$		−	873.000i		−	1.10088i
$794$		−	418.607i		−	0.527213i
$795$		0			0
$796$	294.000			0.369347
$797$	−1074.80			−1.34856			−0.674280	−	0.738476i	$-0.735546\pi$
$797$	−1074.80			−1.34856			−0.674280	+	0.738476i	$0.735546\pi$
$798$		0			0
$799$	256.000			0.320401
$800$		−	141.421i		−	0.176777i
$801$		0			0
$802$		−	550.000i		−	0.685786i
$803$	24.0416			0.0299398
$804$		0			0
$805$	25.0000	−	25.0000i	0.0310559	−	0.0310559i
$806$		−	509.117i		−	0.631659i
$807$		0			0
$808$			256.000i			0.316832i
$809$			1255.82i			1.55231i	0.630540	+	0.776157i	$0.282833\pi$
$809$			1255.82i			1.55231i	−0.630540	+	0.776157i	$0.717167\pi$
$810$		0			0
$811$	−752.000			−0.927250			−0.463625	−	0.886031i	$-0.653452\pi$
$811$	−752.000			−0.927250			−0.463625	+	0.886031i	$0.653452\pi$
$812$	−381.838			−0.470243
$813$		0			0
$814$	−50.0000			−0.0614251
$815$	717.713	−	717.713i	0.880630	−	0.880630i
$816$		0			0
$817$			1344.00i			1.64504i
$818$	−205.061			−0.250686
$819$		0			0
$820$	370.000	−	370.000i	0.451220	−	0.451220i
$821$		−	497.803i		−	0.606338i	−0.952937	−	0.303169i	$-0.901955\pi$
$821$		−	497.803i		−	0.606338i	0.952937	−	0.303169i	$-0.0980446\pi$
$822$		0			0
$823$		−	531.000i		−	0.645200i	−0.946535	−	0.322600i	$-0.895443\pi$
$823$		−	531.000i		−	0.645200i	0.946535	−	0.322600i	$-0.104557\pi$
$824$			36.7696i			0.0446232i
$825$		0			0
$826$	640.000			0.774818
$827$	350.725			0.424093			0.212047	−	0.977260i	$-0.431987\pi$
$827$	350.725			0.424093			0.212047	+	0.977260i	$0.431987\pi$
$828$		0			0
$829$	−705.000			−0.850422			−0.425211	−	0.905094i	$-0.639800\pi$
$829$	−705.000			−0.850422			−0.425211	+	0.905094i	$0.639800\pi$
$830$	−289.914	−	289.914i	−0.349294	−	0.349294i
$831$		0			0
$832$			72.0000i			0.0865385i
$833$	271.529			0.325965
$834$		0			0
$835$	360.000	+	360.000i	0.431138	+	0.431138i
$836$			59.3970i			0.0710490i
$837$		0			0
$838$			992.000i			1.18377i
$839$			1617.86i			1.92832i	0.265322	+	0.964160i	$0.414522\pi$
$839$			1617.86i			1.92832i	−0.265322	+	0.964160i	$0.585478\pi$
$840$		0			0
$841$	−617.000			−0.733650
$842$	−714.178			−0.848192
$843$		0			0
$844$	282.000			0.334123
$845$	−311.127	−	311.127i	−0.368198	−	0.368198i
$846$		0			0
$847$			595.000i			0.702479i
$848$	288.500			0.340212
$849$		0			0
$850$		−	400.000i		−	0.470588i
$851$		−	35.3553i		−	0.0415456i
$852$		0			0
$853$		−	1639.00i		−	1.92145i	−0.277496	−	0.960727i	$-0.589504\pi$
$853$		−	1639.00i		−	1.92145i	0.277496	−	0.960727i	$-0.410496\pi$
$854$			685.894i			0.803154i
$855$		0			0
$856$	348.000			0.406542
$857$	988.535			1.15348			0.576742	−	0.816927i	$-0.304324\pi$
$857$	988.535			1.15348			0.576742	+	0.816927i	$0.304324\pi$
$858$		0			0
$859$	355.000			0.413271			0.206636	−	0.978418i	$-0.433749\pi$
$859$	355.000			0.413271			0.206636	+	0.978418i	$0.433749\pi$
$860$	452.548	−	452.548i	0.526219	−	0.526219i
$861$		0			0
$862$		−	62.0000i		−	0.0719258i
$863$	1459.47			1.69116			0.845578	−	0.533851i	$-0.179256\pi$
$863$	1459.47			1.69116			0.845578	+	0.533851i	$0.179256\pi$
$864$		0			0
$865$	−40.0000	−	40.0000i	−0.0462428	−	0.0462428i
$866$			45.2548i			0.0522573i
$867$		0			0
$868$			400.000i			0.460829i
$869$		−	165.463i		−	0.190406i
$870$		0			0
$871$	1179.00			1.35362
$872$	−22.6274			−0.0259489
$873$		0			0
$874$	−42.0000			−0.0480549
$875$	441.942	+	441.942i	0.505076	+	0.505076i
$876$		0			0
$877$			1129.00i			1.28734i	0.765302	+	0.643672i	$0.222590\pi$
$877$			1129.00i			1.28734i	−0.765302	+	0.643672i	$0.777410\pi$
$878$	712.764			0.811804
$879$		0			0
$880$	20.0000	−	20.0000i	0.0227273	−	0.0227273i
$881$		−	165.463i		−	0.187813i	−0.995581	−	0.0939063i	$-0.970065\pi$
$881$		−	165.463i		−	0.187813i	0.995581	−	0.0939063i	$-0.0299354\pi$
$882$		0			0
$883$			1227.00i			1.38958i	0.719212	+	0.694790i	$0.244503\pi$
$883$			1227.00i			1.38958i	−0.719212	+	0.694790i	$0.755497\pi$
$884$			203.647i			0.230370i
$885$		0			0
$886$	336.000			0.379233
$887$	−237.588			−0.267856			−0.133928	−	0.990991i	$-0.542759\pi$
$887$	−237.588			−0.267856			−0.133928	+	0.990991i	$0.542759\pi$
$888$		0			0
$889$	−40.0000			−0.0449944
$890$	−735.391	+	735.391i	−0.826282	+	0.826282i
$891$		0			0
$892$		−	16.0000i		−	0.0179372i
$893$	475.176			0.532112
$894$		0			0
$895$	−445.000	+	445.000i	−0.497207	+	0.497207i
$896$		−	56.5685i		−	0.0631345i
$897$		0			0
$898$			96.0000i			0.106904i
$899$			1527.35i			1.69894i
$900$		0			0
$901$	816.000			0.905660
$902$	104.652			0.116022
$903$		0			0
$904$	108.000			0.119469
$905$	449.013	+	449.013i	0.496147	+	0.496147i
$906$		0			0
$907$		−	1005.00i		−	1.10805i	−0.832501	−	0.554024i	$-0.813091\pi$
$907$		−	1005.00i		−	1.10805i	0.832501	−	0.554024i	$-0.186909\pi$
$908$	−138.593			−0.152635
$909$		0			0
$910$	−225.000	−	225.000i	−0.247253	−	0.247253i
$911$		−	780.646i		−	0.856911i	−0.903563	−	0.428455i	$-0.859058\pi$
$911$		−	780.646i		−	0.856911i	0.903563	−	0.428455i	$-0.140942\pi$
$912$		0			0
$913$		−	82.0000i		−	0.0898138i
$914$		−	1063.49i		−	1.16355i
$915$		0			0
$916$	−16.0000			−0.0174672
$917$	678.823			0.740264
$918$		0			0
$919$	600.000			0.652884			0.326442	−	0.945217i	$-0.394150\pi$
$919$	600.000			0.652884			0.326442	+	0.945217i	$0.394150\pi$
$920$	14.1421	+	14.1421i	0.0153719	+	0.0153719i
$921$		0			0
$922$			864.000i			0.937093i
$923$	801.859			0.868753
$924$		0			0
$925$	625.000			0.675676
$926$		−	844.285i		−	0.911755i
$927$		0			0
$928$		−	216.000i		−	0.232759i
$929$			1260.06i			1.35637i	0.734893	+	0.678183i	$0.237232\pi$
$929$			1260.06i			1.35637i	−0.734893	+	0.678183i	$0.762768\pi$
$930$		0			0
$931$	504.000			0.541353
$932$	633.568			0.679794
$933$		0			0
$934$	1200.00			1.28480
$935$	56.5685	−	56.5685i	0.0605011	−	0.0605011i
$936$		0			0
$937$			1465.00i			1.56350i	0.623592	+	0.781750i	$0.285673\pi$
$937$			1465.00i			1.56350i	−0.623592	+	0.781750i	$0.714327\pi$
$938$	−926.310			−0.987537
$939$		0			0
$940$	−160.000	−	160.000i	−0.170213	−	0.170213i
$941$			255.973i			0.272022i	0.990707	+	0.136011i	$0.0434282\pi$
$941$			255.973i			0.272022i	−0.990707	+	0.136011i	$0.956572\pi$
$942$		0			0
$943$			74.0000i			0.0784730i
$944$			362.039i			0.383516i
$945$		0			0
$946$	128.000			0.135307
$947$	−407.294			−0.430088			−0.215044	−	0.976604i	$-0.568990\pi$
$947$	−407.294			−0.430088			−0.215044	+	0.976604i	$0.568990\pi$
$948$		0			0
$949$	153.000			0.161222
$950$		−	742.462i		−	0.781539i
$951$		0			0
$952$		−	160.000i		−	0.168067i
$953$	−497.803			−0.522354			−0.261177	−	0.965291i	$-0.584111\pi$
$953$	−497.803			−0.522354			−0.261177	+	0.965291i	$0.584111\pi$
$954$		0			0
$955$	−360.000	+	360.000i	−0.376963	+	0.376963i
$956$		−	410.122i		−	0.428998i
$957$		0			0
$958$		−	238.000i		−	0.248434i
$959$			1336.43i			1.39357i
$960$		0			0
$961$	639.000			0.664932
$962$	−318.198			−0.330767
$963$		0			0
$964$	158.000			0.163900
$965$	958.130	−	958.130i	0.992881	−	0.992881i
$966$		0			0
$967$		−	491.000i		−	0.507756i	−0.967236	−	0.253878i	$-0.918294\pi$
$967$		−	491.000i		−	0.507756i	0.967236	−	0.253878i	$-0.0817062\pi$
$968$	−336.583			−0.347710
$969$		0			0
$970$	205.000	−	205.000i	0.211340	−	0.211340i
$971$			509.117i			0.524322i	0.965024	+	0.262161i	$0.0844352\pi$
$971$			509.117i			0.524322i	−0.965024	+	0.262161i	$0.915565\pi$
$972$		0			0
$973$		−	185.000i		−	0.190134i
$974$			717.006i			0.736146i
$975$		0			0
$976$	−388.000			−0.397541
$977$	470.933			0.482020			0.241010	−	0.970523i	$-0.422521\pi$
$977$	470.933			0.482020			0.241010	+	0.970523i	$0.422521\pi$
$978$		0			0
$979$	−208.000			−0.212462
$980$	−169.706	−	169.706i	−0.173169	−	0.173169i
$981$		0			0
$982$			606.000i			0.617108i
$983$	601.041			0.611435			0.305718	−	0.952122i	$-0.401104\pi$
$983$	601.041			0.611435			0.305718	+	0.952122i	$0.401104\pi$
$984$		0			0
$985$	1120.00	+	1120.00i	1.13706	+	1.13706i
$986$		−	610.940i		−	0.619615i
$987$		0			0
$988$			378.000i			0.382591i
$989$			90.5097i			0.0915163i
$990$		0			0
$991$	−755.000			−0.761857			−0.380928	−	0.924605i	$-0.624396\pi$
$991$	−755.000			−0.761857			−0.380928	+	0.924605i	$0.624396\pi$
$992$	−226.274			−0.228099
$993$		0			0
$994$	−630.000			−0.633803
$995$	−519.723	−	519.723i	−0.522335	−	0.522335i
$996$		0			0
$997$			24.0000i			0.0240722i	0.999928	+	0.0120361i	$0.00383130\pi$
$997$			24.0000i			0.0240722i	−0.999928	+	0.0120361i	$0.996169\pi$
$998$	−1230.37			−1.23283
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.b.c.269.1	✓	4
3.2	odd	2	inner	270.3.b.c.269.4	yes	4
4.3	odd	2		2160.3.c.i.1889.1		4
5.2	odd	4		1350.3.d.f.701.1		2
5.3	odd	4		1350.3.d.g.701.2		2
5.4	even	2	inner	270.3.b.c.269.3	yes	4
9.2	odd	6		810.3.j.e.539.1		8
9.4	even	3		810.3.j.e.269.3		8
9.5	odd	6		810.3.j.e.269.2		8
9.7	even	3		810.3.j.e.539.4		8
12.11	even	2		2160.3.c.i.1889.4		4
15.2	even	4		1350.3.d.f.701.2		2
15.8	even	4		1350.3.d.g.701.1		2
15.14	odd	2	inner	270.3.b.c.269.2	yes	4
20.19	odd	2		2160.3.c.i.1889.3		4
45.4	even	6		810.3.j.e.269.1		8
45.14	odd	6		810.3.j.e.269.4		8
45.29	odd	6		810.3.j.e.539.3		8
45.34	even	6		810.3.j.e.539.2		8
60.59	even	2		2160.3.c.i.1889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.c.269.1	✓	4	1.1	even	1	trivial
270.3.b.c.269.2	yes	4	15.14	odd	2	inner
270.3.b.c.269.3	yes	4	5.4	even	2	inner
270.3.b.c.269.4	yes	4	3.2	odd	2	inner
810.3.j.e.269.1		8	45.4	even	6
810.3.j.e.269.2		8	9.5	odd	6
810.3.j.e.269.3		8	9.4	even	3
810.3.j.e.269.4		8	45.14	odd	6
810.3.j.e.539.1		8	9.2	odd	6
810.3.j.e.539.2		8	45.34	even	6
810.3.j.e.539.3		8	45.29	odd	6
810.3.j.e.539.4		8	9.7	even	3
1350.3.d.f.701.1		2	5.2	odd	4
1350.3.d.f.701.2		2	15.2	even	4
1350.3.d.g.701.1		2	15.8	even	4
1350.3.d.g.701.2		2	5.3	odd	4
2160.3.c.i.1889.1		4	4.3	odd	2
2160.3.c.i.1889.2		4	60.59	even	2
2160.3.c.i.1889.3		4	20.19	odd	2
2160.3.c.i.1889.4		4	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	270.3.b.c.269.1

Level	$270$
Weight	$3$
Character	270.269
Analytic conductor	$7.357$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$