LMFDB - Embedded newform 1200.3.l.w.401.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (of order $2$, degree $1$, not 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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.574198272.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 $ x^6 - 2x^5 - 7x^4 - 34x^3 + 81x^2 + 156*x + 198
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.6
Root			$-1.77752 - 2.54797i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.w.401.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.77752 + 1.13375i) q^{3} +5.85843 q^{7} +(6.42922 + 6.29803i) q^{9} +O(q^{10})$	$q+(2.77752 + 1.13375i) q^{3} +5.85843 q^{7} +(6.42922 + 6.29803i) q^{9} -12.4594i q^{11} +11.8584 q^{13} -29.2932i q^{17} -3.19332 q^{19} +(16.2719 + 6.64200i) q^{21} -19.5353i q^{23} +(10.7169 + 24.7820i) q^{27} -30.3022i q^{29} -3.57529 q^{31} +(14.1258 - 34.6061i) q^{33} -42.7639 q^{37} +(32.9370 + 13.4445i) q^{39} +6.39241i q^{41} +62.6224 q^{43} +69.3534i q^{47} -14.6788 q^{49} +(33.2112 - 81.3625i) q^{51} -57.7856i q^{53} +(-8.86951 - 3.62043i) q^{57} +78.9226i q^{59} +68.5189 q^{61} +(37.6651 + 36.8966i) q^{63} -90.5235 q^{67} +(22.1481 - 54.2596i) q^{69} +26.5398i q^{71} +40.0851 q^{73} -72.9923i q^{77} +148.858 q^{79} +(1.66962 + 80.9828i) q^{81} +9.36722i q^{83} +(34.3551 - 84.1649i) q^{87} -109.706i q^{89} +69.4718 q^{91} +(-9.93044 - 4.05349i) q^{93} -161.849 q^{97} +(78.4695 - 80.1039i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9	$ 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9} + 26 q^{13} - 50 q^{19} - 18 q^{21} - 26 q^{27} + 114 q^{31} + 82 q^{33} + 76 q^{37} + 6 q^{39} - 2 q^{43} + 76 q^{49} + 6 q^{51} + 172 q^{57} + 62 q^{61} + 150 q^{63} - 422 q^{67} + 156 q^{69} + 72 q^{73} + 76 q^{79} - 224 q^{81} + 172 q^{87} + 310 q^{91} + 110 q^{93} - 470 q^{97} + 250 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9 + 26 * q^13 - 50 * q^19 - 18 * q^21 - 26 * q^27 + 114 * q^31 + 82 * q^33 + 76 * q^37 + 6 * q^39 - 2 * q^43 + 76 * q^49 + 6 * q^51 + 172 * q^57 + 62 * q^61 + 150 * q^63 - 422 * q^67 + 156 * q^69 + 72 * q^73 + 76 * q^79 - 224 * q^81 + 172 * q^87 + 310 * q^91 + 110 * q^93 - 470 * q^97 + 250 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.77752	+	1.13375i	0.925839	+	0.377917i
$3$	2.77752	+	1.13375i	0.925839	+	0.377917i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.85843			0.836919			0.418459	−	0.908236i	$-0.362570\pi$
$7$	5.85843			0.836919			0.418459	+	0.908236i	$0.362570\pi$
$8$		0			0
$9$	6.42922	+	6.29803i	0.714357	+	0.699781i
$10$		0			0
$11$		−	12.4594i		−	1.13267i	−0.824175	−	0.566335i	$-0.808361\pi$
$11$		−	12.4594i		−	1.13267i	0.824175	−	0.566335i	$-0.191639\pi$
$12$		0			0
$13$	11.8584			0.912187			0.456093	−	0.889932i	$-0.349248\pi$
$13$	11.8584			0.912187			0.456093	+	0.889932i	$0.349248\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	29.2932i		−	1.72313i	−0.507646	−	0.861566i	$-0.669484\pi$
$17$		−	29.2932i		−	1.72313i	0.507646	−	0.861566i	$-0.330516\pi$
$18$		0			0
$19$	−3.19332			−0.168070			−0.0840348	−	0.996463i	$-0.526781\pi$
$19$	−3.19332			−0.168070			−0.0840348	+	0.996463i	$0.526781\pi$
$20$		0			0
$21$	16.2719	+	6.64200i	0.774852	+	0.316286i
$22$		0			0
$23$		−	19.5353i		−	0.849360i	−0.905344	−	0.424680i	$-0.860387\pi$
$23$		−	19.5353i		−	0.849360i	0.905344	−	0.424680i	$-0.139613\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	10.7169	+	24.7820i	0.396921	+	0.917853i
$28$		0			0
$29$		−	30.3022i		−	1.04490i	−0.852669	−	0.522451i	$-0.825018\pi$
$29$		−	30.3022i		−	1.04490i	0.852669	−	0.522451i	$-0.174982\pi$
$30$		0			0
$31$	−3.57529			−0.115332			−0.0576660	−	0.998336i	$-0.518366\pi$
$31$	−3.57529			−0.115332			−0.0576660	+	0.998336i	$0.518366\pi$
$32$		0			0
$33$	14.1258	−	34.6061i	0.428055	−	1.04867i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−42.7639			−1.15578			−0.577891	−	0.816114i	$-0.696124\pi$
$37$	−42.7639			−1.15578			−0.577891	+	0.816114i	$0.696124\pi$
$38$		0			0
$39$	32.9370	+	13.4445i	0.844539	+	0.344731i
$40$		0			0
$41$			6.39241i			0.155913i	0.996957	+	0.0779563i	$0.0248395\pi$
$41$			6.39241i			0.155913i	−0.996957	+	0.0779563i	$0.975161\pi$
$42$		0			0
$43$	62.6224			1.45633			0.728167	−	0.685400i	$-0.240372\pi$
$43$	62.6224			1.45633			0.728167	+	0.685400i	$0.240372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			69.3534i			1.47560i	0.675017	+	0.737802i	$0.264136\pi$
$47$			69.3534i			1.47560i	−0.675017	+	0.737802i	$0.735864\pi$
$48$		0			0
$49$	−14.6788			−0.299567
$50$		0			0
$51$	33.2112	−	81.3625i	0.651201	−	1.59534i
$52$		0			0
$53$		−	57.7856i		−	1.09029i	−0.838340	−	0.545147i	$-0.816474\pi$
$53$		−	57.7856i		−	1.09029i	0.838340	−	0.545147i	$-0.183526\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.86951	−	3.62043i	−0.155605	−	0.0635164i
$58$		0			0
$59$			78.9226i			1.33767i	0.743411	+	0.668835i	$0.233207\pi$
$59$			78.9226i			1.33767i	−0.743411	+	0.668835i	$0.766793\pi$
$60$		0			0
$61$	68.5189			1.12326			0.561630	−	0.827388i	$-0.310174\pi$
$61$	68.5189			1.12326			0.561630	+	0.827388i	$0.310174\pi$
$62$		0			0
$63$	37.6651	+	36.8966i	0.597859	+	0.585660i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−90.5235			−1.35110			−0.675549	−	0.737315i	$-0.736093\pi$
$67$	−90.5235			−1.35110			−0.675549	+	0.737315i	$0.736093\pi$
$68$		0			0
$69$	22.1481	−	54.2596i	0.320988	−	0.786371i
$70$		0			0
$71$			26.5398i			0.373799i	0.982379	+	0.186900i	$0.0598439\pi$
$71$			26.5398i			0.373799i	−0.982379	+	0.186900i	$0.940156\pi$
$72$		0			0
$73$	40.0851			0.549112			0.274556	−	0.961571i	$-0.411469\pi$
$73$	40.0851			0.549112			0.274556	+	0.961571i	$0.411469\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	72.9923i		−	0.947952i
$78$		0			0
$79$	148.858			1.88428			0.942140	−	0.335220i	$-0.108811\pi$
$79$	148.858			1.88428			0.942140	+	0.335220i	$0.108811\pi$
$80$		0			0
$81$	1.66962	+	80.9828i	0.0206125	+	0.999788i
$82$		0			0
$83$			9.36722i			0.112858i	0.998407	+	0.0564290i	$0.0179715\pi$
$83$			9.36722i			0.112858i	−0.998407	+	0.0564290i	$0.982029\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	34.3551	−	84.1649i	0.394887	−	0.967412i
$88$		0			0
$89$		−	109.706i		−	1.23265i	−0.787490	−	0.616327i	$-0.788620\pi$
$89$		−	109.706i		−	1.23265i	0.787490	−	0.616327i	$-0.211380\pi$
$90$		0			0
$91$	69.4718			0.763426
$92$		0			0
$93$	−9.93044	−	4.05349i	−0.106779	−	0.0435859i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−161.849			−1.66855			−0.834274	−	0.551351i	$-0.814113\pi$
$97$	−161.849			−1.66855			−0.834274	+	0.551351i	$0.814113\pi$
$98$		0			0
$99$	78.4695	−	80.1039i	0.792621	−	0.809131i
$100$		0			0
$101$		−	139.273i		−	1.37894i	−0.724315	−	0.689470i	$-0.757844\pi$
$101$		−	139.273i		−	1.37894i	0.724315	−	0.689470i	$-0.242156\pi$
$102$		0			0
$103$	1.22672			0.0119099			0.00595493	−	0.999982i	$-0.498104\pi$
$103$	1.22672			0.0119099			0.00595493	+	0.999982i	$0.498104\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			199.298i			1.86260i	0.364259	+	0.931298i	$0.381322\pi$
$107$			199.298i			1.86260i	−0.364259	+	0.931298i	$0.618678\pi$
$108$		0			0
$109$	209.480			1.92184			0.960920	−	0.276828i	$-0.0892831\pi$
$109$	209.480			1.92184			0.960920	+	0.276828i	$0.0892831\pi$
$110$		0			0
$111$	−118.778	−	48.4837i	−1.07007	−	0.436790i
$112$		0			0
$113$			126.007i			1.11510i	0.830143	+	0.557551i	$0.188259\pi$
$113$			126.007i			1.11510i	−0.830143	+	0.557551i	$0.811741\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	76.2404	+	74.6848i	0.651627	+	0.638331i
$118$		0			0
$119$		−	171.612i		−	1.44212i
$120$		0			0
$121$	−34.2357			−0.282940
$122$		0			0
$123$	−7.24741	+	17.7550i	−0.0589220	+	0.144350i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−49.4064			−0.389026			−0.194513	−	0.980900i	$-0.562313\pi$
$127$	−49.4064			−0.389026			−0.194513	+	0.980900i	$0.562313\pi$
$128$		0			0
$129$	173.935	+	70.9982i	1.34833	+	0.550374i
$130$		0			0
$131$			102.351i			0.781304i	0.920538	+	0.390652i	$0.127750\pi$
$131$			102.351i			0.781304i	−0.920538	+	0.390652i	$0.872250\pi$
$132$		0			0
$133$	−18.7078			−0.140661
$134$		0			0
$135$		0			0
$136$		0			0
$137$			123.181i			0.899129i	0.893248	+	0.449565i	$0.148421\pi$
$137$			123.181i			0.899129i	−0.893248	+	0.449565i	$0.851579\pi$
$138$		0			0
$139$	−35.9285			−0.258479			−0.129239	−	0.991613i	$-0.541254\pi$
$139$	−35.9285			−0.258479			−0.129239	+	0.991613i	$0.541254\pi$
$140$		0			0
$141$	−78.6295	+	192.630i	−0.557656	+	1.36617i
$142$		0			0
$143$		−	147.748i		−	1.03321i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−40.7706	−	16.6421i	−0.277351	−	0.113212i
$148$		0			0
$149$			19.2425i			0.129144i	0.997913	+	0.0645722i	$0.0205683\pi$
$149$			19.2425i			0.129144i	−0.997913	+	0.0645722i	$0.979432\pi$
$150$		0			0
$151$	67.6695			0.448142			0.224071	−	0.974573i	$-0.428065\pi$
$151$	67.6695			0.448142			0.224071	+	0.974573i	$0.428065\pi$
$152$		0			0
$153$	184.490	−	188.332i	1.20581	−	1.23093i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−27.0003			−0.171977			−0.0859883	−	0.996296i	$-0.527405\pi$
$157$	−27.0003			−0.171977			−0.0859883	+	0.996296i	$0.527405\pi$
$158$		0			0
$159$	65.5145	−	160.501i	0.412041	−	1.00944i
$160$		0			0
$161$		−	114.446i		−	0.710845i
$162$		0			0
$163$	−150.362			−0.922466			−0.461233	−	0.887279i	$-0.652593\pi$
$163$	−150.362			−0.922466			−0.461233	+	0.887279i	$0.652593\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	252.006i		−	1.50902i	−0.656290	−	0.754509i	$-0.727875\pi$
$167$		−	252.006i		−	1.50902i	0.656290	−	0.754509i	$-0.272125\pi$
$168$		0			0
$169$	−28.3776			−0.167915
$170$		0			0
$171$	−20.5305	−	20.1116i	−0.120062	−	0.117612i
$172$		0			0
$173$			295.790i			1.70977i	0.518820	+	0.854884i	$0.326372\pi$
$173$			295.790i			1.70977i	−0.518820	+	0.854884i	$0.673628\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−89.4786	+	219.209i	−0.505529	+	1.23847i
$178$		0			0
$179$		−	13.6377i		−	0.0761884i	−0.999274	−	0.0380942i	$-0.987871\pi$
$179$		−	13.6377i		−	0.0761884i	0.999274	−	0.0380942i	$-0.0121287\pi$
$180$		0			0
$181$	323.377			1.78661			0.893307	−	0.449448i	$-0.148379\pi$
$181$	323.377			1.78661			0.893307	+	0.449448i	$0.148379\pi$
$182$		0			0
$183$	190.312	+	77.6834i	1.03996	+	0.424499i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−364.975			−1.95174
$188$		0			0
$189$	62.7840	+	145.184i	0.332190	+	0.768168i
$190$		0			0
$191$			326.893i			1.71148i	0.517406	+	0.855740i	$0.326898\pi$
$191$			326.893i			1.71148i	−0.517406	+	0.855740i	$0.673102\pi$
$192$		0			0
$193$	−23.8007			−0.123319			−0.0616597	−	0.998097i	$-0.519639\pi$
$193$	−23.8007			−0.123319			−0.0616597	+	0.998097i	$0.519639\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.71030i			0.0442147i	0.999756	+	0.0221074i	$0.00703756\pi$
$197$			8.71030i			0.0442147i	−0.999756	+	0.0221074i	$0.992962\pi$
$198$		0			0
$199$	−86.9997			−0.437184			−0.218592	−	0.975816i	$-0.570146\pi$
$199$	−86.9997			−0.437184			−0.218592	+	0.975816i	$0.570146\pi$
$200$		0			0
$201$	−251.431	−	102.631i	−1.25090	−	0.510603i
$202$		0			0
$203$		−	177.523i		−	0.874499i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	123.034	−	125.596i	0.594366	−	0.606746i
$208$		0			0
$209$			39.7867i			0.190367i
$210$		0			0
$211$	248.258			1.17658			0.588290	−	0.808650i	$-0.299801\pi$
$211$	248.258			1.17658			0.588290	+	0.808650i	$0.299801\pi$
$212$		0			0
$213$	−30.0895	+	73.7147i	−0.141265	+	0.346078i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−20.9456			−0.0965235
$218$		0			0
$219$	111.337	+	45.4466i	0.508389	+	0.207519i
$220$		0			0
$221$		−	347.372i		−	1.57182i
$222$		0			0
$223$	8.54624			0.0383239			0.0191620	−	0.999816i	$-0.493900\pi$
$223$	8.54624			0.0383239			0.0191620	+	0.999816i	$0.493900\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	79.4623i		−	0.350054i	−0.984564	−	0.175027i	$-0.943999\pi$
$227$		−	79.4623i		−	0.350054i	0.984564	−	0.175027i	$-0.0560013\pi$
$228$		0			0
$229$	−167.301			−0.730574			−0.365287	−	0.930895i	$-0.619029\pi$
$229$	−167.301			−0.730574			−0.365287	+	0.930895i	$0.619029\pi$
$230$		0			0
$231$	82.7551	−	202.737i	0.358247	−	0.877651i
$232$		0			0
$233$		−	227.399i		−	0.975963i	−0.872854	−	0.487982i	$-0.837733\pi$
$233$		−	227.399i		−	0.975963i	0.872854	−	0.487982i	$-0.162267\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	413.456	+	168.768i	1.74454	+	0.712102i
$238$		0			0
$239$			227.491i			0.951846i	0.879487	+	0.475923i	$0.157886\pi$
$239$			227.491i			0.951846i	−0.879487	+	0.475923i	$0.842114\pi$
$240$		0			0
$241$	−285.556			−1.18488			−0.592440	−	0.805614i	$-0.701835\pi$
$241$	−285.556			−1.18488			−0.592440	+	0.805614i	$0.701835\pi$
$242$		0			0
$243$	−87.1770	+	226.824i	−0.358753	+	0.933433i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−37.8678			−0.153311
$248$		0			0
$249$	−10.6201	+	26.0176i	−0.0426510	+	0.104488i
$250$		0			0
$251$			63.3323i			0.252320i	0.992010	+	0.126160i	$0.0402653\pi$
$251$			63.3323i			0.252320i	−0.992010	+	0.126160i	$0.959735\pi$
$252$		0			0
$253$	−243.397			−0.962044
$254$		0			0
$255$		0			0
$256$		0			0
$257$			37.9769i			0.147770i	0.997267	+	0.0738851i	$0.0235398\pi$
$257$			37.9769i			0.147770i	−0.997267	+	0.0738851i	$0.976460\pi$
$258$		0			0
$259$	−250.530			−0.967296
$260$		0			0
$261$	190.844	−	194.819i	0.731203	−	0.746434i
$262$		0			0
$263$		−	112.740i		−	0.428670i	−0.976760	−	0.214335i	$-0.931242\pi$
$263$		−	112.740i		−	0.428670i	0.976760	−	0.214335i	$-0.0687584\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	124.380	−	304.711i	0.465841	−	1.14124i
$268$		0			0
$269$		−	95.8931i		−	0.356480i	−0.983987	−	0.178240i	$-0.942960\pi$
$269$		−	95.8931i		−	0.356480i	0.983987	−	0.178240i	$-0.0570403\pi$
$270$		0			0
$271$	73.8204			0.272400			0.136200	−	0.990681i	$-0.456511\pi$
$271$	73.8204			0.272400			0.136200	+	0.990681i	$0.456511\pi$
$272$		0			0
$273$	192.959	+	78.7637i	0.706810	+	0.288512i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	44.5583			0.160860			0.0804301	−	0.996760i	$-0.474371\pi$
$277$	44.5583			0.160860			0.0804301	+	0.996760i	$0.474371\pi$
$278$		0			0
$279$	−22.9863	−	22.5173i	−0.0823882	−	0.0807071i
$280$		0			0
$281$			179.919i			0.640280i	0.947370	+	0.320140i	$0.103730\pi$
$281$			179.919i			0.640280i	−0.947370	+	0.320140i	$0.896270\pi$
$282$		0			0
$283$	106.070			0.374806			0.187403	−	0.982283i	$-0.439993\pi$
$283$	106.070			0.374806			0.187403	+	0.982283i	$0.439993\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			37.4495i			0.130486i
$288$		0			0
$289$	−569.093			−1.96918
$290$		0			0
$291$	−449.539	−	183.497i	−1.54481	−	0.630573i
$292$		0			0
$293$		−	198.829i		−	0.678599i	−0.940678	−	0.339299i	$-0.889810\pi$
$293$		−	198.829i		−	0.678599i	0.940678	−	0.339299i	$-0.110190\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	308.768	−	133.525i	1.03962	−	0.449580i
$298$		0			0
$299$		−	231.658i		−	0.774775i
$300$		0			0
$301$	366.869			1.21883
$302$		0			0
$303$	157.901	−	386.833i	0.521125	−	1.27668i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	245.107			0.798395			0.399197	−	0.916865i	$-0.369289\pi$
$307$	245.107			0.798395			0.399197	+	0.916865i	$0.369289\pi$
$308$		0			0
$309$	3.40723	+	1.39079i	0.0110266	+	0.00450094i
$310$		0			0
$311$		−	6.45769i		−	0.0207643i	−0.999946	−	0.0103821i	$-0.996695\pi$
$311$		−	6.45769i		−	0.0207643i	0.999946	−	0.0103821i	$-0.00330480\pi$
$312$		0			0
$313$	401.243			1.28193			0.640963	−	0.767572i	$-0.278535\pi$
$313$	401.243			1.28193			0.640963	+	0.767572i	$0.278535\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	321.613i		−	1.01455i	−0.861783	−	0.507277i	$-0.830652\pi$
$317$		−	321.613i		−	1.01455i	0.861783	−	0.507277i	$-0.169348\pi$
$318$		0			0
$319$	−377.546			−1.18353
$320$		0			0
$321$	−225.954	+	553.553i	−0.703907	+	1.72446i
$322$		0			0
$323$			93.5427i			0.289606i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	581.836	+	237.499i	1.77931	+	0.726296i
$328$		0			0
$329$			406.302i			1.23496i
$330$		0			0
$331$	−28.3052			−0.0855141			−0.0427571	−	0.999085i	$-0.513614\pi$
$331$	−28.3052			−0.0855141			−0.0427571	+	0.999085i	$0.513614\pi$
$332$		0			0
$333$	−274.939	−	269.329i	−0.825641	−	0.808795i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−453.867			−1.34679			−0.673393	−	0.739285i	$-0.735164\pi$
$337$	−453.867			−1.34679			−0.673393	+	0.739285i	$0.735164\pi$
$338$		0			0
$339$	−142.860	+	349.985i	−0.421416	+	1.03241i
$340$		0			0
$341$			44.5458i			0.130633i
$342$		0			0
$343$	−373.058			−1.08763
$344$		0			0
$345$		0			0
$346$		0			0
$347$			148.725i			0.428602i	0.976768	+	0.214301i	$0.0687473\pi$
$347$			148.725i			0.428602i	−0.976768	+	0.214301i	$0.931253\pi$
$348$		0			0
$349$	141.120			0.404355			0.202177	−	0.979349i	$-0.435198\pi$
$349$	141.120			0.404355			0.202177	+	0.979349i	$0.435198\pi$
$350$		0			0
$351$	127.085	+	293.876i	0.362066	+	0.837253i
$352$		0			0
$353$			209.069i			0.592263i	0.955147	+	0.296132i	$0.0956967\pi$
$353$			209.069i			0.592263i	−0.955147	+	0.296132i	$0.904303\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	194.566	−	476.656i	0.545002	−	1.33517i
$358$		0			0
$359$			457.136i			1.27336i	0.771128	+	0.636680i	$0.219693\pi$
$359$			457.136i			1.27336i	−0.771128	+	0.636680i	$0.780307\pi$
$360$		0			0
$361$	−350.803			−0.971753
$362$		0			0
$363$	−95.0904	−	38.8148i	−0.261957	−	0.106928i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−343.509			−0.935990			−0.467995	−	0.883731i	$-0.655024\pi$
$367$	−343.509			−0.935990			−0.467995	+	0.883731i	$0.655024\pi$
$368$		0			0
$369$	−40.2596	+	41.0982i	−0.109105	+	0.111377i
$370$		0			0
$371$		−	338.533i		−	0.912488i
$372$		0			0
$373$	−385.038			−1.03227			−0.516137	−	0.856506i	$-0.672631\pi$
$373$	−385.038			−1.03227			−0.516137	+	0.856506i	$0.672631\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	359.336i		−	0.953147i
$378$		0			0
$379$	−316.721			−0.835676			−0.417838	−	0.908522i	$-0.637212\pi$
$379$	−316.721			−0.835676			−0.417838	+	0.908522i	$0.637212\pi$
$380$		0			0
$381$	−137.227	−	56.0145i	−0.360176	−	0.147020i
$382$		0			0
$383$			576.465i			1.50513i	0.658518	+	0.752565i	$0.271184\pi$
$383$			576.465i			1.50513i	−0.658518	+	0.752565i	$0.728816\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	402.613	+	394.398i	1.04034	+	1.01912i
$388$		0			0
$389$			54.0496i			0.138945i	0.997584	+	0.0694725i	$0.0221316\pi$
$389$			54.0496i			0.138945i	−0.997584	+	0.0694725i	$0.977868\pi$
$390$		0			0
$391$	−572.251			−1.46356
$392$		0			0
$393$	−116.040	+	284.281i	−0.295268	+	0.723362i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	27.0211			0.0680631			0.0340316	−	0.999421i	$-0.489165\pi$
$397$	27.0211			0.0680631			0.0340316	+	0.999421i	$0.489165\pi$
$398$		0			0
$399$	−51.9614	−	21.2101i	−0.130229	−	0.0531580i
$400$		0			0
$401$			282.008i			0.703262i	0.936139	+	0.351631i	$0.114373\pi$
$401$			282.008i			0.703262i	−0.936139	+	0.351631i	$0.885627\pi$
$402$		0			0
$403$	−42.3973			−0.105204
$404$		0			0
$405$		0			0
$406$		0			0
$407$			532.811i			1.30912i
$408$		0			0
$409$	−704.603			−1.72274			−0.861372	−	0.507974i	$-0.830395\pi$
$409$	−704.603			−1.72274			−0.861372	+	0.507974i	$0.830395\pi$
$410$		0			0
$411$	−139.656	+	342.137i	−0.339796	+	0.832449i
$412$		0			0
$413$			462.362i			1.11952i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−99.7922	−	40.7340i	−0.239310	−	0.0976835i
$418$		0			0
$419$			345.764i			0.825212i	0.910909	+	0.412606i	$0.135381\pi$
$419$			345.764i			0.825212i	−0.910909	+	0.412606i	$0.864619\pi$
$420$		0			0
$421$	585.603			1.39098			0.695490	−	0.718536i	$-0.255187\pi$
$421$	585.603			1.39098			0.695490	+	0.718536i	$0.255187\pi$
$422$		0			0
$423$	−436.790	+	445.888i	−1.03260	+	1.05411i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	401.413			0.940077
$428$		0			0
$429$	167.510	−	410.374i	0.390466	−	0.956583i
$430$		0			0
$431$		−	824.304i		−	1.91254i	−0.292487	−	0.956269i	$-0.594483\pi$
$431$		−	824.304i		−	1.91254i	0.292487	−	0.956269i	$-0.405517\pi$
$432$		0			0
$433$	−685.949			−1.58418			−0.792089	−	0.610406i	$-0.791006\pi$
$433$	−685.949			−1.58418			−0.792089	+	0.610406i	$0.791006\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			62.3824i			0.142751i
$438$		0			0
$439$	−166.154			−0.378482			−0.189241	−	0.981931i	$-0.560603\pi$
$439$	−166.154			−0.378482			−0.189241	+	0.981931i	$0.560603\pi$
$440$		0			0
$441$	−94.3731	−	92.4475i	−0.213998	−	0.209632i
$442$		0			0
$443$			241.838i			0.545910i	0.962027	+	0.272955i	$0.0880010\pi$
$443$			241.838i			0.545910i	−0.962027	+	0.272955i	$0.911999\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.8162	+	53.4464i	−0.0488059	+	0.119567i
$448$		0			0
$449$		−	265.195i		−	0.590634i	−0.955399	−	0.295317i	$-0.904575\pi$
$449$		−	265.195i		−	0.590634i	0.955399	−	0.295317i	$-0.0954253\pi$
$450$		0			0
$451$	79.6454			0.176597
$452$		0			0
$453$	187.953	+	76.7203i	0.414908	+	0.169361i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−603.375			−1.32029			−0.660147	−	0.751136i	$-0.729506\pi$
$457$	−603.375			−1.32029			−0.660147	+	0.751136i	$0.729506\pi$
$458$		0			0
$459$	725.946	−	313.931i	1.58158	−	0.683946i
$460$		0			0
$461$			772.098i			1.67483i	0.546566	+	0.837416i	$0.315935\pi$
$461$			772.098i			1.67483i	−0.546566	+	0.837416i	$0.684065\pi$
$462$		0			0
$463$	−643.789			−1.39047			−0.695236	−	0.718781i	$-0.744700\pi$
$463$	−643.789			−1.39047			−0.695236	+	0.718781i	$0.744700\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	503.823i		−	1.07885i	−0.842034	−	0.539425i	$-0.818642\pi$
$467$		−	503.823i		−	1.07885i	0.842034	−	0.539425i	$-0.181358\pi$
$468$		0			0
$469$	−530.326			−1.13076
$470$		0			0
$471$	−74.9939	−	30.6117i	−0.159223	−	0.0649929i
$472$		0			0
$473$		−	780.235i		−	1.64955i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	363.936	−	371.516i	0.762968	−	0.778860i
$478$		0			0
$479$		−	248.720i		−	0.519249i	−0.965710	−	0.259624i	$-0.916401\pi$
$479$		−	248.720i		−	0.519249i	0.965710	−	0.259624i	$-0.0835987\pi$
$480$		0			0
$481$	−507.113			−1.05429
$482$		0			0
$483$	129.753	−	317.876i	0.268640	−	0.658128i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−586.988			−1.20531			−0.602657	−	0.798001i	$-0.705891\pi$
$487$	−586.988			−1.20531			−0.602657	+	0.798001i	$0.705891\pi$
$488$		0			0
$489$	−417.633	−	170.473i	−0.854055	−	0.348616i
$490$		0			0
$491$			106.210i			0.216314i	0.994134	+	0.108157i	$0.0344949\pi$
$491$			106.210i			0.216314i	−0.994134	+	0.108157i	$0.965505\pi$
$492$		0			0
$493$	−887.649			−1.80050
$494$		0			0
$495$		0			0
$496$		0			0
$497$			155.481i			0.312840i
$498$		0			0
$499$	−404.643			−0.810908			−0.405454	−	0.914115i	$-0.632887\pi$
$499$	−404.643			−0.810908			−0.405454	+	0.914115i	$0.632887\pi$
$500$		0			0
$501$	285.712	−	699.951i	0.570284	−	1.39711i
$502$		0			0
$503$			457.971i			0.910478i	0.890369	+	0.455239i	$0.150446\pi$
$503$			457.971i			0.910478i	−0.890369	+	0.455239i	$0.849554\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−78.8194	−	32.1732i	−0.155462	−	0.0634579i
$508$		0			0
$509$		−	913.401i		−	1.79450i	−0.441522	−	0.897250i	$-0.645561\pi$
$509$		−	913.401i		−	1.79450i	0.441522	−	0.897250i	$-0.354439\pi$
$510$		0			0
$511$	234.836			0.459562
$512$		0			0
$513$	−34.2224	−	79.1370i	−0.0667103	−	0.154263i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	864.099			1.67137
$518$		0			0
$519$	−335.352	+	821.561i	−0.646150	+	1.58297i
$520$		0			0
$521$			482.739i			0.926563i	0.886211	+	0.463282i	$0.153328\pi$
$521$			482.739i			0.926563i	−0.886211	+	0.463282i	$0.846672\pi$
$522$		0			0
$523$	334.636			0.639839			0.319919	−	0.947445i	$-0.396344\pi$
$523$	334.636			0.639839			0.319919	+	0.947445i	$0.396344\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			104.732i			0.198732i
$528$		0			0
$529$	147.373			0.278588
$530$		0			0
$531$	−497.057	+	507.410i	−0.936077	+	0.955575i
$532$		0			0
$533$			75.8040i			0.142221i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	15.4618	−	37.8790i	0.0287929	−	0.0705383i
$538$		0			0
$539$			182.888i			0.339311i
$540$		0			0
$541$	−82.4307			−0.152367			−0.0761836	−	0.997094i	$-0.524274\pi$
$541$	−82.4307			−0.152367			−0.0761836	+	0.997094i	$0.524274\pi$
$542$		0			0
$543$	898.185	+	366.629i	1.65412	+	0.675192i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	663.343			1.21269			0.606346	−	0.795201i	$-0.292635\pi$
$547$	663.343			1.21269			0.606346	+	0.795201i	$0.292635\pi$
$548$		0			0
$549$	440.523	+	431.534i	0.802409	+	0.786036i
$550$		0			0
$551$			96.7646i			0.175616i
$552$		0			0
$553$	872.075			1.57699
$554$		0			0
$555$		0			0
$556$		0			0
$557$			487.179i			0.874648i	0.899304	+	0.437324i	$0.144074\pi$
$557$			487.179i			0.874648i	−0.899304	+	0.437324i	$0.855926\pi$
$558$		0			0
$559$	742.603			1.32845
$560$		0			0
$561$	−1013.72	−	413.791i	−1.80700	−	0.737595i
$562$		0			0
$563$		−	29.6248i		−	0.0526196i	−0.999654	−	0.0263098i	$-0.991624\pi$
$563$		−	29.6248i		−	0.0526196i	0.999654	−	0.0263098i	$-0.00837563\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9.78133	+	474.432i	0.0172510	+	0.836741i
$568$		0			0
$569$			946.809i			1.66399i	0.554784	+	0.831994i	$0.312801\pi$
$569$			946.809i			1.66399i	−0.554784	+	0.831994i	$0.687199\pi$
$570$		0			0
$571$	385.718			0.675514			0.337757	−	0.941233i	$-0.390332\pi$
$571$	385.718			0.675514			0.337757	+	0.941233i	$0.390332\pi$
$572$		0			0
$573$	−370.615	+	907.951i	−0.646798	+	1.58456i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−216.888			−0.375889			−0.187944	−	0.982180i	$-0.560182\pi$
$577$	−216.888			−0.375889			−0.187944	+	0.982180i	$0.560182\pi$
$578$		0			0
$579$	−66.1067	−	26.9840i	−0.114174	−	0.0466045i
$580$		0			0
$581$			54.8772i			0.0944530i
$582$		0			0
$583$	−719.972			−1.23494
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	499.326i		−	0.850641i	−0.905043	−	0.425321i	$-0.860161\pi$
$587$		−	499.326i		−	0.850641i	0.905043	−	0.425321i	$-0.139839\pi$
$588$		0			0
$589$	11.4171			0.0193838
$590$		0			0
$591$	−9.87531	+	24.1930i	−0.0167095	+	0.0409357i
$592$		0			0
$593$		−	339.358i		−	0.572274i	−0.958189	−	0.286137i	$-0.907629\pi$
$593$		−	339.358i		−	0.572274i	0.958189	−	0.286137i	$-0.0923712\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−241.643	−	98.6360i	−0.404762	−	0.165219i
$598$		0			0
$599$		−	641.821i		−	1.07149i	−0.844381	−	0.535744i	$-0.820031\pi$
$599$		−	641.821i		−	1.07149i	0.844381	−	0.535744i	$-0.179969\pi$
$600$		0			0
$601$	121.025			0.201372			0.100686	−	0.994918i	$-0.467896\pi$
$601$	121.025			0.201372			0.100686	+	0.994918i	$0.467896\pi$
$602$		0			0
$603$	−581.995	−	570.120i	−0.965166	−	0.945473i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−218.456			−0.359895			−0.179947	−	0.983676i	$-0.557593\pi$
$607$	−218.456			−0.359895			−0.179947	+	0.983676i	$0.557593\pi$
$608$		0			0
$609$	201.267	−	493.074i	0.330488	−	0.809645i
$610$		0			0
$611$			822.422i			1.34603i
$612$		0			0
$613$	201.848			0.329279			0.164639	−	0.986354i	$-0.447354\pi$
$613$	201.848			0.329279			0.164639	+	0.986354i	$0.447354\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			745.812i			1.20877i	0.796692	+	0.604386i	$0.206582\pi$
$617$			745.812i			1.20877i	−0.796692	+	0.604386i	$0.793418\pi$
$618$		0			0
$619$	474.849			0.767124			0.383562	−	0.923515i	$-0.374697\pi$
$619$	474.849			0.767124			0.383562	+	0.923515i	$0.374697\pi$
$620$		0			0
$621$	484.124	−	209.357i	0.779587	−	0.337128i
$622$		0			0
$623$		−	642.706i		−	1.03163i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−45.1083	+	110.508i	−0.0719430	+	0.176249i
$628$		0			0
$629$			1252.69i			1.99156i
$630$		0			0
$631$	−207.368			−0.328634			−0.164317	−	0.986408i	$-0.552542\pi$
$631$	−207.368			−0.328634			−0.164317	+	0.986408i	$0.552542\pi$
$632$		0			0
$633$	689.542	+	281.463i	1.08932	+	0.444650i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−174.067			−0.273261
$638$		0			0
$639$	−167.148	+	170.630i	−0.261578	+	0.267026i
$640$		0			0
$641$			371.542i			0.579629i	0.957083	+	0.289815i	$0.0935937\pi$
$641$			371.542i			0.579629i	−0.957083	+	0.289815i	$0.906406\pi$
$642$		0			0
$643$	779.511			1.21230			0.606151	−	0.795349i	$-0.292713\pi$
$643$	779.511			1.21230			0.606151	+	0.795349i	$0.292713\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1129.46i		−	1.74568i	−0.488006	−	0.872840i	$-0.662276\pi$
$647$		−	1129.46i		−	1.74568i	0.488006	−	0.872840i	$-0.337724\pi$
$648$		0			0
$649$	983.325			1.51514
$650$		0			0
$651$	−58.1768	−	23.7471i	−0.0893652	−	0.0364779i
$652$		0			0
$653$		−	327.316i		−	0.501250i	−0.968084	−	0.250625i	$-0.919364\pi$
$653$		−	327.316i		−	0.501250i	0.968084	−	0.250625i	$-0.0806361\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	257.716	+	252.457i	0.392262	+	0.384258i
$658$		0			0
$659$		−	949.895i		−	1.44142i	−0.693237	−	0.720710i	$-0.743816\pi$
$659$		−	949.895i		−	1.44142i	0.693237	−	0.720710i	$-0.256184\pi$
$660$		0			0
$661$	736.462			1.11416			0.557082	−	0.830458i	$-0.311921\pi$
$661$	736.462			1.11416			0.557082	+	0.830458i	$0.311921\pi$
$662$		0			0
$663$	393.833	−	964.831i	0.594017	−	1.45525i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−591.961			−0.887498
$668$		0			0
$669$	23.7373	+	9.68931i	0.0354818	+	0.0144833i
$670$		0			0
$671$		−	853.701i		−	1.27228i
$672$		0			0
$673$	602.296			0.894943			0.447471	−	0.894298i	$-0.352325\pi$
$673$	602.296			0.894943			0.447471	+	0.894298i	$0.352325\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	830.123i		−	1.22618i	−0.790014	−	0.613089i	$-0.789927\pi$
$677$		−	830.123i		−	1.22618i	0.790014	−	0.613089i	$-0.210073\pi$
$678$		0			0
$679$	−948.182			−1.39644
$680$		0			0
$681$	90.0905	−	220.708i	0.132291	−	0.324094i
$682$		0			0
$683$			276.640i			0.405037i	0.979278	+	0.202518i	$0.0649126\pi$
$683$			276.640i			0.405037i	−0.979278	+	0.202518i	$0.935087\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−464.683	−	189.678i	−0.676394	−	0.276097i
$688$		0			0
$689$		−	685.247i		−	0.994553i
$690$		0			0
$691$	247.973			0.358861			0.179431	−	0.983771i	$-0.442574\pi$
$691$	247.973			0.358861			0.179431	+	0.983771i	$0.442574\pi$
$692$		0			0
$693$	459.708	−	469.283i	0.663359	−	0.677176i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	187.254			0.268658
$698$		0			0
$699$	257.814	−	631.606i	0.368833	−	0.903585i
$700$		0			0
$701$		−	1213.96i		−	1.73175i	−0.500263	−	0.865874i	$-0.666763\pi$
$701$		−	1213.96i		−	1.73175i	0.500263	−	0.865874i	$-0.333237\pi$
$702$		0			0
$703$	136.559			0.194252
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	815.920i		−	1.15406i
$708$		0			0
$709$	745.830			1.05195			0.525973	−	0.850501i	$-0.323701\pi$
$709$	745.830			1.05195			0.525973	+	0.850501i	$0.323701\pi$
$710$		0			0
$711$	957.041	+	937.513i	1.34605	+	1.31858i
$712$		0			0
$713$			69.8443i			0.0979583i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−257.919	+	631.861i	−0.359719	+	0.881257i
$718$		0			0
$719$		−	551.198i		−	0.766618i	−0.923620	−	0.383309i	$-0.874784\pi$
$719$		−	551.198i		−	0.766618i	0.923620	−	0.383309i	$-0.125216\pi$
$720$		0			0
$721$	7.18663			0.00996759
$722$		0			0
$723$	−793.138	−	323.750i	−1.09701	−	0.447787i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−63.5439			−0.0874056			−0.0437028	−	0.999045i	$-0.513915\pi$
$727$	−63.5439			−0.0874056			−0.0437028	+	0.999045i	$0.513915\pi$
$728$		0			0
$729$	−499.298	+	531.171i	−0.684908	+	0.728630i
$730$		0			0
$731$		−	1834.41i		−	2.50945i
$732$		0			0
$733$	37.5413			0.0512159			0.0256080	−	0.999672i	$-0.491848\pi$
$733$	37.5413			0.0512159			0.0256080	+	0.999672i	$0.491848\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1127.87i			1.53035i
$738$		0			0
$739$	−501.470			−0.678579			−0.339289	−	0.940682i	$-0.610187\pi$
$739$	−501.470			−0.678579			−0.339289	+	0.940682i	$0.610187\pi$
$740$		0			0
$741$	−105.178	−	42.9326i	−0.141941	−	0.0579388i
$742$		0			0
$743$		−	150.710i		−	0.202840i	−0.994844	−	0.101420i	$-0.967661\pi$
$743$		−	150.710i		−	0.202840i	0.994844	−	0.101420i	$-0.0323386\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−58.9950	+	60.2238i	−0.0789759	+	0.0806209i
$748$		0			0
$749$			1167.57i			1.55884i
$750$		0			0
$751$	1349.71			1.79722			0.898609	−	0.438751i	$-0.144579\pi$
$751$	1349.71			1.79722			0.898609	+	0.438751i	$0.144579\pi$
$752$		0			0
$753$	−71.8031	+	175.907i	−0.0953561	+	0.233608i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	205.791			0.271851			0.135925	−	0.990719i	$-0.456599\pi$
$757$	205.791			0.271851			0.135925	+	0.990719i	$0.456599\pi$
$758$		0			0
$759$	−676.040	−	275.952i	−0.890698	−	0.363573i
$760$		0			0
$761$		−	738.457i		−	0.970377i	−0.874410	−	0.485189i	$-0.838751\pi$
$761$		−	738.457i		−	0.970377i	0.874410	−	0.485189i	$-0.161249\pi$
$762$		0			0
$763$	1227.23			1.60842
$764$		0			0
$765$		0			0
$766$		0			0
$767$			935.898i			1.22021i
$768$		0			0
$769$	467.952			0.608520			0.304260	−	0.952589i	$-0.401591\pi$
$769$	467.952			0.608520			0.304260	+	0.952589i	$0.401591\pi$
$770$		0			0
$771$	−43.0564	+	105.482i	−0.0558449	+	0.136812i
$772$		0			0
$773$		−	233.233i		−	0.301724i	−0.988555	−	0.150862i	$-0.951795\pi$
$773$		−	233.233i		−	0.301724i	0.988555	−	0.150862i	$-0.0482049\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−695.850	−	284.038i	−0.895560	−	0.365558i
$778$		0			0
$779$		−	20.4130i		−	0.0262041i
$780$		0			0
$781$	330.668			0.423391
$782$		0			0
$783$	750.950	−	324.744i	0.959067	−	0.414744i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−254.788			−0.323746			−0.161873	−	0.986812i	$-0.551753\pi$
$787$	−254.788			−0.323746			−0.161873	+	0.986812i	$0.551753\pi$
$788$		0			0
$789$	127.819	−	313.138i	0.162002	−	0.396880i
$790$		0			0
$791$			738.201i			0.933250i
$792$		0			0
$793$	812.526			1.02462
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1196.67i			1.50147i	0.660602	+	0.750737i	$0.270301\pi$
$797$			1196.67i			1.50147i	−0.660602	+	0.750737i	$0.729699\pi$
$798$		0			0
$799$	2031.58			2.54266
$800$		0			0
$801$	690.933	−	705.325i	0.862589	−	0.880556i
$802$		0			0
$803$		−	499.435i		−	0.621962i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	108.719	−	266.345i	0.134720	−	0.330043i
$808$		0			0
$809$		−	561.420i		−	0.693968i	−0.937871	−	0.346984i	$-0.887206\pi$
$809$		−	561.420i		−	0.693968i	0.937871	−	0.346984i	$-0.112794\pi$
$810$		0			0
$811$	−388.881			−0.479508			−0.239754	−	0.970834i	$-0.577067\pi$
$811$	−388.881			−0.479508			−0.239754	+	0.970834i	$0.577067\pi$
$812$		0			0
$813$	205.037	+	83.6939i	0.252199	+	0.102945i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−199.973			−0.244765
$818$		0			0
$819$	446.649	+	437.535i	0.545359	+	0.534231i
$820$		0			0
$821$		−	309.777i		−	0.377317i	−0.982043	−	0.188658i	$-0.939586\pi$
$821$		−	309.777i		−	0.377317i	0.982043	−	0.188658i	$-0.0604139\pi$
$822$		0			0
$823$	178.579			0.216985			0.108493	−	0.994097i	$-0.465398\pi$
$823$	178.579			0.216985			0.108493	+	0.994097i	$0.465398\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			165.734i			0.200404i	0.994967	+	0.100202i	$0.0319489\pi$
$827$			165.734i			0.200404i	−0.994967	+	0.100202i	$0.968051\pi$
$828$		0			0
$829$	173.278			0.209021			0.104510	−	0.994524i	$-0.466672\pi$
$829$	173.278			0.209021			0.104510	+	0.994524i	$0.466672\pi$
$830$		0			0
$831$	123.761	+	50.5180i	0.148931	+	0.0607919i
$832$		0			0
$833$			429.989i			0.516194i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−38.3159	−	88.6030i	−0.0457777	−	0.105858i
$838$		0			0
$839$			815.931i			0.972504i	0.873819	+	0.486252i	$0.161636\pi$
$839$			815.931i			0.972504i	−0.873819	+	0.486252i	$0.838364\pi$
$840$		0			0
$841$	−77.2223			−0.0918220
$842$		0			0
$843$	−203.983	+	499.727i	−0.241973	+	0.592796i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−200.568			−0.236798
$848$		0			0
$849$	294.612	+	120.257i	0.347010	+	0.141646i
$850$		0			0
$851$			835.405i			0.981675i
$852$		0			0
$853$	631.549			0.740386			0.370193	−	0.928955i	$-0.379292\pi$
$853$	631.549			0.740386			0.370193	+	0.928955i	$0.379292\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	300.140i		−	0.350221i	−0.984549	−	0.175111i	$-0.943972\pi$
$857$		−	300.140i		−	0.350221i	0.984549	−	0.175111i	$-0.0560283\pi$
$858$		0			0
$859$	1243.32			1.44741			0.723703	−	0.690111i	$-0.242438\pi$
$859$	1243.32			1.44741			0.723703	+	0.690111i	$0.242438\pi$
$860$		0			0
$861$	−42.4584	+	104.017i	−0.0493129	+	0.120809i
$862$		0			0
$863$		−	134.903i		−	0.156319i	−0.996941	−	0.0781596i	$-0.975096\pi$
$863$		−	134.903i		−	0.156319i	0.996941	−	0.0781596i	$-0.0249044\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1580.67	−	645.210i	−1.82314	−	0.744187i
$868$		0			0
$869$		−	1854.68i		−	2.13427i
$870$		0			0
$871$	−1073.47			−1.23245
$872$		0			0
$873$	−1040.56	−	1019.33i	−1.19194	−	1.16762i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.4694			−0.0164988			−0.00824940	−	0.999966i	$-0.502626\pi$
$877$	−14.4694			−0.0164988			−0.00824940	+	0.999966i	$0.502626\pi$
$878$		0			0
$879$	225.423	−	552.252i	0.256454	−	0.628273i
$880$		0			0
$881$			1281.36i			1.45444i	0.686404	+	0.727221i	$0.259188\pi$
$881$			1281.36i			1.45444i	−0.686404	+	0.727221i	$0.740812\pi$
$882$		0			0
$883$	−565.182			−0.640070			−0.320035	−	0.947406i	$-0.603695\pi$
$883$	−565.182			−0.640070			−0.320035	+	0.947406i	$0.603695\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	517.280i		−	0.583179i	−0.956543	−	0.291590i	$-0.905816\pi$
$887$		−	517.280i		−	0.583179i	0.956543	−	0.291590i	$-0.0941842\pi$
$888$		0			0
$889$	−289.444			−0.325583
$890$		0			0
$891$	1008.99	−	20.8023i	1.13243	−	0.0233472i
$892$		0			0
$893$		−	221.468i		−	0.248004i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	262.642	−	643.433i	0.292801	−	0.717317i
$898$		0			0
$899$			108.339i			0.120511i
$900$		0			0
$901$	−1692.73			−1.87872
$902$		0			0
$903$	1018.98	+	415.938i	1.12844	+	0.460618i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−591.131			−0.651743			−0.325872	−	0.945414i	$-0.605658\pi$
$907$	−591.131			−0.651743			−0.325872	+	0.945414i	$0.605658\pi$
$908$		0			0
$909$	877.145	−	895.415i	0.964956	−	0.985055i
$910$		0			0
$911$			328.905i			0.361038i	0.983572	+	0.180519i	$0.0577777\pi$
$911$			328.905i			0.361038i	−0.983572	+	0.180519i	$0.942222\pi$
$912$		0			0
$913$	116.710			0.127831
$914$		0			0
$915$		0			0
$916$		0			0
$917$			599.615i			0.653888i
$918$		0			0
$919$	739.366			0.804533			0.402266	−	0.915523i	$-0.368223\pi$
$919$	739.366			0.804533			0.402266	+	0.915523i	$0.368223\pi$
$920$		0			0
$921$	680.790	+	277.891i	0.739185	+	0.301727i
$922$		0			0
$923$			314.720i			0.340975i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.88682	+	7.72590i	0.00850790	+	0.00833430i
$928$		0			0
$929$			1163.63i			1.25256i	0.779599	+	0.626279i	$0.215423\pi$
$929$			1163.63i			1.25256i	−0.779599	+	0.626279i	$0.784577\pi$
$930$		0			0
$931$	46.8741			0.0503481
$932$		0			0
$933$	7.32142	−	17.9364i	0.00784718	−	0.0192244i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−379.177			−0.404671			−0.202336	−	0.979316i	$-0.564853\pi$
$937$	−379.177			−0.404671			−0.202336	+	0.979316i	$0.564853\pi$
$938$		0			0
$939$	1114.46	+	454.910i	1.18686	+	0.484462i
$940$		0			0
$941$			1102.24i			1.17135i	0.810545	+	0.585676i	$0.199171\pi$
$941$			1102.24i			1.17135i	−0.810545	+	0.585676i	$0.800829\pi$
$942$		0			0
$943$	124.878			0.132426
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1585.60i			1.67434i	0.546940	+	0.837172i	$0.315793\pi$
$947$			1585.60i			1.67434i	−0.546940	+	0.837172i	$0.684207\pi$
$948$		0			0
$949$	475.347			0.500892
$950$		0			0
$951$	364.630	−	893.287i	0.383417	−	0.939314i
$952$		0			0
$953$		−	23.3120i		−	0.0244617i	−0.999925	−	0.0122308i	$-0.996107\pi$
$953$		−	23.3120i		−	0.0244617i	0.999925	−	0.0122308i	$-0.00389330\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1048.64	−	428.043i	−1.09576	−	0.447276i
$958$		0			0
$959$			721.646i			0.752498i
$960$		0			0
$961$	−948.217			−0.986699
$962$		0			0
$963$	−1255.18	+	1281.33i	−1.30341	+	1.33056i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1446.24			1.49560			0.747798	−	0.663926i	$-0.231111\pi$
$967$	1446.24			1.49560			0.747798	+	0.663926i	$0.231111\pi$
$968$		0			0
$969$	−106.054	+	259.816i	−0.109447	+	0.268128i
$970$		0			0
$971$			168.397i			0.173427i	0.996233	+	0.0867134i	$0.0276365\pi$
$971$			168.397i			0.173427i	−0.996233	+	0.0867134i	$0.972364\pi$
$972$		0			0
$973$	−210.485			−0.216326
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	623.561i		−	0.638241i	−0.947714	−	0.319120i	$-0.896613\pi$
$977$		−	623.561i		−	0.638241i	0.947714	−	0.319120i	$-0.103387\pi$
$978$		0			0
$979$	−1366.87			−1.39619
$980$		0			0
$981$	1346.80	+	1319.31i	1.37288	+	1.34487i
$982$		0			0
$983$		−	378.106i		−	0.384645i	−0.981332	−	0.192323i	$-0.938398\pi$
$983$		−	378.106i		−	0.384645i	0.981332	−	0.192323i	$-0.0616020\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−460.645	+	1128.51i	−0.466713	+	1.14337i
$988$		0			0
$989$		−	1223.34i		−	1.23695i
$990$		0			0
$991$	−1753.42			−1.76934			−0.884672	−	0.466215i	$-0.845617\pi$
$991$	−1753.42			−1.76934			−0.884672	+	0.466215i	$0.845617\pi$
$992$		0			0
$993$	−78.6181	−	32.0910i	−0.0791723	−	0.0323173i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	839.151			0.841676			0.420838	−	0.907136i	$-0.361736\pi$
$997$	839.151			0.841676			0.420838	+	0.907136i	$0.361736\pi$
$998$		0			0
$999$	−458.295	−	1059.78i	−0.458754	−	1.06084i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.w.401.6		6
3.2	odd	2	inner	1200.3.l.w.401.5		6
4.3	odd	2		600.3.l.d.401.1	✓	6
5.2	odd	4		1200.3.c.l.449.7		12
5.3	odd	4		1200.3.c.l.449.6		12
5.4	even	2		1200.3.l.v.401.1		6
12.11	even	2		600.3.l.d.401.2	yes	6
15.2	even	4		1200.3.c.l.449.5		12
15.8	even	4		1200.3.c.l.449.8		12
15.14	odd	2		1200.3.l.v.401.2		6
20.3	even	4		600.3.c.c.449.7		12
20.7	even	4		600.3.c.c.449.6		12
20.19	odd	2		600.3.l.e.401.6	yes	6
60.23	odd	4		600.3.c.c.449.5		12
60.47	odd	4		600.3.c.c.449.8		12
60.59	even	2		600.3.l.e.401.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.5		12	60.23	odd	4
600.3.c.c.449.6		12	20.7	even	4
600.3.c.c.449.7		12	20.3	even	4
600.3.c.c.449.8		12	60.47	odd	4
600.3.l.d.401.1	✓	6	4.3	odd	2
600.3.l.d.401.2	yes	6	12.11	even	2
600.3.l.e.401.5	yes	6	60.59	even	2
600.3.l.e.401.6	yes	6	20.19	odd	2
1200.3.c.l.449.5		12	15.2	even	4
1200.3.c.l.449.6		12	5.3	odd	4
1200.3.c.l.449.7		12	5.2	odd	4
1200.3.c.l.449.8		12	15.8	even	4
1200.3.l.v.401.1		6	5.4	even	2
1200.3.l.v.401.2		6	15.14	odd	2
1200.3.l.w.401.5		6	3.2	odd	2	inner
1200.3.l.w.401.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.77752 + 1.13375i) q^{3} +5.85843 q^{7} +(6.42922 + 6.29803i) q^{9} +O(q^{10})\)	\(q+(2.77752 + 1.13375i) q^{3} +5.85843 q^{7} +(6.42922 + 6.29803i) q^{9} -12.4594i q^{11} +11.8584 q^{13} -29.2932i q^{17} -3.19332 q^{19} +(16.2719 + 6.64200i) q^{21} -19.5353i q^{23} +(10.7169 + 24.7820i) q^{27} -30.3022i q^{29} -3.57529 q^{31} +(14.1258 - 34.6061i) q^{33} -42.7639 q^{37} +(32.9370 + 13.4445i) q^{39} +6.39241i q^{41} +62.6224 q^{43} +69.3534i q^{47} -14.6788 q^{49} +(33.2112 - 81.3625i) q^{51} -57.7856i q^{53} +(-8.86951 - 3.62043i) q^{57} +78.9226i q^{59} +68.5189 q^{61} +(37.6651 + 36.8966i) q^{63} -90.5235 q^{67} +(22.1481 - 54.2596i) q^{69} +26.5398i q^{71} +40.0851 q^{73} -72.9923i q^{77} +148.858 q^{79} +(1.66962 + 80.9828i) q^{81} +9.36722i q^{83} +(34.3551 - 84.1649i) q^{87} -109.706i q^{89} +69.4718 q^{91} +(-9.93044 - 4.05349i) q^{93} -161.849 q^{97} +(78.4695 - 80.1039i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9	\( 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9} + 26 q^{13} - 50 q^{19} - 18 q^{21} - 26 q^{27} + 114 q^{31} + 82 q^{33} + 76 q^{37} + 6 q^{39} - 2 q^{43} + 76 q^{49} + 6 q^{51} + 172 q^{57} + 62 q^{61} + 150 q^{63} - 422 q^{67} + 156 q^{69} + 72 q^{73} + 76 q^{79} - 224 q^{81} + 172 q^{87} + 310 q^{91} + 110 q^{93} - 470 q^{97} + 250 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9 + 26 * q^13 - 50 * q^19 - 18 * q^21 - 26 * q^27 + 114 * q^31 + 82 * q^33 + 76 * q^37 + 6 * q^39 - 2 * q^43 + 76 * q^49 + 6 * q^51 + 172 * q^57 + 62 * q^61 + 150 * q^63 - 422 * q^67 + 156 * q^69 + 72 * q^73 + 76 * q^79 - 224 * q^81 + 172 * q^87 + 310 * q^91 + 110 * q^93 - 470 * q^97 + 250 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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