LMFDB - Embedded newform 1120.3.c.g.209.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,3,Mod(209,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1120.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1120.c (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:	$30.5177896084$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.3
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.25300i q^{3} +(3.15168 - 3.88161i) q^{5} +(-5.63086 + 4.15854i) q^{7} +7.43000 q^{9} +O(q^{10})$	\(q-1.25300i q^{3} +(3.15168 - 3.88161i) q^{5} +(-5.63086 + 4.15854i) q^{7} +7.43000 q^{9} +20.2442i q^{11} -1.87714i q^{13} +(-4.86364 - 3.94905i) q^{15} -24.6339 q^{17} -25.5929 q^{19} +(5.21064 + 7.05545i) q^{21} -21.6975i q^{23} +(-5.13378 - 24.4672i) q^{25} -20.5867i q^{27} -30.6111i q^{29} -53.2186i q^{31} +25.3659 q^{33} +(-1.60486 + 34.9632i) q^{35} -18.6006 q^{37} -2.35205 q^{39} +26.3649i q^{41} +12.9132 q^{43} +(23.4170 - 28.8403i) q^{45} -34.2377 q^{47} +(14.4131 - 46.8323i) q^{49} +30.8662i q^{51} -49.5854 q^{53} +(78.5799 + 63.8032i) q^{55} +32.0679i q^{57} -81.5010 q^{59} -38.9363 q^{61} +(-41.8373 + 30.8979i) q^{63} +(-7.28631 - 5.91614i) q^{65} -71.3076 q^{67} -27.1868 q^{69} +44.9698 q^{71} +21.6236 q^{73} +(-30.6573 + 6.43261i) q^{75} +(-84.1861 - 113.992i) q^{77} -71.5348 q^{79} +41.0749 q^{81} -33.2821i q^{83} +(-77.6383 + 95.6192i) q^{85} -38.3556 q^{87} +29.8239i q^{89} +(7.80615 + 10.5699i) q^{91} -66.6827 q^{93} +(-80.6609 + 99.3418i) q^{95} +111.145 q^{97} +150.414i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 224 q^{9}+O(q^{10}) $ 80 * q - 224 * q^9	$ 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) $ 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$		−	1.25300i		−	0.417666i	−0.977951	−	0.208833i	$-0.933034\pi$
$3$		−	1.25300i		−	0.417666i	0.977951	−	0.208833i	$-0.0669665\pi$
$4$		0			0
$5$	3.15168	−	3.88161i	0.630337	−	0.776322i
$5$	3.15168	−	3.88161i	0.630337	−	0.776322i
$6$		0			0
$7$	−5.63086	+	4.15854i	−0.804408	+	0.594077i
$7$	−5.63086	+	4.15854i	−0.804408	+	0.594077i
$8$		0			0
$9$	7.43000			0.825555
$10$		0			0
$11$			20.2442i			1.84038i	0.391474	+	0.920189i	$0.371965\pi$
$11$			20.2442i			1.84038i	−0.391474	+	0.920189i	$0.628035\pi$
$12$		0			0
$13$		−	1.87714i		−	0.144395i	−0.997390	−	0.0721976i	$-0.976999\pi$
$13$		−	1.87714i		−	0.144395i	0.997390	−	0.0721976i	$-0.0230012\pi$
$14$		0			0
$15$	−4.86364	−	3.94905i	−0.324243	−	0.263270i
$16$		0			0
$17$	−24.6339			−1.44905			−0.724527	−	0.689247i	$-0.757942\pi$
$17$	−24.6339			−1.44905			−0.724527	+	0.689247i	$0.757942\pi$
$18$		0			0
$19$	−25.5929			−1.34700			−0.673499	−	0.739188i	$-0.735209\pi$
$19$	−25.5929			−1.34700			−0.673499	+	0.739188i	$0.735209\pi$
$20$		0			0
$21$	5.21064	+	7.05545i	0.248126	+	0.335974i
$22$		0			0
$23$		−	21.6975i		−	0.943367i	−0.881768	−	0.471684i	$-0.843646\pi$
$23$		−	21.6975i		−	0.943367i	0.881768	−	0.471684i	$-0.156354\pi$
$24$		0			0
$25$	−5.13378	−	24.4672i	−0.205351	−	0.978688i
$26$		0			0
$27$		−	20.5867i		−	0.762472i
$28$		0			0
$29$		−	30.6111i		−	1.05556i	−0.849382	−	0.527778i	$-0.823025\pi$
$29$		−	30.6111i		−	1.05556i	0.849382	−	0.527778i	$-0.176975\pi$
$30$		0			0
$31$		−	53.2186i		−	1.71673i	−0.513041	−	0.858364i	$-0.671481\pi$
$31$		−	53.2186i		−	1.71673i	0.513041	−	0.858364i	$-0.328519\pi$
$32$		0			0
$33$	25.3659			0.768663
$34$		0			0
$35$	−1.60486	+	34.9632i	−0.0458530	+	0.998948i
$36$		0			0
$37$	−18.6006			−0.502719			−0.251359	−	0.967894i	$-0.580878\pi$
$37$	−18.6006			−0.502719			−0.251359	+	0.967894i	$0.580878\pi$
$38$		0			0
$39$	−2.35205			−0.0603089
$40$		0			0
$41$			26.3649i			0.643047i	0.946902	+	0.321523i	$0.104195\pi$
$41$			26.3649i			0.643047i	−0.946902	+	0.321523i	$0.895805\pi$
$42$		0			0
$43$	12.9132			0.300308			0.150154	−	0.988663i	$-0.452023\pi$
$43$	12.9132			0.300308			0.150154	+	0.988663i	$0.452023\pi$
$44$		0			0
$45$	23.4170	−	28.8403i	0.520378	−	0.640897i
$46$		0			0
$47$	−34.2377			−0.728461			−0.364230	−	0.931309i	$-0.618668\pi$
$47$	−34.2377			−0.728461			−0.364230	+	0.931309i	$0.618668\pi$
$48$		0			0
$49$	14.4131	−	46.8323i	0.294145	−	0.955761i
$50$		0			0
$51$			30.8662i			0.605220i
$52$		0			0
$53$	−49.5854			−0.935573			−0.467786	−	0.883842i	$-0.654948\pi$
$53$	−49.5854			−0.935573			−0.467786	+	0.883842i	$0.654948\pi$
$54$		0			0
$55$	78.5799	+	63.8032i	1.42873	+	1.16006i
$56$		0			0
$57$			32.0679i			0.562595i
$58$		0			0
$59$	−81.5010			−1.38137			−0.690687	−	0.723154i	$-0.742692\pi$
$59$	−81.5010			−1.38137			−0.690687	+	0.723154i	$0.742692\pi$
$60$		0			0
$61$	−38.9363			−0.638299			−0.319150	−	0.947704i	$-0.603397\pi$
$61$	−38.9363			−0.638299			−0.319150	+	0.947704i	$0.603397\pi$
$62$		0			0
$63$	−41.8373	+	30.8979i	−0.664083	+	0.490444i
$64$		0			0
$65$	−7.28631	−	5.91614i	−0.112097	−	0.0910175i
$66$		0			0
$67$	−71.3076			−1.06429			−0.532147	−	0.846652i	$-0.678614\pi$
$67$	−71.3076			−1.06429			−0.532147	+	0.846652i	$0.678614\pi$
$68$		0			0
$69$	−27.1868			−0.394012
$70$		0			0
$71$	44.9698			0.633378			0.316689	−	0.948529i	$-0.397429\pi$
$71$	44.9698			0.633378			0.316689	+	0.948529i	$0.397429\pi$
$72$		0			0
$73$	21.6236			0.296214			0.148107	−	0.988971i	$-0.452682\pi$
$73$	21.6236			0.296214			0.148107	+	0.988971i	$0.452682\pi$
$74$		0			0
$75$	−30.6573	+	6.43261i	−0.408765	+	0.0857681i
$76$		0			0
$77$	−84.1861	−	113.992i	−1.09333	−	1.48042i
$78$		0			0
$79$	−71.5348			−0.905504			−0.452752	−	0.891636i	$-0.649558\pi$
$79$	−71.5348			−0.905504			−0.452752	+	0.891636i	$0.649558\pi$
$80$		0			0
$81$	41.0749			0.507097
$82$		0			0
$83$		−	33.2821i		−	0.400990i	−0.979695	−	0.200495i	$-0.935745\pi$
$83$		−	33.2821i		−	0.400990i	0.979695	−	0.200495i	$-0.0642550\pi$
$84$		0			0
$85$	−77.6383	+	95.6192i	−0.913392	+	1.12493i
$86$		0			0
$87$	−38.3556			−0.440869
$88$		0			0
$89$			29.8239i			0.335100i	0.985864	+	0.167550i	$0.0535855\pi$
$89$			29.8239i			0.335100i	−0.985864	+	0.167550i	$0.946415\pi$
$90$		0			0
$91$	7.80615	+	10.5699i	0.0857818	+	0.116153i
$92$		0			0
$93$	−66.6827			−0.717019
$94$		0			0
$95$	−80.6609	+	99.3418i	−0.849062	+	1.04570i
$96$		0			0
$97$	111.145			1.14583			0.572914	−	0.819615i	$-0.305813\pi$
$97$	111.145			1.14583			0.572914	+	0.819615i	$0.305813\pi$
$98$		0			0
$99$			150.414i			1.51933i
$100$		0			0
$101$	−9.26688			−0.0917512			−0.0458756	−	0.998947i	$-0.514608\pi$
$101$	−9.26688			−0.0917512			−0.0458756	+	0.998947i	$0.514608\pi$
$102$		0			0
$103$	105.365			1.02296			0.511482	−	0.859294i	$-0.329097\pi$
$103$	105.365			1.02296			0.511482	+	0.859294i	$0.329097\pi$
$104$		0			0
$105$	43.8088	+	2.01088i	0.417226	+	0.0191512i
$106$		0			0
$107$	46.8956			0.438277			0.219138	−	0.975694i	$-0.429675\pi$
$107$	46.8956			0.438277			0.219138	+	0.975694i	$0.429675\pi$
$108$		0			0
$109$			116.324i			1.06720i	0.845738	+	0.533599i	$0.179161\pi$
$109$			116.324i			1.06720i	−0.845738	+	0.533599i	$0.820839\pi$
$110$		0			0
$111$			23.3065i			0.209968i
$112$		0			0
$113$		−	54.4860i		−	0.482177i	−0.970503	−	0.241089i	$-0.922496\pi$
$113$		−	54.4860i		−	0.482177i	0.970503	−	0.241089i	$-0.0775044\pi$
$114$		0			0
$115$	−84.2210	−	68.3835i	−0.732357	−	0.594639i
$116$		0			0
$117$		−	13.9471i		−	0.119206i
$118$		0			0
$119$	138.710	−	102.441i	1.16563	−	0.860850i
$120$		0			0
$121$	−288.826			−2.38699
$122$		0			0
$123$	33.0352			0.268579
$124$		0			0
$125$	−111.152	−	57.1856i	−0.889217	−	0.457485i
$126$		0			0
$127$		−	134.141i		−	1.05623i	−0.849174	−	0.528113i	$-0.822900\pi$
$127$		−	134.141i		−	1.05623i	0.849174	−	0.528113i	$-0.177100\pi$
$128$		0			0
$129$		−	16.1802i		−	0.125428i
$130$		0			0
$131$	64.3999			0.491603			0.245801	−	0.969320i	$-0.420949\pi$
$131$	64.3999			0.491603			0.245801	+	0.969320i	$0.420949\pi$
$132$		0			0
$133$	144.110	−	106.429i	1.08354	−	0.800220i
$134$		0			0
$135$	−79.9097	−	64.8829i	−0.591923	−	0.480614i
$136$		0			0
$137$		−	55.7969i		−	0.407276i	−0.979046	−	0.203638i	$-0.934723\pi$
$137$		−	55.7969i		−	0.407276i	0.979046	−	0.203638i	$-0.0652766\pi$
$138$		0			0
$139$	177.788			1.27905			0.639526	−	0.768770i	$-0.279131\pi$
$139$	177.788			1.27905			0.639526	+	0.768770i	$0.279131\pi$
$140$		0			0
$141$			42.8997i			0.304253i
$142$		0			0
$143$	38.0010			0.265742
$144$		0			0
$145$	−118.820	−	96.4766i	−0.819451	−	0.665356i
$146$		0			0
$147$	−58.6807	−	18.0596i	−0.399188	−	0.122854i
$148$		0			0
$149$			38.4754i			0.258224i	0.991630	+	0.129112i	$0.0412127\pi$
$149$			38.4754i			0.258224i	−0.991630	+	0.129112i	$0.958787\pi$
$150$		0			0
$151$	19.3404			0.128082			0.0640409	−	0.997947i	$-0.479601\pi$
$151$	19.3404			0.128082			0.0640409	+	0.997947i	$0.479601\pi$
$152$		0			0
$153$	−183.030			−1.19627
$154$		0			0
$155$	−206.574	−	167.728i	−1.33273	−	1.08212i
$156$		0			0
$157$			211.491i			1.34707i	0.739154	+	0.673537i	$0.235226\pi$
$157$			211.491i			1.34707i	−0.739154	+	0.673537i	$0.764774\pi$
$158$		0			0
$159$			62.1303i			0.390757i
$160$		0			0
$161$	90.2297	+	122.175i	0.560433	+	0.758852i
$162$		0			0
$163$	−291.274			−1.78696			−0.893478	−	0.449107i	$-0.851742\pi$
$163$	−291.274			−1.78696			−0.893478	+	0.449107i	$0.851742\pi$
$164$		0			0
$165$	79.9452	−	98.4604i	0.484516	−	0.596730i
$166$		0			0
$167$	−164.016			−0.982132			−0.491066	−	0.871122i	$-0.663393\pi$
$167$	−164.016			−0.982132			−0.491066	+	0.871122i	$0.663393\pi$
$168$		0			0
$169$	165.476			0.979150
$170$		0			0
$171$	−190.156			−1.11202
$172$		0			0
$173$			150.234i			0.868404i	0.900815	+	0.434202i	$0.142970\pi$
$173$			150.234i			0.868404i	−0.900815	+	0.434202i	$0.857030\pi$
$174$		0			0
$175$	130.655	+	116.422i	0.746602	+	0.665271i
$176$		0			0
$177$			102.121i			0.576952i
$178$		0			0
$179$		−	79.3837i		−	0.443484i	−0.975105	−	0.221742i	$-0.928826\pi$
$179$		−	79.3837i		−	0.443484i	0.975105	−	0.221742i	$-0.0711743\pi$
$180$		0			0
$181$	−275.472			−1.52195			−0.760973	−	0.648783i	$-0.775278\pi$
$181$	−275.472			−1.52195			−0.760973	+	0.648783i	$0.775278\pi$
$182$		0			0
$183$			48.7870i			0.266596i
$184$		0			0
$185$	−58.6232	+	72.2002i	−0.316882	+	0.390271i
$186$		0			0
$187$		−	498.693i		−	2.66681i
$188$		0			0
$189$	85.6108	+	115.921i	0.452967	+	0.613339i
$190$		0			0
$191$	−10.3836			−0.0543646			−0.0271823	−	0.999630i	$-0.508653\pi$
$191$	−10.3836			−0.0543646			−0.0271823	+	0.999630i	$0.508653\pi$
$192$		0			0
$193$			180.586i			0.935676i	0.883814	+	0.467838i	$0.154967\pi$
$193$			180.586i			0.935676i	−0.883814	+	0.467838i	$0.845033\pi$
$194$		0			0
$195$	−7.41291	+	9.12972i	−0.0380149	+	0.0468191i
$196$		0			0
$197$	−116.058			−0.589129			−0.294564	−	0.955632i	$-0.595175\pi$
$197$	−116.058			−0.589129			−0.294564	+	0.955632i	$0.595175\pi$
$198$		0			0
$199$		−	107.219i		−	0.538788i	−0.963030	−	0.269394i	$-0.913177\pi$
$199$		−	107.219i		−	0.538788i	0.963030	−	0.269394i	$-0.0868234\pi$
$200$		0			0
$201$			89.3483i			0.444519i
$202$		0			0
$203$	127.298	+	172.367i	0.627081	+	0.849098i
$204$		0			0
$205$	102.338	+	83.0939i	0.499211	+	0.405336i
$206$		0			0
$207$		−	161.212i		−	0.778802i
$208$		0			0
$209$		−	518.108i		−	2.47898i
$210$		0			0
$211$			17.1240i			0.0811565i	0.999176	+	0.0405782i	$0.0129200\pi$
$211$			17.1240i			0.0811565i	−0.999176	+	0.0405782i	$0.987080\pi$
$212$		0			0
$213$		−	56.3471i		−	0.264540i
$214$		0			0
$215$	40.6984	−	50.1241i	0.189295	−	0.233135i
$216$		0			0
$217$	221.312	+	299.666i	1.01987	+	1.38095i
$218$		0			0
$219$		−	27.0944i		−	0.123719i
$220$		0			0
$221$			46.2412i			0.209236i
$222$		0			0
$223$	49.3141			0.221139			0.110570	−	0.993868i	$-0.464732\pi$
$223$	49.3141			0.221139			0.110570	+	0.993868i	$0.464732\pi$
$224$		0			0
$225$	−38.1440	−	181.791i	−0.169529	−	0.807961i
$226$		0			0
$227$			289.142i			1.27375i	0.770965	+	0.636877i	$0.219774\pi$
$227$			289.142i			1.27375i	−0.770965	+	0.636877i	$0.780226\pi$
$228$		0			0
$229$	307.715			1.34373			0.671866	−	0.740673i	$-0.265493\pi$
$229$	307.715			1.34373			0.671866	+	0.740673i	$0.265493\pi$
$230$		0			0
$231$	−142.832	+	105.485i	−0.618319	+	0.456645i
$232$		0			0
$233$		−	317.304i		−	1.36182i	−0.732367	−	0.680910i	$-0.761585\pi$
$233$		−	317.304i		−	1.36182i	0.732367	−	0.680910i	$-0.238415\pi$
$234$		0			0
$235$	−107.906	+	132.897i	−0.459176	+	0.565520i
$236$		0			0
$237$			89.6329i			0.378198i
$238$		0			0
$239$	59.5865			0.249316			0.124658	−	0.992200i	$-0.460217\pi$
$239$	59.5865			0.249316			0.124658	+	0.992200i	$0.460217\pi$
$240$		0			0
$241$			177.653i			0.737149i	0.929598	+	0.368575i	$0.120154\pi$
$241$			177.653i			0.737149i	−0.929598	+	0.368575i	$0.879846\pi$
$242$		0			0
$243$		−	236.747i		−	0.974269i
$244$		0			0
$245$	−136.359	−	203.547i	−0.556568	−	0.830802i
$246$		0			0
$247$			48.0415i			0.194500i
$248$		0			0
$249$	−41.7024			−0.167480
$250$		0			0
$251$	213.192			0.849369			0.424685	−	0.905341i	$-0.360385\pi$
$251$	213.192			0.849369			0.424685	+	0.905341i	$0.360385\pi$
$252$		0			0
$253$	439.247			1.73615
$254$		0			0
$255$	119.811	+	97.2806i	0.469845	+	0.381492i
$256$		0			0
$257$	144.197			0.561077			0.280538	−	0.959843i	$-0.409487\pi$
$257$	144.197			0.561077			0.280538	+	0.959843i	$0.409487\pi$
$258$		0			0
$259$	104.737	−	77.3513i	0.404391	−	0.298654i
$260$		0			0
$261$		−	227.441i		−	0.871420i
$262$		0			0
$263$			77.4776i			0.294592i	0.989092	+	0.147296i	$0.0470570\pi$
$263$			77.4776i			0.294592i	−0.989092	+	0.147296i	$0.952943\pi$
$264$		0			0
$265$	−156.277	+	192.471i	−0.589726	+	0.726306i
$266$		0			0
$267$	37.3692			0.139960
$268$		0			0
$269$	−225.255			−0.837380			−0.418690	−	0.908129i	$-0.637511\pi$
$269$	−225.255			−0.837380			−0.418690	+	0.908129i	$0.637511\pi$
$270$		0			0
$271$		−	115.941i		−	0.427828i	−0.976853	−	0.213914i	$-0.931379\pi$
$271$		−	115.941i		−	0.427828i	0.976853	−	0.213914i	$-0.0686212\pi$
$272$		0			0
$273$	13.2440	−	9.78108i	0.0485130	−	0.0358281i
$274$		0			0
$275$	495.318	−	103.929i	1.80116	−	0.377924i
$276$		0			0
$277$	−5.14570			−0.0185766			−0.00928828	−	0.999957i	$-0.502957\pi$
$277$	−5.14570			−0.0185766			−0.00928828	+	0.999957i	$0.502957\pi$
$278$		0			0
$279$		−	395.414i		−	1.41725i
$280$		0			0
$281$	−252.334			−0.897986			−0.448993	−	0.893535i	$-0.648217\pi$
$281$	−252.334			−0.897986			−0.448993	+	0.893535i	$0.648217\pi$
$282$		0			0
$283$			340.732i			1.20400i	0.798496	+	0.602000i	$0.205629\pi$
$283$			340.732i			1.20400i	−0.798496	+	0.602000i	$0.794371\pi$
$284$		0			0
$285$	124.475	+	101.068i	0.436754	+	0.354624i
$286$		0			0
$287$	−109.640	−	148.457i	−0.382019	−	0.517272i
$288$		0			0
$289$	317.830			1.09976
$290$		0			0
$291$		−	139.265i		−	0.478573i
$292$		0			0
$293$			91.0228i			0.310658i	0.987863	+	0.155329i	$0.0496437\pi$
$293$			91.0228i			0.310658i	−0.987863	+	0.155329i	$0.950356\pi$
$294$		0			0
$295$	−256.865	+	316.355i	−0.870730	+	1.07239i
$296$		0			0
$297$	416.761			1.40324
$298$		0			0
$299$	−40.7291			−0.136218
$300$		0			0
$301$	−72.7126	+	53.7002i	−0.241570	+	0.178406i
$302$		0			0
$303$			11.6114i			0.0383213i
$304$		0			0
$305$	−122.715	+	151.135i	−0.402343	+	0.495526i
$306$		0			0
$307$			344.181i			1.12111i	0.828116	+	0.560556i	$0.189413\pi$
$307$			344.181i			1.12111i	−0.828116	+	0.560556i	$0.810587\pi$
$308$		0			0
$309$		−	132.022i		−	0.427257i
$310$		0			0
$311$			138.241i			0.444506i	0.974989	+	0.222253i	$0.0713410\pi$
$311$			138.241i			0.444506i	−0.974989	+	0.222253i	$0.928659\pi$
$312$		0			0
$313$	−387.619			−1.23840			−0.619200	−	0.785233i	$-0.712543\pi$
$313$	−387.619			−1.23840			−0.619200	+	0.785233i	$0.712543\pi$
$314$		0			0
$315$	−11.9241	+	259.776i	−0.0378542	+	0.824687i
$316$		0			0
$317$	429.090			1.35360			0.676798	−	0.736169i	$-0.263367\pi$
$317$	429.090			1.35360			0.676798	+	0.736169i	$0.263367\pi$
$318$		0			0
$319$	619.696			1.94262
$320$		0			0
$321$		−	58.7601i		−	0.183053i
$322$		0			0
$323$	630.454			1.95187
$324$		0			0
$325$	−45.9283	+	9.63680i	−0.141318	+	0.0296517i
$326$		0			0
$327$	145.754			0.445732
$328$		0			0
$329$	192.787	−	142.379i	0.585980	−	0.432762i
$330$		0			0
$331$		−	45.1558i		−	0.136422i	−0.997671	−	0.0682112i	$-0.978271\pi$
$331$		−	45.1558i		−	0.136422i	0.997671	−	0.0682112i	$-0.0217292\pi$
$332$		0			0
$333$	−138.202			−0.415022
$334$		0			0
$335$	−224.739	+	276.788i	−0.670863	+	0.826234i
$336$		0			0
$337$			33.4834i			0.0993574i	0.998765	+	0.0496787i	$0.0158197\pi$
$337$			33.4834i			0.0993574i	−0.998765	+	0.0496787i	$0.984180\pi$
$338$		0			0
$339$	−68.2708			−0.201389
$340$		0			0
$341$	1077.37			3.15943
$342$		0			0
$343$	113.596	+	323.643i	0.331183	+	0.943566i
$344$		0			0
$345$	−85.6843	+	105.529i	−0.248360	+	0.305880i
$346$		0			0
$347$	478.158			1.37798			0.688988	−	0.724773i	$-0.258055\pi$
$347$	478.158			1.37798			0.688988	+	0.724773i	$0.258055\pi$
$348$		0			0
$349$	−350.359			−1.00389			−0.501947	−	0.864898i	$-0.667383\pi$
$349$	−350.359			−1.00389			−0.501947	+	0.864898i	$0.667383\pi$
$350$		0			0
$351$	−38.6441			−0.110097
$352$		0			0
$353$	−226.898			−0.642771			−0.321386	−	0.946948i	$-0.604149\pi$
$353$	−226.898			−0.642771			−0.321386	+	0.946948i	$0.604149\pi$
$354$		0			0
$355$	141.731	−	174.555i	0.399241	−	0.491705i
$356$		0			0
$357$	−128.358	−	173.803i	−0.359547	−	0.486844i
$358$		0			0
$359$	205.730			0.573064			0.286532	−	0.958071i	$-0.407498\pi$
$359$	205.730			0.573064			0.286532	+	0.958071i	$0.407498\pi$
$360$		0			0
$361$	293.999			0.814402
$362$		0			0
$363$			361.898i			0.996964i
$364$		0			0
$365$	68.1509	−	83.9345i	0.186715	−	0.229958i
$366$		0			0
$367$	−414.799			−1.13024			−0.565122	−	0.825007i	$-0.691171\pi$
$367$	−414.799			−1.13024			−0.565122	+	0.825007i	$0.691171\pi$
$368$		0			0
$369$			195.891i			0.530871i
$370$		0			0
$371$	279.208	−	206.203i	0.752582	−	0.555802i
$372$		0			0
$373$	−514.862			−1.38033			−0.690164	−	0.723653i	$-0.742462\pi$
$373$	−514.862			−1.38033			−0.690164	+	0.723653i	$0.742462\pi$
$374$		0			0
$375$	−71.6534	+	139.273i	−0.191076	+	0.371396i
$376$		0			0
$377$	−57.4612			−0.152417
$378$		0			0
$379$		−	483.791i		−	1.27649i	−0.769832	−	0.638247i	$-0.779660\pi$
$379$		−	483.791i		−	1.27649i	0.769832	−	0.638247i	$-0.220340\pi$
$380$		0			0
$381$	−168.078			−0.441149
$382$		0			0
$383$	−34.0598			−0.0889290			−0.0444645	−	0.999011i	$-0.514158\pi$
$383$	−34.0598			−0.0889290			−0.0444645	+	0.999011i	$0.514158\pi$
$384$		0			0
$385$	−707.800	−	32.4890i	−1.83844	−	0.0843869i
$386$		0			0
$387$	95.9453			0.247921
$388$		0			0
$389$		−	503.110i		−	1.29334i	−0.762770	−	0.646670i	$-0.776161\pi$
$389$		−	503.110i		−	1.29334i	0.762770	−	0.646670i	$-0.223839\pi$
$390$		0			0
$391$			534.493i			1.36699i
$392$		0			0
$393$		−	80.6929i		−	0.205326i
$394$		0			0
$395$	−225.455	+	277.670i	−0.570773	+	0.702963i
$396$		0			0
$397$			324.083i			0.816330i	0.912908	+	0.408165i	$0.133831\pi$
$397$			324.083i			0.816330i	−0.912908	+	0.408165i	$0.866169\pi$
$398$		0			0
$399$	−133.356	−	180.570i	−0.334225	−	0.452556i
$400$		0			0
$401$	288.570			0.719627			0.359813	−	0.933024i	$-0.382840\pi$
$401$	288.570			0.719627			0.359813	+	0.933024i	$0.382840\pi$
$402$		0			0
$403$	−99.8986			−0.247887
$404$		0			0
$405$	129.455	−	159.437i	0.319642	−	0.393671i
$406$		0			0
$407$		−	376.553i		−	0.925192i
$408$		0			0
$409$		−	723.203i		−	1.76822i	−0.467277	−	0.884111i	$-0.654765\pi$
$409$		−	723.203i		−	1.76822i	0.467277	−	0.884111i	$-0.345235\pi$
$410$		0			0
$411$	−69.9133			−0.170105
$412$		0			0
$413$	458.921	−	338.925i	1.11119	−	0.820642i
$414$		0			0
$415$	−129.188	−	104.895i	−0.311297	−	0.252758i
$416$		0			0
$417$		−	222.768i		−	0.534216i
$418$		0			0
$419$	−3.44359			−0.00821860			−0.00410930	−	0.999992i	$-0.501308\pi$
$419$	−3.44359			−0.00821860			−0.00410930	+	0.999992i	$0.501308\pi$
$420$		0			0
$421$		−	613.823i		−	1.45801i	−0.684508	−	0.729006i	$-0.739983\pi$
$421$		−	613.823i		−	1.45801i	0.684508	−	0.729006i	$-0.260017\pi$
$422$		0			0
$423$	−254.386			−0.601385
$424$		0			0
$425$	126.465	+	602.723i	0.297565	+	1.41817i
$426$		0			0
$427$	219.244	−	161.918i	0.513453	−	0.379199i
$428$		0			0
$429$		−	47.6152i		−	0.110991i
$430$		0			0
$431$	−697.462			−1.61824			−0.809121	−	0.587643i	$-0.800056\pi$
$431$	−697.462			−1.61824			−0.809121	+	0.587643i	$0.800056\pi$
$432$		0			0
$433$	722.396			1.66835			0.834176	−	0.551499i	$-0.185944\pi$
$433$	722.396			1.66835			0.834176	+	0.551499i	$0.185944\pi$
$434$		0			0
$435$	−120.885	+	148.882i	−0.277896	+	0.342257i
$436$		0			0
$437$			555.302i			1.27071i
$438$		0			0
$439$		−	641.760i		−	1.46187i	−0.682448	−	0.730934i	$-0.739085\pi$
$439$		−	641.760i		−	1.46187i	0.682448	−	0.730934i	$-0.260915\pi$
$440$		0			0
$441$	107.089	−	347.964i	0.242833	−	0.789034i
$442$		0			0
$443$	−209.765			−0.473511			−0.236756	−	0.971569i	$-0.576084\pi$
$443$	−209.765			−0.473511			−0.236756	+	0.971569i	$0.576084\pi$
$444$		0			0
$445$	115.765	+	93.9954i	0.260145	+	0.211226i
$446$		0			0
$447$	48.2095			0.107851
$448$		0			0
$449$	8.26585			0.0184095			0.00920473	−	0.999958i	$-0.497070\pi$
$449$	8.26585			0.0184095			0.00920473	+	0.999958i	$0.497070\pi$
$450$		0			0
$451$	−533.736			−1.18345
$452$		0			0
$453$		−	24.2334i		−	0.0534954i
$454$		0			0
$455$	65.6307	+	3.01253i	0.144243	+	0.00662095i
$456$		0			0
$457$		−	7.84458i		−	0.0171654i	−0.999963	−	0.00858269i	$-0.997268\pi$
$457$		−	7.84458i		−	0.0171654i	0.999963	−	0.00858269i	$-0.00273199\pi$
$458$		0			0
$459$			507.132i			1.10486i
$460$		0			0
$461$	−12.9168			−0.0280190			−0.0140095	−	0.999902i	$-0.504460\pi$
$461$	−12.9168			−0.0280190			−0.0140095	+	0.999902i	$0.504460\pi$
$462$		0			0
$463$		−	332.355i		−	0.717829i	−0.933370	−	0.358915i	$-0.883147\pi$
$463$		−	332.355i		−	0.717829i	0.933370	−	0.358915i	$-0.116853\pi$
$464$		0			0
$465$	−210.163	+	258.836i	−0.451963	+	0.556637i
$466$		0			0
$467$			175.773i			0.376387i	0.982132	+	0.188194i	$0.0602632\pi$
$467$			175.773i			0.376387i	−0.982132	+	0.188194i	$0.939737\pi$
$468$		0			0
$469$	401.523	−	296.536i	0.856126	−	0.632272i
$470$		0			0
$471$	264.997			0.562626
$472$		0			0
$473$			261.417i			0.552680i
$474$		0			0
$475$	131.388	+	626.188i	0.276607	+	1.31829i
$476$		0			0
$477$	−368.419			−0.772367
$478$		0			0
$479$		−	540.022i		−	1.12740i	−0.825981	−	0.563698i	$-0.809378\pi$
$479$		−	540.022i		−	1.12740i	0.825981	−	0.563698i	$-0.190622\pi$
$480$		0			0
$481$			34.9158i			0.0725901i
$482$		0			0
$483$	153.085	−	113.058i	0.316947	−	0.234074i
$484$		0			0
$485$	350.295	−	431.423i	0.722258	−	0.889532i
$486$		0			0
$487$		−	252.162i		−	0.517786i	−0.965906	−	0.258893i	$-0.916642\pi$
$487$		−	252.162i		−	0.517786i	0.965906	−	0.258893i	$-0.0833577\pi$
$488$		0			0
$489$			364.965i			0.746350i
$490$		0			0
$491$			426.337i			0.868303i	0.900840	+	0.434152i	$0.142952\pi$
$491$			426.337i			0.868303i	−0.900840	+	0.434152i	$0.857048\pi$
$492$		0			0
$493$			754.072i			1.52956i
$494$		0			0
$495$	583.849	+	474.058i	1.17949	+	0.957692i
$496$		0			0
$497$	−253.219	+	187.009i	−0.509494	+	0.376275i
$498$		0			0
$499$			105.582i			0.211586i	0.994388	+	0.105793i	$0.0337381\pi$
$499$			105.582i			0.211586i	−0.994388	+	0.105793i	$0.966262\pi$
$500$		0			0
$501$			205.512i			0.410203i
$502$		0			0
$503$	138.518			0.275385			0.137692	−	0.990475i	$-0.456031\pi$
$503$	138.518			0.275385			0.137692	+	0.990475i	$0.456031\pi$
$504$		0			0
$505$	−29.2063	+	35.9704i	−0.0578342	+	0.0712285i
$506$		0			0
$507$		−	207.341i		−	0.408957i
$508$		0			0
$509$	581.779			1.14298			0.571492	−	0.820608i	$-0.306365\pi$
$509$	581.779			1.14298			0.571492	+	0.820608i	$0.306365\pi$
$510$		0			0
$511$	−121.760	+	89.9228i	−0.238277	+	0.175974i
$512$		0			0
$513$			526.875i			1.02705i
$514$		0			0
$515$	332.078	−	408.987i	0.644812	−	0.794150i
$516$		0			0
$517$		−	693.113i		−	1.34064i
$518$		0			0
$519$	188.243			0.362703
$520$		0			0
$521$		−	325.445i		−	0.624654i	−0.949975	−	0.312327i	$-0.898892\pi$
$521$		−	325.445i		−	0.624654i	0.949975	−	0.312327i	$-0.101108\pi$
$522$		0			0
$523$		−	938.001i		−	1.79350i	−0.442536	−	0.896751i	$-0.645921\pi$
$523$		−	938.001i		−	1.79350i	0.442536	−	0.896751i	$-0.354079\pi$
$524$		0			0
$525$	145.877	−	163.711i	0.277861	−	0.311830i
$526$		0			0
$527$			1310.98i			2.48763i
$528$		0			0
$529$	58.2206			0.110058
$530$		0			0
$531$	−605.552			−1.14040
$532$		0			0
$533$	49.4906			0.0928528
$534$		0			0
$535$	147.800	−	182.030i	0.276262	−	0.340244i
$536$		0			0
$537$	−99.4675			−0.185228
$538$		0			0
$539$	948.080	+	291.781i	1.75896	+	0.541338i
$540$		0			0
$541$			579.022i			1.07028i	0.844763	+	0.535141i	$0.179741\pi$
$541$			579.022i			1.07028i	−0.844763	+	0.535141i	$0.820259\pi$
$542$		0			0
$543$			345.166i			0.635665i
$544$		0			0
$545$	451.526	+	366.618i	0.828488	+	0.672694i
$546$		0			0
$547$	−177.049			−0.323674			−0.161837	−	0.986818i	$-0.551742\pi$
$547$	−177.049			−0.323674			−0.161837	+	0.986818i	$0.551742\pi$
$548$		0			0
$549$	−289.296			−0.526951
$550$		0			0
$551$			783.429i			1.42183i
$552$		0			0
$553$	402.802	−	297.480i	0.728395	−	0.537939i
$554$		0			0
$555$	90.4667	+	73.4547i	0.163003	+	0.132351i
$556$		0			0
$557$	819.737			1.47170			0.735850	−	0.677144i	$-0.236783\pi$
$557$	819.737			1.47170			0.735850	+	0.677144i	$0.236783\pi$
$558$		0			0
$559$		−	24.2399i		−	0.0433630i
$560$		0			0
$561$	−624.861			−1.11383
$562$		0			0
$563$			299.392i			0.531781i	0.964003	+	0.265890i	$0.0856659\pi$
$563$			299.392i			0.531781i	−0.964003	+	0.265890i	$0.914334\pi$
$564$		0			0
$565$	−211.493	−	171.723i	−0.374325	−	0.303934i
$566$		0			0
$567$	−231.287	+	170.811i	−0.407913	+	0.301255i
$568$		0			0
$569$	756.264			1.32911			0.664555	−	0.747239i	$-0.268621\pi$
$569$	756.264			1.32911			0.664555	+	0.747239i	$0.268621\pi$
$570$		0			0
$571$		−	612.323i		−	1.07237i	−0.844100	−	0.536185i	$-0.819865\pi$
$571$		−	612.323i		−	1.07237i	0.844100	−	0.536185i	$-0.180135\pi$
$572$		0			0
$573$			13.0107i			0.0227062i
$574$		0			0
$575$	−530.876	+	111.390i	−0.923263	+	0.193722i
$576$		0			0
$577$	−1086.54			−1.88309			−0.941547	−	0.336883i	$-0.890627\pi$
$577$	−1086.54			−1.88309			−0.941547	+	0.336883i	$0.890627\pi$
$578$		0			0
$579$	226.273			0.390800
$580$		0			0
$581$	138.405	+	187.407i	0.238219	+	0.322559i
$582$		0			0
$583$		−	1003.81i		−	1.72181i
$584$		0			0
$585$	−54.1373	−	43.9569i	−0.0925423	−	0.0751400i
$586$		0			0
$587$			293.714i			0.500365i	0.968199	+	0.250182i	$0.0804906\pi$
$587$			293.714i			0.500365i	−0.968199	+	0.250182i	$0.919509\pi$
$588$		0			0
$589$			1362.02i			2.31243i
$590$		0			0
$591$			145.421i			0.246059i
$592$		0			0
$593$	306.654			0.517123			0.258562	−	0.965995i	$-0.416752\pi$
$593$	306.654			0.517123			0.258562	+	0.965995i	$0.416752\pi$
$594$		0			0
$595$	39.5339	−	861.280i	0.0664435	−	1.44753i
$596$		0			0
$597$	−134.345			−0.225033
$598$		0			0
$599$	879.523			1.46832			0.734160	−	0.678977i	$-0.237576\pi$
$599$	879.523			1.46832			0.734160	+	0.678977i	$0.237576\pi$
$600$		0			0
$601$			344.758i			0.573641i	0.957984	+	0.286821i	$0.0925984\pi$
$601$			344.758i			0.573641i	−0.957984	+	0.286821i	$0.907402\pi$
$602$		0			0
$603$	−529.816			−0.878633
$604$		0			0
$605$	−910.288	+	1121.11i	−1.50461	+	1.85307i
$606$		0			0
$607$	−297.560			−0.490214			−0.245107	−	0.969496i	$-0.578823\pi$
$607$	−297.560			−0.490214			−0.245107	+	0.969496i	$0.578823\pi$
$608$		0			0
$609$	215.975	−	159.503i	0.354639	−	0.261910i
$610$		0			0
$611$			64.2688i			0.105186i
$612$		0			0
$613$	−372.563			−0.607771			−0.303885	−	0.952709i	$-0.598284\pi$
$613$	−372.563			−0.607771			−0.303885	+	0.952709i	$0.598284\pi$
$614$		0			0
$615$	104.116	−	128.230i	0.169295	−	0.208503i
$616$		0			0
$617$			933.370i			1.51276i	0.654135	+	0.756378i	$0.273033\pi$
$617$			933.370i			1.51276i	−0.654135	+	0.756378i	$0.726967\pi$
$618$		0			0
$619$	−383.452			−0.619471			−0.309735	−	0.950823i	$-0.600240\pi$
$619$	−383.452			−0.619471			−0.309735	+	0.950823i	$0.600240\pi$
$620$		0			0
$621$	−446.680			−0.719291
$622$		0			0
$623$	−124.024	−	167.934i	−0.199075	−	0.269557i
$624$		0			0
$625$	−572.289	+	251.218i	−0.915662	+	0.401949i
$626$		0			0
$627$	−649.187			−1.03539
$628$		0			0
$629$	458.205			0.728466
$630$		0			0
$631$	68.2956			0.108234			0.0541170	−	0.998535i	$-0.482766\pi$
$631$	68.2956			0.108234			0.0541170	+	0.998535i	$0.482766\pi$
$632$		0			0
$633$	21.4563			0.0338963
$634$		0			0
$635$	−520.682	−	422.769i	−0.819971	−	0.665778i
$636$		0			0
$637$	−87.9106	−	27.0553i	−0.138007	−	0.0424731i
$638$		0			0
$639$	334.126			0.522888
$640$		0			0
$641$	−215.545			−0.336264			−0.168132	−	0.985764i	$-0.553774\pi$
$641$	−215.545			−0.336264			−0.168132	+	0.985764i	$0.553774\pi$
$642$		0			0
$643$		−	371.623i		−	0.577951i	−0.957336	−	0.288976i	$-0.906685\pi$
$643$		−	371.623i		−	0.577951i	0.957336	−	0.288976i	$-0.0933147\pi$
$644$		0			0
$645$	−62.8054	−	50.9950i	−0.0973727	−	0.0790620i
$646$		0			0
$647$	762.196			1.17805			0.589023	−	0.808116i	$-0.299513\pi$
$647$	762.196			1.17805			0.589023	+	0.808116i	$0.299513\pi$
$648$		0			0
$649$		−	1649.92i		−	2.54225i
$650$		0			0
$651$	375.481	−	277.303i	0.576776	−	0.425964i
$652$		0			0
$653$	−193.799			−0.296782			−0.148391	−	0.988929i	$-0.547409\pi$
$653$	−193.799			−0.296782			−0.148391	+	0.988929i	$0.547409\pi$
$654$		0			0
$655$	202.968	−	249.975i	0.309875	−	0.381642i
$656$		0			0
$657$	160.664			0.244541
$658$		0			0
$659$			37.2651i			0.0565480i	0.999600	+	0.0282740i	$0.00900109\pi$
$659$			37.2651i			0.0565480i	−0.999600	+	0.0282740i	$0.990999\pi$
$660$		0			0
$661$	−1168.25			−1.76740			−0.883701	−	0.468052i	$-0.844956\pi$
$661$	−1168.25			−1.76740			−0.883701	+	0.468052i	$0.844956\pi$
$662$		0			0
$663$	57.9401			0.0873908
$664$		0			0
$665$	41.0730	−	894.811i	0.0617639	−	1.34558i
$666$		0			0
$667$	−664.183			−0.995777
$668$		0			0
$669$		−	61.7904i		−	0.0923623i
$670$		0			0
$671$		−	788.232i		−	1.17471i
$672$		0			0
$673$		−	680.892i		−	1.01173i	−0.862614	−	0.505863i	$-0.831174\pi$
$673$		−	680.892i		−	1.01173i	0.862614	−	0.505863i	$-0.168826\pi$
$674$		0			0
$675$	−503.700	+	105.688i	−0.746222	+	0.156574i
$676$		0			0
$677$		−	668.119i		−	0.986882i	−0.869779	−	0.493441i	$-0.835739\pi$
$677$		−	668.119i		−	0.986882i	0.869779	−	0.493441i	$-0.164261\pi$
$678$		0			0
$679$	−625.844	+	462.202i	−0.921714	+	0.680710i
$680$		0			0
$681$	362.294			0.532003
$682$		0			0
$683$	−567.050			−0.830234			−0.415117	−	0.909768i	$-0.636259\pi$
$683$	−567.050			−0.830234			−0.415117	+	0.909768i	$0.636259\pi$
$684$		0			0
$685$	−216.582	−	175.854i	−0.316177	−	0.256721i
$686$		0			0
$687$		−	385.565i		−	0.561231i
$688$		0			0
$689$			93.0785i			0.135092i
$690$		0			0
$691$	1156.80			1.67409			0.837046	−	0.547133i	$-0.184281\pi$
$691$	1156.80			1.67409			0.837046	+	0.547133i	$0.184281\pi$
$692$		0			0
$693$	−625.503	−	846.960i	−0.902602	−	1.22216i
$694$		0			0
$695$	560.332	−	690.104i	0.806233	−	0.992956i
$696$		0			0
$697$		−	649.471i		−	0.931810i
$698$		0			0
$699$	−397.581			−0.568785
$700$		0			0
$701$		−	31.0475i		−	0.0442903i	−0.999755	−	0.0221451i	$-0.992950\pi$
$701$		−	31.0475i		−	0.0442903i	0.999755	−	0.0221451i	$-0.00704959\pi$
$702$		0			0
$703$	476.044			0.677161
$704$		0			0
$705$	166.520	+	135.206i	0.236198	+	0.191782i
$706$		0			0
$707$	52.1804	−	38.5367i	0.0738054	−	0.0545073i
$708$		0			0
$709$			798.804i			1.12666i	0.826231	+	0.563332i	$0.190481\pi$
$709$			798.804i			1.12666i	−0.826231	+	0.563332i	$0.809519\pi$
$710$		0			0
$711$	−531.504			−0.747544
$712$		0			0
$713$	−1154.71			−1.61951
$714$		0			0
$715$	119.767	−	147.505i	0.167507	−	0.206301i
$716$		0			0
$717$		−	74.6618i		−	0.104131i
$718$		0			0
$719$		−	929.695i		−	1.29304i	−0.762897	−	0.646520i	$-0.776224\pi$
$719$		−	929.695i		−	1.29304i	0.762897	−	0.646520i	$-0.223776\pi$
$720$		0			0
$721$	−593.297	+	438.166i	−0.822881	+	0.607720i
$722$		0			0
$723$	222.599			0.307882
$724$		0			0
$725$	−748.969	+	157.151i	−1.03306	+	0.216760i
$726$		0			0
$727$	−395.797			−0.544425			−0.272212	−	0.962237i	$-0.587755\pi$
$727$	−395.797			−0.544425			−0.272212	+	0.962237i	$0.587755\pi$
$728$		0			0
$729$	73.0301			0.100178
$730$		0			0
$731$	−318.103			−0.435162
$732$		0			0
$733$		−	707.573i		−	0.965311i	−0.875810	−	0.482655i	$-0.839672\pi$
$733$		−	707.573i		−	0.965311i	0.875810	−	0.482655i	$-0.160328\pi$
$734$		0			0
$735$	−255.043	+	170.858i	−0.346998	+	0.232459i
$736$		0			0
$737$		−	1443.56i		−	1.95870i
$738$		0			0
$739$			939.005i			1.27064i	0.772248	+	0.635321i	$0.219132\pi$
$739$			939.005i			1.27064i	−0.772248	+	0.635321i	$0.780868\pi$
$740$		0			0
$741$	60.1958			0.0812359
$742$		0			0
$743$			596.728i			0.803133i	0.915830	+	0.401566i	$0.131534\pi$
$743$			596.728i			0.803133i	−0.915830	+	0.401566i	$0.868466\pi$
$744$		0			0
$745$	149.346	+	121.262i	0.200465	+	0.162768i
$746$		0			0
$747$		−	247.286i		−	0.331039i
$748$		0			0
$749$	−264.062	+	195.017i	−0.352553	+	0.260370i
$750$		0			0
$751$	−93.9107			−0.125048			−0.0625238	−	0.998043i	$-0.519915\pi$
$751$	−93.9107			−0.125048			−0.0625238	+	0.998043i	$0.519915\pi$
$752$		0			0
$753$		−	267.129i		−	0.354752i
$754$		0			0
$755$	60.9547	−	75.0717i	0.0807347	−	0.0994327i
$756$		0			0
$757$	−220.662			−0.291496			−0.145748	−	0.989322i	$-0.546559\pi$
$757$	−220.662			−0.291496			−0.145748	+	0.989322i	$0.546559\pi$
$758$		0			0
$759$		−	550.375i		−	0.725131i
$760$		0			0
$761$			1174.42i			1.54326i	0.636073	+	0.771629i	$0.280558\pi$
$761$			1174.42i			1.54326i	−0.636073	+	0.771629i	$0.719442\pi$
$762$		0			0
$763$	−483.740	−	655.007i	−0.633997	−	0.858462i
$764$		0			0
$765$	−576.852	+	710.451i	−0.754056	+	0.928694i
$766$		0			0
$767$			152.989i			0.199464i
$768$		0			0
$769$		−	852.450i		−	1.10852i	−0.832344	−	0.554259i	$-0.813002\pi$
$769$		−	852.450i		−	1.10852i	0.832344	−	0.554259i	$-0.186998\pi$
$770$		0			0
$771$		−	180.678i		−	0.234343i
$772$		0			0
$773$			338.769i			0.438252i	0.975697	+	0.219126i	$0.0703206\pi$
$773$			338.769i			0.438252i	−0.975697	+	0.219126i	$0.929679\pi$
$774$		0			0
$775$	−1302.11	+	273.212i	−1.68014	+	0.352532i
$776$		0			0
$777$	−96.9209	−	131.235i	−0.124737	−	0.168900i
$778$		0			0
$779$		−	674.756i		−	0.866183i
$780$		0			0
$781$			910.376i			1.16565i
$782$		0			0
$783$	−630.183			−0.804832
$784$		0			0
$785$	820.924	+	666.551i	1.04576	+	0.849110i
$786$		0			0
$787$		−	220.828i		−	0.280594i	−0.990109	−	0.140297i	$-0.955194\pi$
$787$		−	220.828i		−	0.280594i	0.990109	−	0.140297i	$-0.0448058\pi$
$788$		0			0
$789$	97.0792			0.123041
$790$		0			0
$791$	226.582	+	306.803i	0.286450	+	0.387867i
$792$		0			0
$793$			73.0887i			0.0921673i
$794$		0			0
$795$	241.166	+	195.815i	0.303353	+	0.246308i
$796$		0			0
$797$			571.390i			0.716925i	0.933544	+	0.358463i	$0.116699\pi$
$797$			571.390i			0.716925i	−0.933544	+	0.358463i	$0.883301\pi$
$798$		0			0
$799$	843.407			1.05558
$800$		0			0
$801$			221.591i			0.276643i
$802$		0			0
$803$			437.753i			0.545146i
$804$		0			0
$805$	758.612	+	34.8213i	0.942375	+	0.0432563i
$806$		0			0
$807$			282.244i			0.349745i
$808$		0			0
$809$	−610.054			−0.754085			−0.377042	−	0.926196i	$-0.623059\pi$
$809$	−610.054			−0.754085			−0.377042	+	0.926196i	$0.623059\pi$
$810$		0			0
$811$	−957.214			−1.18029			−0.590144	−	0.807298i	$-0.700929\pi$
$811$	−957.214			−1.18029			−0.590144	+	0.807298i	$0.700929\pi$
$812$		0			0
$813$	−145.274			−0.178689
$814$		0			0
$815$	−918.003	+	1130.61i	−1.12638	+	1.38725i
$816$		0			0
$817$	−330.488			−0.404514
$818$		0			0
$819$	57.9997	+	78.5342i	0.0708176	+	0.0958904i
$820$		0			0
$821$		−	1059.13i		−	1.29005i	−0.764163	−	0.645023i	$-0.776848\pi$
$821$		−	1059.13i		−	1.29005i	0.764163	−	0.645023i	$-0.223152\pi$
$822$		0			0
$823$			40.1520i			0.0487874i	0.999702	+	0.0243937i	$0.00776553\pi$
$823$			40.1520i			0.0487874i	−0.999702	+	0.0243937i	$0.992234\pi$
$824$		0			0
$825$	−130.223	−	620.632i	−0.157846	−	0.752281i
$826$		0			0
$827$	827.416			1.00050			0.500251	−	0.865880i	$-0.333241\pi$
$827$	827.416			1.00050			0.500251	+	0.865880i	$0.333241\pi$
$828$		0			0
$829$	−1391.07			−1.67801			−0.839007	−	0.544121i	$-0.816863\pi$
$829$	−1391.07			−1.67801			−0.839007	+	0.544121i	$0.816863\pi$
$830$		0			0
$831$			6.44755i			0.00775879i
$832$		0			0
$833$	−355.051	+	1153.66i	−0.426232	+	1.38495i
$834$		0			0
$835$	−516.927	+	636.646i	−0.619074	+	0.762451i
$836$		0			0
$837$	−1095.60			−1.30896
$838$		0			0
$839$		−	300.081i		−	0.357665i	−0.983880	−	0.178832i	$-0.942768\pi$
$839$		−	300.081i		−	0.357665i	0.983880	−	0.178832i	$-0.0572320\pi$
$840$		0			0
$841$	−96.0405			−0.114198
$842$		0			0
$843$			316.174i			0.375058i
$844$		0			0
$845$	521.529	−	642.315i	0.617194	−	0.760136i
$846$		0			0
$847$	1626.34	−	1201.09i	1.92012	−	1.41806i
$848$		0			0
$849$	426.936			0.502870
$850$		0			0
$851$			403.585i			0.474248i
$852$		0			0
$853$		−	317.790i		−	0.372556i	−0.982497	−	0.186278i	$-0.940358\pi$
$853$		−	317.790i		−	0.372556i	0.982497	−	0.186278i	$-0.0596425\pi$
$854$		0			0
$855$	−599.310	+	738.110i	−0.700948	+	0.863286i
$856$		0			0
$857$	395.762			0.461799			0.230900	−	0.972978i	$-0.425833\pi$
$857$	395.762			0.461799			0.230900	+	0.972978i	$0.425833\pi$
$858$		0			0
$859$	349.345			0.406688			0.203344	−	0.979107i	$-0.434819\pi$
$859$	349.345			0.406688			0.203344	+	0.979107i	$0.434819\pi$
$860$		0			0
$861$	−186.016	+	137.378i	−0.216047	+	0.159556i
$862$		0			0
$863$			357.972i			0.414799i	0.978256	+	0.207400i	$0.0665000\pi$
$863$			357.972i			0.414799i	−0.978256	+	0.207400i	$0.933500\pi$
$864$		0			0
$865$	583.149	+	473.490i	0.674161	+	0.547387i
$866$		0			0
$867$		−	398.240i		−	0.459331i
$868$		0			0
$869$		−	1448.16i		−	1.66647i
$870$		0			0
$871$			133.854i			0.153679i
$872$		0			0
$873$	825.810			0.945945
$874$		0			0
$875$	863.691	−	140.227i	0.987075	−	0.160259i
$876$		0			0
$877$	768.096			0.875822			0.437911	−	0.899018i	$-0.355718\pi$
$877$	768.096			0.875822			0.437911	+	0.899018i	$0.355718\pi$
$878$		0			0
$879$	114.051			0.129751
$880$		0			0
$881$		−	353.881i		−	0.401681i	−0.979624	−	0.200841i	$-0.935633\pi$
$881$		−	353.881i		−	0.401681i	0.979624	−	0.200841i	$-0.0643674\pi$
$882$		0			0
$883$	−1219.92			−1.38156			−0.690780	−	0.723065i	$-0.742733\pi$
$883$	−1219.92			−1.38156			−0.690780	+	0.723065i	$0.742733\pi$
$884$		0			0
$885$	396.392	+	321.852i	0.447900	+	0.363674i
$886$		0			0
$887$	900.425			1.01514			0.507568	−	0.861612i	$-0.330545\pi$
$887$	900.425			1.01514			0.507568	+	0.861612i	$0.330545\pi$
$888$		0			0
$889$	557.829	+	755.327i	0.627480	+	0.849637i
$890$		0			0
$891$			831.526i			0.933250i
$892$		0			0
$893$	876.243			0.981235
$894$		0			0
$895$	−308.136	−	250.192i	−0.344286	−	0.279544i
$896$		0			0
$897$			51.0334i			0.0568934i
$898$		0			0
$899$	−1629.08			−1.81210
$900$		0			0
$901$	1221.48			1.35570
$902$		0			0
$903$	67.2862	+	91.1086i	0.0745140	+	0.100895i
$904$		0			0
$905$	−868.202	+	1069.28i	−0.959339	+	1.18152i
$906$		0			0
$907$	−436.196			−0.480922			−0.240461	−	0.970659i	$-0.577299\pi$
$907$	−436.196			−0.480922			−0.240461	+	0.970659i	$0.577299\pi$
$908$		0			0
$909$	−68.8529			−0.0757457
$910$		0			0
$911$	474.803			0.521189			0.260594	−	0.965448i	$-0.416082\pi$
$911$	474.803			0.521189			0.260594	+	0.965448i	$0.416082\pi$
$912$		0			0
$913$	673.769			0.737972
$914$		0			0
$915$	189.372	+	153.761i	0.206964	+	0.168045i
$916$		0			0
$917$	−362.627	+	267.810i	−0.395449	+	0.292050i
$918$		0			0
$919$	−1234.65			−1.34347			−0.671733	−	0.740793i	$-0.734450\pi$
$919$	−1234.65			−1.34347			−0.671733	+	0.740793i	$0.734450\pi$
$920$		0			0
$921$	431.258			0.468250
$922$		0			0
$923$		−	84.4145i		−	0.0914567i
$924$		0			0
$925$	95.4913	+	455.105i	0.103234	+	0.492005i
$926$		0			0
$927$	782.865			0.844514
$928$		0			0
$929$			286.886i			0.308812i	0.988008	+	0.154406i	$0.0493463\pi$
$929$			286.886i			0.308812i	−0.988008	+	0.154406i	$0.950654\pi$
$930$		0			0
$931$	−368.874	+	1198.58i	−0.396212	+	1.28741i
$932$		0			0
$933$	173.216			0.185655
$934$		0			0
$935$	−1935.73	−	1571.72i	−2.07030	−	1.68099i
$936$		0			0
$937$	22.6680			0.0241922			0.0120961	−	0.999927i	$-0.496150\pi$
$937$	22.6680			0.0241922			0.0120961	+	0.999927i	$0.496150\pi$
$938$		0			0
$939$			485.686i			0.517237i
$940$		0			0
$941$	−8.61500			−0.00915515			−0.00457758	−	0.999990i	$-0.501457\pi$
$941$	−8.61500			−0.00915515			−0.00457758	+	0.999990i	$0.501457\pi$
$942$		0			0
$943$	572.052			0.606630
$944$		0			0
$945$	719.778	+	33.0388i	0.761670	+	0.0349616i
$946$		0			0
$947$	−1034.37			−1.09226			−0.546130	−	0.837700i	$-0.683900\pi$
$947$	−1034.37			−1.09226			−0.546130	+	0.837700i	$0.683900\pi$
$948$		0			0
$949$		−	40.5905i		−	0.0427719i
$950$		0			0
$951$		−	537.649i		−	0.565351i
$952$		0			0
$953$		−	1094.92i		−	1.14892i	−0.818533	−	0.574459i	$-0.805212\pi$
$953$		−	1094.92i		−	1.14892i	0.818533	−	0.574459i	$-0.194788\pi$
$954$		0			0
$955$	−32.7260	+	40.3052i	−0.0342680	+	0.0422044i
$956$		0			0
$957$		−	776.478i		−	0.811366i
$958$		0			0
$959$	232.033	+	314.184i	0.241954	+	0.327616i
$960$		0			0
$961$	−1871.22			−1.94716
$962$		0			0
$963$	348.434			0.361822
$964$		0			0
$965$	700.962	+	569.149i	0.726386	+	0.589791i
$966$		0			0
$967$			1664.66i			1.72147i	0.509057	+	0.860733i	$0.329994\pi$
$967$			1664.66i			1.72147i	−0.509057	+	0.860733i	$0.670006\pi$
$968$		0			0
$969$		−	789.958i		−	0.815230i
$970$		0			0
$971$	666.438			0.686342			0.343171	−	0.939273i	$-0.388499\pi$
$971$	666.438			0.686342			0.343171	+	0.939273i	$0.388499\pi$
$972$		0			0
$973$	−1001.10	+	739.339i	−1.02888	+	0.759855i
$974$		0			0
$975$	12.0749	+	57.5480i	0.0123845	+	0.0590236i
$976$		0			0
$977$		−	596.765i		−	0.610814i	−0.952222	−	0.305407i	$-0.901208\pi$
$977$		−	596.765i		−	0.610814i	0.952222	−	0.305407i	$-0.0987924\pi$
$978$		0			0
$979$	−603.759			−0.616710
$980$		0			0
$981$			864.291i			0.881030i
$982$		0			0
$983$	−896.429			−0.911932			−0.455966	−	0.889997i	$-0.650706\pi$
$983$	−896.429			−0.911932			−0.455966	+	0.889997i	$0.650706\pi$
$984$		0			0
$985$	−365.779	+	450.493i	−0.371349	+	0.457353i
$986$		0			0
$987$	−178.400	−	241.562i	−0.180750	−	0.244744i
$988$		0			0
$989$		−	280.184i		−	0.283300i
$990$		0			0
$991$	−251.503			−0.253787			−0.126893	−	0.991916i	$-0.540501\pi$
$991$	−251.503			−0.253787			−0.126893	+	0.991916i	$0.540501\pi$
$992$		0			0
$993$	−56.5801			−0.0569789
$994$		0			0
$995$	−416.182	−	337.920i	−0.418273	−	0.339618i
$996$		0			0
$997$		−	428.914i		−	0.430205i	−0.976592	−	0.215102i	$-0.930991\pi$
$997$		−	428.914i		−	0.430205i	0.976592	−	0.215102i	$-0.0690085\pi$
$998$		0			0
$999$			382.925i			0.383309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.3		80
4.3	odd	2		280.3.c.g.69.44	yes	80
5.4	even	2	inner	1120.3.c.g.209.18		80
7.6	odd	2	inner	1120.3.c.g.209.80		80
8.3	odd	2		280.3.c.g.69.39	yes	80
8.5	even	2	inner	1120.3.c.g.209.50		80
20.19	odd	2		280.3.c.g.69.37	✓	80
28.27	even	2		280.3.c.g.69.43	yes	80
35.34	odd	2	inner	1120.3.c.g.209.49		80
40.19	odd	2		280.3.c.g.69.42	yes	80
40.29	even	2	inner	1120.3.c.g.209.79		80
56.13	odd	2	inner	1120.3.c.g.209.17		80
56.27	even	2		280.3.c.g.69.40	yes	80
140.139	even	2		280.3.c.g.69.38	yes	80
280.69	odd	2	inner	1120.3.c.g.209.4		80
280.139	even	2		280.3.c.g.69.41	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.37	✓	80	20.19	odd	2
280.3.c.g.69.38	yes	80	140.139	even	2
280.3.c.g.69.39	yes	80	8.3	odd	2
280.3.c.g.69.40	yes	80	56.27	even	2
280.3.c.g.69.41	yes	80	280.139	even	2
280.3.c.g.69.42	yes	80	40.19	odd	2
280.3.c.g.69.43	yes	80	28.27	even	2
280.3.c.g.69.44	yes	80	4.3	odd	2
1120.3.c.g.209.3		80	1.1	even	1	trivial
1120.3.c.g.209.4		80	280.69	odd	2	inner
1120.3.c.g.209.17		80	56.13	odd	2	inner
1120.3.c.g.209.18		80	5.4	even	2	inner
1120.3.c.g.209.49		80	35.34	odd	2	inner
1120.3.c.g.209.50		80	8.5	even	2	inner
1120.3.c.g.209.79		80	40.29	even	2	inner
1120.3.c.g.209.80		80	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1120.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.25300i q^{3} +(3.15168 - 3.88161i) q^{5} +(-5.63086 + 4.15854i) q^{7} +7.43000 q^{9} +O(q^{10})\)	\(q-1.25300i q^{3} +(3.15168 - 3.88161i) q^{5} +(-5.63086 + 4.15854i) q^{7} +7.43000 q^{9} +20.2442i q^{11} -1.87714i q^{13} +(-4.86364 - 3.94905i) q^{15} -24.6339 q^{17} -25.5929 q^{19} +(5.21064 + 7.05545i) q^{21} -21.6975i q^{23} +(-5.13378 - 24.4672i) q^{25} -20.5867i q^{27} -30.6111i q^{29} -53.2186i q^{31} +25.3659 q^{33} +(-1.60486 + 34.9632i) q^{35} -18.6006 q^{37} -2.35205 q^{39} +26.3649i q^{41} +12.9132 q^{43} +(23.4170 - 28.8403i) q^{45} -34.2377 q^{47} +(14.4131 - 46.8323i) q^{49} +30.8662i q^{51} -49.5854 q^{53} +(78.5799 + 63.8032i) q^{55} +32.0679i q^{57} -81.5010 q^{59} -38.9363 q^{61} +(-41.8373 + 30.8979i) q^{63} +(-7.28631 - 5.91614i) q^{65} -71.3076 q^{67} -27.1868 q^{69} +44.9698 q^{71} +21.6236 q^{73} +(-30.6573 + 6.43261i) q^{75} +(-84.1861 - 113.992i) q^{77} -71.5348 q^{79} +41.0749 q^{81} -33.2821i q^{83} +(-77.6383 + 95.6192i) q^{85} -38.3556 q^{87} +29.8239i q^{89} +(7.80615 + 10.5699i) q^{91} -66.6827 q^{93} +(-80.6609 + 99.3418i) q^{95} +111.145 q^{97} +150.414i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 80 * q - 224 * q^9	\( 80 q - 224 q^{9} + 72 q^{15} - 104 q^{25} + 112 q^{39} + 192 q^{49} + 472 q^{65} - 800 q^{71} - 480 q^{79} - 896 q^{81} - 1176 q^{95}+O(q^{100}) \) 80 * q - 224 * q^9 + 72 * q^15 - 104 * q^25 + 112 * q^39 + 192 * q^49 + 472 * q^65 - 800 * q^71 - 480 * q^79 - 896 * q^81 - 1176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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