LMFDB - Embedded newform 1120.3.c.g.209.53

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			209.53
Character	$\chi$	$=$	1120.209
Dual form			1120.3.c.g.209.54

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.20162i q^{3} +(1.26609 + 4.83705i) q^{5} +(2.96368 + 6.34166i) q^{7} +7.55612 q^{9} +O(q^{10})$	$q-1.20162i q^{3} +(1.26609 + 4.83705i) q^{5} +(2.96368 + 6.34166i) q^{7} +7.55612 q^{9} -6.35173i q^{11} +11.4646i q^{13} +(5.81227 - 1.52135i) q^{15} +17.8708 q^{17} +3.65270 q^{19} +(7.62023 - 3.56121i) q^{21} +1.47479i q^{23} +(-21.7940 + 12.2483i) q^{25} -19.8941i q^{27} +13.9833i q^{29} -31.1953i q^{31} -7.63234 q^{33} +(-26.9226 + 22.3646i) q^{35} -10.8744 q^{37} +13.7760 q^{39} +60.7473i q^{41} +34.9718 q^{43} +(9.56671 + 36.5493i) q^{45} +54.6600 q^{47} +(-31.4332 + 37.5893i) q^{49} -21.4738i q^{51} -50.2489 q^{53} +(30.7236 - 8.04185i) q^{55} -4.38915i q^{57} -48.8348 q^{59} -13.3248 q^{61} +(22.3939 + 47.9183i) q^{63} +(-55.4547 + 14.5152i) q^{65} -105.220 q^{67} +1.77213 q^{69} -65.2930 q^{71} +97.4832 q^{73} +(14.7177 + 26.1881i) q^{75} +(40.2805 - 18.8245i) q^{77} +12.8701 q^{79} +44.1000 q^{81} +140.293i q^{83} +(22.6260 + 86.4417i) q^{85} +16.8025 q^{87} -25.4993i q^{89} +(-72.7044 + 33.9774i) q^{91} -37.4848 q^{93} +(4.62465 + 17.6683i) q^{95} +117.389 q^{97} -47.9944i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.20162i		−	0.400539i	−0.979741	−	0.200269i	$-0.935818\pi$
$3$		−	1.20162i		−	0.400539i	0.979741	−	0.200269i	$-0.0641817\pi$
$4$		0			0
$5$	1.26609	+	4.83705i	0.253218	+	0.967409i
$5$	1.26609	+	4.83705i	0.253218	+	0.967409i
$6$		0			0
$7$	2.96368	+	6.34166i	0.423383	+	0.905951i
$7$	2.96368	+	6.34166i	0.423383	+	0.905951i
$8$		0			0
$9$	7.55612			0.839569
$10$		0			0
$11$		−	6.35173i		−	0.577430i	−0.957415	−	0.288715i	$-0.906772\pi$
$11$		−	6.35173i		−	0.577430i	0.957415	−	0.288715i	$-0.0932280\pi$
$12$		0			0
$13$			11.4646i			0.881890i	0.897534	+	0.440945i	$0.145357\pi$
$13$			11.4646i			0.881890i	−0.897534	+	0.440945i	$0.854643\pi$
$14$		0			0
$15$	5.81227	−	1.52135i	0.387485	−	0.101423i
$16$		0			0
$17$	17.8708			1.05122			0.525611	−	0.850725i	$-0.323837\pi$
$17$	17.8708			1.05122			0.525611	+	0.850725i	$0.323837\pi$
$18$		0			0
$19$	3.65270			0.192248			0.0961238	−	0.995369i	$-0.469356\pi$
$19$	3.65270			0.192248			0.0961238	+	0.995369i	$0.469356\pi$
$20$		0			0
$21$	7.62023	−	3.56121i	0.362868	−	0.169581i
$22$		0			0
$23$			1.47479i			0.0641214i	0.999486	+	0.0320607i	$0.0102070\pi$
$23$			1.47479i			0.0641214i	−0.999486	+	0.0320607i	$0.989793\pi$
$24$		0			0
$25$	−21.7940	+	12.2483i	−0.871762	+	0.489930i
$26$		0			0
$27$		−	19.8941i		−	0.736818i
$28$		0			0
$29$			13.9833i			0.482182i	0.970503	+	0.241091i	$0.0775052\pi$
$29$			13.9833i			0.482182i	−0.970503	+	0.241091i	$0.922495\pi$
$30$		0			0
$31$		−	31.1953i		−	1.00630i	−0.864199	−	0.503150i	$-0.832174\pi$
$31$		−	31.1953i		−	1.00630i	0.864199	−	0.503150i	$-0.167826\pi$
$32$		0			0
$33$	−7.63234			−0.231283
$34$		0			0
$35$	−26.9226	+	22.3646i	−0.769217	+	0.638988i
$36$		0			0
$37$	−10.8744			−0.293902			−0.146951	−	0.989144i	$-0.546946\pi$
$37$	−10.8744			−0.293902			−0.146951	+	0.989144i	$0.546946\pi$
$38$		0			0
$39$	13.7760			0.353231
$40$		0			0
$41$			60.7473i			1.48164i	0.671703	+	0.740821i	$0.265563\pi$
$41$			60.7473i			1.48164i	−0.671703	+	0.740821i	$0.734437\pi$
$42$		0			0
$43$	34.9718			0.813299			0.406649	−	0.913584i	$-0.366697\pi$
$43$	34.9718			0.813299			0.406649	+	0.913584i	$0.366697\pi$
$44$		0			0
$45$	9.56671	+	36.5493i	0.212594	+	0.812207i
$46$		0			0
$47$	54.6600			1.16298			0.581490	−	0.813554i	$-0.302470\pi$
$47$	54.6600			1.16298			0.581490	+	0.813554i	$0.302470\pi$
$48$		0			0
$49$	−31.4332	+	37.5893i	−0.641493	+	0.767129i
$50$		0			0
$51$		−	21.4738i		−	0.421055i
$52$		0			0
$53$	−50.2489			−0.948093			−0.474047	−	0.880500i	$-0.657207\pi$
$53$	−50.2489			−0.948093			−0.474047	+	0.880500i	$0.657207\pi$
$54$		0			0
$55$	30.7236	−	8.04185i	0.558611	−	0.146215i
$56$		0			0
$57$		−	4.38915i		−	0.0770026i
$58$		0			0
$59$	−48.8348			−0.827709			−0.413855	−	0.910343i	$-0.635818\pi$
$59$	−48.8348			−0.827709			−0.413855	+	0.910343i	$0.635818\pi$
$60$		0			0
$61$	−13.3248			−0.218440			−0.109220	−	0.994018i	$-0.534835\pi$
$61$	−13.3248			−0.218440			−0.109220	+	0.994018i	$0.534835\pi$
$62$		0			0
$63$	22.3939	+	47.9183i	0.355459	+	0.760608i
$64$		0			0
$65$	−55.4547	+	14.5152i	−0.853149	+	0.223310i
$66$		0			0
$67$	−105.220			−1.57045			−0.785224	−	0.619212i	$-0.787452\pi$
$67$	−105.220			−1.57045			−0.785224	+	0.619212i	$0.787452\pi$
$68$		0			0
$69$	1.77213			0.0256831
$70$		0			0
$71$	−65.2930			−0.919620			−0.459810	−	0.888017i	$-0.652083\pi$
$71$	−65.2930			−0.919620			−0.459810	+	0.888017i	$0.652083\pi$
$72$		0			0
$73$	97.4832			1.33539			0.667693	−	0.744437i	$-0.267282\pi$
$73$	97.4832			1.33539			0.667693	+	0.744437i	$0.267282\pi$
$74$		0			0
$75$	14.7177	+	26.1881i	0.196236	+	0.349174i
$76$		0			0
$77$	40.2805	−	18.8245i	0.523123	−	0.244474i
$78$		0			0
$79$	12.8701			0.162913			0.0814563	−	0.996677i	$-0.474043\pi$
$79$	12.8701			0.162913			0.0814563	+	0.996677i	$0.474043\pi$
$80$		0			0
$81$	44.1000			0.544445
$82$		0			0
$83$			140.293i			1.69027i	0.534550	+	0.845137i	$0.320481\pi$
$83$			140.293i			1.69027i	−0.534550	+	0.845137i	$0.679519\pi$
$84$		0			0
$85$	22.6260	+	86.4417i	0.266188	+	1.01696i
$86$		0			0
$87$	16.8025			0.193132
$88$		0			0
$89$		−	25.4993i		−	0.286509i	−0.989686	−	0.143255i	$-0.954243\pi$
$89$		−	25.4993i		−	0.286509i	0.989686	−	0.143255i	$-0.0457568\pi$
$90$		0			0
$91$	−72.7044	+	33.9774i	−0.798949	+	0.373378i
$92$		0			0
$93$	−37.4848			−0.403062
$94$		0			0
$95$	4.62465	+	17.6683i	0.0486805	+	0.185982i
$96$		0			0
$97$	117.389			1.21019			0.605097	−	0.796152i	$-0.293134\pi$
$97$	117.389			1.21019			0.605097	+	0.796152i	$0.293134\pi$
$98$		0			0
$99$		−	47.9944i		−	0.484792i
$100$		0			0
$101$	83.2390			0.824148			0.412074	−	0.911150i	$-0.364804\pi$
$101$	83.2390			0.824148			0.412074	+	0.911150i	$0.364804\pi$
$102$		0			0
$103$	157.514			1.52926			0.764631	−	0.644468i	$-0.222921\pi$
$103$	157.514			1.52926			0.764631	+	0.644468i	$0.222921\pi$
$104$		0			0
$105$	26.8736	+	32.3506i	0.255939	+	0.308101i
$106$		0			0
$107$	−89.9587			−0.840735			−0.420368	−	0.907354i	$-0.638099\pi$
$107$	−89.9587			−0.840735			−0.420368	+	0.907354i	$0.638099\pi$
$108$		0			0
$109$		−	38.4998i		−	0.353210i	−0.984282	−	0.176605i	$-0.943489\pi$
$109$		−	38.4998i		−	0.353210i	0.984282	−	0.176605i	$-0.0565115\pi$
$110$		0			0
$111$			13.0668i			0.117719i
$112$		0			0
$113$			146.901i			1.30001i	0.759930	+	0.650005i	$0.225233\pi$
$113$			146.901i			1.30001i	−0.759930	+	0.650005i	$0.774767\pi$
$114$		0			0
$115$	−7.13363	+	1.86722i	−0.0620316	+	0.0162367i
$116$		0			0
$117$			86.6277i			0.740408i
$118$		0			0
$119$	52.9632	+	113.330i	0.445069	+	0.952354i
$120$		0			0
$121$	80.6555			0.666575
$122$		0			0
$123$	72.9949			0.593455
$124$		0			0
$125$	−86.8386	−	89.9114i	−0.694709	−	0.719291i
$126$		0			0
$127$			162.061i			1.27607i	0.770008	+	0.638034i	$0.220252\pi$
$127$			162.061i			1.27607i	−0.770008	+	0.638034i	$0.779748\pi$
$128$		0			0
$129$		−	42.0227i		−	0.325757i
$130$		0			0
$131$	17.0555			0.130195			0.0650975	−	0.997879i	$-0.479264\pi$
$131$	17.0555			0.130195			0.0650975	+	0.997879i	$0.479264\pi$
$132$		0			0
$133$	10.8255	+	23.1642i	0.0813944	+	0.174167i
$134$		0			0
$135$	96.2287	−	25.1877i	0.712805	−	0.186575i
$136$		0			0
$137$		−	122.950i		−	0.897447i	−0.893671	−	0.448724i	$-0.851879\pi$
$137$		−	122.950i		−	0.897447i	0.893671	−	0.448724i	$-0.148121\pi$
$138$		0			0
$139$	233.445			1.67946			0.839730	−	0.543004i	$-0.182713\pi$
$139$	233.445			1.67946			0.839730	+	0.543004i	$0.182713\pi$
$140$		0			0
$141$		−	65.6803i		−	0.465818i
$142$		0			0
$143$	72.8199			0.509230
$144$		0			0
$145$	−67.6378	+	17.7041i	−0.466467	+	0.122097i
$146$		0			0
$147$	45.1679	+	37.7706i	0.307265	+	0.256943i
$148$		0			0
$149$		−	213.221i		−	1.43101i	−0.698607	−	0.715505i	$-0.746196\pi$
$149$		−	213.221i		−	1.43101i	0.698607	−	0.715505i	$-0.253804\pi$
$150$		0			0
$151$	−87.8507			−0.581793			−0.290896	−	0.956755i	$-0.593953\pi$
$151$	−87.8507			−0.581793			−0.290896	+	0.956755i	$0.593953\pi$
$152$		0			0
$153$	135.034			0.882572
$154$		0			0
$155$	150.893	−	39.4960i	0.973504	−	0.254813i
$156$		0			0
$157$			104.488i			0.665530i	0.943010	+	0.332765i	$0.107982\pi$
$157$			104.488i			0.665530i	−0.943010	+	0.332765i	$0.892018\pi$
$158$		0			0
$159$			60.3799i			0.379748i
$160$		0			0
$161$	−9.35262	+	4.37081i	−0.0580908	+	0.0271479i
$162$		0			0
$163$	−110.004			−0.674873			−0.337437	−	0.941348i	$-0.609560\pi$
$163$	−110.004			−0.674873			−0.337437	+	0.941348i	$0.609560\pi$
$164$		0			0
$165$	−9.66321	−	36.9180i	−0.0585649	−	0.223745i
$166$		0			0
$167$	−114.859			−0.687779			−0.343889	−	0.939010i	$-0.611745\pi$
$167$	−114.859			−0.687779			−0.343889	+	0.939010i	$0.611745\pi$
$168$		0			0
$169$	37.5635			0.222269
$170$		0			0
$171$	27.6003			0.161405
$172$		0			0
$173$			81.3690i			0.470341i	0.971954	+	0.235170i	$0.0755648\pi$
$173$			81.3690i			0.470341i	−0.971954	+	0.235170i	$0.924435\pi$
$174$		0			0
$175$	−142.265	−	101.910i	−0.812942	−	0.582345i
$176$		0			0
$177$			58.6807i			0.331529i
$178$		0			0
$179$		−	199.041i		−	1.11196i	−0.831196	−	0.555979i	$-0.812343\pi$
$179$		−	199.041i		−	1.11196i	0.831196	−	0.555979i	$-0.187657\pi$
$180$		0			0
$181$	−271.498			−1.49999			−0.749993	−	0.661445i	$-0.769943\pi$
$181$	−271.498			−1.49999			−0.749993	+	0.661445i	$0.769943\pi$
$182$		0			0
$183$			16.0113i			0.0874935i
$184$		0			0
$185$	−13.7679	−	52.5999i	−0.0744213	−	0.284324i
$186$		0			0
$187$		−	113.510i		−	0.607006i
$188$		0			0
$189$	126.161	−	58.9598i	0.667521	−	0.311956i
$190$		0			0
$191$	−152.273			−0.797240			−0.398620	−	0.917116i	$-0.630511\pi$
$191$	−152.273			−0.797240			−0.398620	+	0.917116i	$0.630511\pi$
$192$		0			0
$193$			178.881i			0.926843i	0.886138	+	0.463422i	$0.153378\pi$
$193$			178.881i			0.926843i	−0.886138	+	0.463422i	$0.846622\pi$
$194$		0			0
$195$	17.4417	+	66.6352i	0.0894444	+	0.341719i
$196$		0			0
$197$	271.801			1.37970			0.689849	−	0.723953i	$-0.257677\pi$
$197$	271.801			1.37970			0.689849	+	0.723953i	$0.257677\pi$
$198$		0			0
$199$			39.6863i			0.199429i	0.995016	+	0.0997144i	$0.0317929\pi$
$199$			39.6863i			0.199429i	−0.995016	+	0.0997144i	$0.968207\pi$
$200$		0			0
$201$			126.434i			0.629025i
$202$		0			0
$203$	−88.6771	+	41.4420i	−0.436833	+	0.204148i
$204$		0			0
$205$	−293.838	+	76.9114i	−1.43335	+	0.375178i
$206$		0			0
$207$			11.1437i			0.0538343i
$208$		0			0
$209$		−	23.2010i		−	0.111010i
$210$		0			0
$211$			415.917i			1.97117i	0.169182	+	0.985585i	$0.445887\pi$
$211$			415.917i			1.97117i	−0.169182	+	0.985585i	$0.554113\pi$
$212$		0			0
$213$			78.4571i			0.368343i
$214$		0			0
$215$	44.2774	+	169.160i	0.205942	+	0.786793i
$216$		0			0
$217$	197.830	−	92.4529i	0.911658	−	0.426050i
$218$		0			0
$219$		−	117.137i		−	0.534874i
$220$		0			0
$221$			204.881i			0.927062i
$222$		0			0
$223$	−80.3875			−0.360482			−0.180241	−	0.983622i	$-0.557688\pi$
$223$	−80.3875			−0.360482			−0.180241	+	0.983622i	$0.557688\pi$
$224$		0			0
$225$	−164.678	+	92.5493i	−0.731904	+	0.411330i
$226$		0			0
$227$		−	115.865i		−	0.510417i	−0.966886	−	0.255208i	$-0.917856\pi$
$227$		−	115.865i		−	0.510417i	0.966886	−	0.255208i	$-0.0821441\pi$
$228$		0			0
$229$	−368.810			−1.61052			−0.805262	−	0.592919i	$-0.797975\pi$
$229$	−368.810			−1.61052			−0.805262	+	0.592919i	$0.797975\pi$
$230$		0			0
$231$	−22.6198	−	48.4016i	−0.0979213	−	0.209531i
$232$		0			0
$233$		−	298.852i		−	1.28263i	−0.767280	−	0.641313i	$-0.778390\pi$
$233$		−	298.852i		−	1.28263i	0.767280	−	0.641313i	$-0.221610\pi$
$234$		0			0
$235$	69.2044	+	264.393i	0.294487	+	1.12508i
$236$		0			0
$237$		−	15.4649i		−	0.0652528i
$238$		0			0
$239$	52.4564			0.219483			0.109741	−	0.993960i	$-0.464998\pi$
$239$	52.4564			0.219483			0.109741	+	0.993960i	$0.464998\pi$
$240$		0			0
$241$		−	389.409i		−	1.61580i	−0.589317	−	0.807902i	$-0.700603\pi$
$241$		−	389.409i		−	1.61580i	0.589317	−	0.807902i	$-0.299397\pi$
$242$		0			0
$243$		−	232.038i		−	0.954889i
$244$		0			0
$245$	−221.618	−	104.452i	−0.904565	−	0.426336i
$246$		0			0
$247$			41.8767i			0.169541i
$248$		0			0
$249$	168.578			0.677020
$250$		0			0
$251$	−21.5558			−0.0858796			−0.0429398	−	0.999078i	$-0.513672\pi$
$251$	−21.5558			−0.0858796			−0.0429398	+	0.999078i	$0.513672\pi$
$252$		0			0
$253$	9.36747			0.0370256
$254$		0			0
$255$	103.870	−	27.1877i	0.407332	−	0.106618i
$256$		0			0
$257$	−287.660			−1.11930			−0.559649	−	0.828729i	$-0.689064\pi$
$257$	−287.660			−1.11930			−0.559649	+	0.828729i	$0.689064\pi$
$258$		0			0
$259$	−32.2282	−	68.9616i	−0.124433	−	0.266261i
$260$		0			0
$261$			105.659i			0.404825i
$262$		0			0
$263$		−	222.984i		−	0.847850i	−0.905697	−	0.423925i	$-0.860652\pi$
$263$		−	222.984i		−	0.847850i	0.905697	−	0.423925i	$-0.139348\pi$
$264$		0			0
$265$	−63.6196	−	243.056i	−0.240074	−	0.917194i
$266$		0			0
$267$	−30.6404			−0.114758
$268$		0			0
$269$	473.218			1.75917			0.879587	−	0.475738i	$-0.157819\pi$
$269$	473.218			1.75917			0.879587	+	0.475738i	$0.157819\pi$
$270$		0			0
$271$			289.845i			1.06954i	0.844998	+	0.534769i	$0.179601\pi$
$271$			289.845i			1.06954i	−0.844998	+	0.534769i	$0.820399\pi$
$272$		0			0
$273$	40.8277	+	87.3627i	0.149552	+	0.320010i
$274$		0			0
$275$	77.7976	+	138.430i	0.282900	+	0.503381i
$276$		0			0
$277$	−76.0683			−0.274615			−0.137307	−	0.990528i	$-0.543845\pi$
$277$	−76.0683			−0.274615			−0.137307	+	0.990528i	$0.543845\pi$
$278$		0			0
$279$		−	235.715i		−	0.844858i
$280$		0			0
$281$	−38.4073			−0.136681			−0.0683403	−	0.997662i	$-0.521770\pi$
$281$	−38.4073			−0.136681			−0.0683403	+	0.997662i	$0.521770\pi$
$282$		0			0
$283$			36.9983i			0.130736i	0.997861	+	0.0653680i	$0.0208221\pi$
$283$			36.9983i			0.130736i	−0.997861	+	0.0653680i	$0.979178\pi$
$284$		0			0
$285$	21.2305	−	5.55705i	0.0744930	−	0.0194984i
$286$		0			0
$287$	−385.238	+	180.036i	−1.34229	+	0.627302i
$288$		0			0
$289$	30.3640			0.105066
$290$		0			0
$291$		−	141.056i		−	0.484729i
$292$		0			0
$293$		−	444.003i		−	1.51537i	−0.652621	−	0.757684i	$-0.726331\pi$
$293$		−	444.003i		−	1.51537i	0.652621	−	0.757684i	$-0.273669\pi$
$294$		0			0
$295$	−61.8292	−	236.216i	−0.209591	−	0.800734i
$296$		0			0
$297$	−126.362			−0.425461
$298$		0			0
$299$	−16.9079			−0.0565480
$300$		0			0
$301$	103.645	+	221.779i	0.344337	+	0.736809i
$302$		0			0
$303$		−	100.021i		−	0.330103i
$304$		0			0
$305$	−16.8704	−	64.4528i	−0.0553128	−	0.211321i
$306$		0			0
$307$			218.485i			0.711676i	0.934548	+	0.355838i	$0.115805\pi$
$307$			218.485i			0.711676i	−0.934548	+	0.355838i	$0.884195\pi$
$308$		0			0
$309$		−	189.271i		−	0.612529i
$310$		0			0
$311$		−	527.773i		−	1.69702i	−0.529179	−	0.848510i	$-0.677500\pi$
$311$		−	527.773i		−	1.69702i	0.529179	−	0.848510i	$-0.322500\pi$
$312$		0			0
$313$	−405.291			−1.29486			−0.647429	−	0.762126i	$-0.724156\pi$
$313$	−405.291			−1.29486			−0.647429	+	0.762126i	$0.724156\pi$
$314$		0			0
$315$	−203.430	+	168.989i	−0.645811	+	0.536474i
$316$		0			0
$317$	548.657			1.73078			0.865390	−	0.501098i	$-0.167071\pi$
$317$	548.657			1.73078			0.865390	+	0.501098i	$0.167071\pi$
$318$		0			0
$319$	88.8180			0.278426
$320$		0			0
$321$			108.096i			0.336747i
$322$		0			0
$323$	65.2766			0.202095
$324$		0			0
$325$	−140.421	−	249.859i	−0.432065	−	0.768798i
$326$		0			0
$327$	−46.2620			−0.141474
$328$		0			0
$329$	161.995	+	346.635i	0.492386	+	1.05360i
$330$		0			0
$331$			77.0081i			0.232653i	0.993211	+	0.116326i	$0.0371119\pi$
$331$			77.0081i			0.232653i	−0.993211	+	0.116326i	$0.962888\pi$
$332$		0			0
$333$	−82.1682			−0.246751
$334$		0			0
$335$	−133.218	−	508.954i	−0.397665	−	1.51927i
$336$		0			0
$337$		−	532.210i		−	1.57926i	−0.613585	−	0.789629i	$-0.710273\pi$
$337$		−	532.210i		−	1.57926i	0.613585	−	0.789629i	$-0.289727\pi$
$338$		0			0
$339$	176.519			0.520704
$340$		0			0
$341$	−198.144			−0.581068
$342$		0			0
$343$	−331.536	−	87.9357i	−0.966578	−	0.256372i
$344$		0			0
$345$	2.24368	+	8.57189i	0.00650341	+	0.0248460i
$346$		0			0
$347$	−538.276			−1.55123			−0.775614	−	0.631207i	$-0.782560\pi$
$347$	−538.276			−1.55123			−0.775614	+	0.631207i	$0.782560\pi$
$348$		0			0
$349$	534.502			1.53153			0.765763	−	0.643123i	$-0.222362\pi$
$349$	534.502			1.53153			0.765763	+	0.643123i	$0.222362\pi$
$350$		0			0
$351$	228.077			0.649793
$352$		0			0
$353$	−547.043			−1.54970			−0.774849	−	0.632147i	$-0.782174\pi$
$353$	−547.043			−1.54970			−0.774849	+	0.632147i	$0.782174\pi$
$354$		0			0
$355$	−82.6667	−	315.825i	−0.232864	−	0.889649i
$356$		0			0
$357$	136.179	−	63.6415i	0.381455	−	0.178267i
$358$		0			0
$359$	522.975			1.45676			0.728378	−	0.685175i	$-0.240274\pi$
$359$	522.975			1.45676			0.728378	+	0.685175i	$0.240274\pi$
$360$		0			0
$361$	−347.658			−0.963041
$362$		0			0
$363$		−	96.9170i		−	0.266989i
$364$		0			0
$365$	123.422	+	471.531i	0.338143	+	1.29187i
$366$		0			0
$367$	−12.4454			−0.0339113			−0.0169556	−	0.999856i	$-0.505397\pi$
$367$	−12.4454			−0.0339113			−0.0169556	+	0.999856i	$0.505397\pi$
$368$		0			0
$369$			459.014i			1.24394i
$370$		0			0
$371$	−148.922	−	318.661i	−0.401407	−	0.858926i
$372$		0			0
$373$	94.6474			0.253746			0.126873	−	0.991919i	$-0.459506\pi$
$373$	94.6474			0.253746			0.126873	+	0.991919i	$0.459506\pi$
$374$		0			0
$375$	−108.039	+	104.347i	−0.288104	+	0.278258i
$376$		0			0
$377$	−160.312			−0.425232
$378$		0			0
$379$		−	585.226i		−	1.54413i	−0.635543	−	0.772066i	$-0.719224\pi$
$379$		−	585.226i		−	1.54413i	0.635543	−	0.772066i	$-0.280776\pi$
$380$		0			0
$381$	194.735			0.511115
$382$		0			0
$383$	−206.190			−0.538356			−0.269178	−	0.963090i	$-0.586752\pi$
$383$	−206.190			−0.538356			−0.269178	+	0.963090i	$0.586752\pi$
$384$		0			0
$385$	142.054	+	171.005i	0.368970	+	0.444169i
$386$		0			0
$387$	264.251			0.682820
$388$		0			0
$389$		−	377.902i		−	0.971470i	−0.874106	−	0.485735i	$-0.838552\pi$
$389$		−	377.902i		−	0.971470i	0.874106	−	0.485735i	$-0.161448\pi$
$390$		0			0
$391$			26.3556i			0.0674057i
$392$		0			0
$393$		−	20.4942i		−	0.0521481i
$394$		0			0
$395$	16.2947	+	62.2532i	0.0412523	+	0.157603i
$396$		0			0
$397$			262.453i			0.661090i	0.943790	+	0.330545i	$0.107233\pi$
$397$			262.453i			0.661090i	−0.943790	+	0.330545i	$0.892767\pi$
$398$		0			0
$399$	27.8345	−	13.0080i	0.0697605	−	0.0326016i
$400$		0			0
$401$	−148.618			−0.370618			−0.185309	−	0.982680i	$-0.559329\pi$
$401$	−148.618			−0.370618			−0.185309	+	0.982680i	$0.559329\pi$
$402$		0			0
$403$	357.641			0.887446
$404$		0			0
$405$	55.8345	+	213.314i	0.137863	+	0.526701i
$406$		0			0
$407$			69.0712i			0.169708i
$408$		0			0
$409$		−	453.526i		−	1.10887i	−0.832228	−	0.554433i	$-0.812935\pi$
$409$		−	453.526i		−	1.10887i	0.832228	−	0.554433i	$-0.187065\pi$
$410$		0			0
$411$	−147.739			−0.359462
$412$		0			0
$413$	−144.731	−	309.694i	−0.350438	−	0.749864i
$414$		0			0
$415$	−678.602	+	177.623i	−1.63519	+	0.428007i
$416$		0			0
$417$		−	280.511i		−	0.672689i
$418$		0			0
$419$	−2.02019			−0.00482145			−0.00241072	−	0.999997i	$-0.500767\pi$
$419$	−2.02019			−0.00482145			−0.00241072	+	0.999997i	$0.500767\pi$
$420$		0			0
$421$		−	357.993i		−	0.850340i	−0.905114	−	0.425170i	$-0.860214\pi$
$421$		−	357.993i		−	0.850340i	0.905114	−	0.425170i	$-0.139786\pi$
$422$		0			0
$423$	413.018			0.976401
$424$		0			0
$425$	−389.476	+	218.886i	−0.916414	+	0.515025i
$426$		0			0
$427$	−39.4905	−	84.5014i	−0.0924837	−	0.197896i
$428$		0			0
$429$		−	87.5015i		−	0.203966i
$430$		0			0
$431$	446.016			1.03484			0.517420	−	0.855732i	$-0.326893\pi$
$431$	446.016			1.03484			0.517420	+	0.855732i	$0.326893\pi$
$432$		0			0
$433$	315.080			0.727667			0.363833	−	0.931464i	$-0.381468\pi$
$433$	315.080			0.727667			0.363833	+	0.931464i	$0.381468\pi$
$434$		0			0
$435$	21.2735	+	81.2746i	0.0489046	+	0.186838i
$436$		0			0
$437$			5.38698i			0.0123272i
$438$		0			0
$439$			526.489i			1.19929i	0.800265	+	0.599646i	$0.204692\pi$
$439$			526.489i			1.19929i	−0.800265	+	0.599646i	$0.795308\pi$
$440$		0			0
$441$	−237.513	+	284.029i	−0.538578	+	0.644057i
$442$		0			0
$443$	504.255			1.13827			0.569136	−	0.822243i	$-0.307278\pi$
$443$	504.255			1.13827			0.569136	+	0.822243i	$0.307278\pi$
$444$		0			0
$445$	123.341	−	32.2844i	0.277172	−	0.0725492i
$446$		0			0
$447$	−256.209			−0.573175
$448$		0			0
$449$	363.198			0.808905			0.404452	−	0.914559i	$-0.367462\pi$
$449$	363.198			0.808905			0.404452	+	0.914559i	$0.367462\pi$
$450$		0			0
$451$	385.850			0.855544
$452$		0			0
$453$			105.563i			0.233030i
$454$		0			0
$455$	−256.400	−	308.656i	−0.563517	−	0.678365i
$456$		0			0
$457$			760.978i			1.66516i	0.553904	+	0.832580i	$0.313137\pi$
$457$			760.978i			1.66516i	−0.553904	+	0.832580i	$0.686863\pi$
$458$		0			0
$459$		−	355.523i		−	0.774559i
$460$		0			0
$461$	430.994			0.934912			0.467456	−	0.884016i	$-0.345171\pi$
$461$	430.994			0.934912			0.467456	+	0.884016i	$0.345171\pi$
$462$		0			0
$463$		−	84.2589i		−	0.181985i	−0.995852	−	0.0909923i	$-0.970996\pi$
$463$		−	84.2589i		−	0.181985i	0.995852	−	0.0909923i	$-0.0290039\pi$
$464$		0			0
$465$	−47.4590	−	181.316i	−0.102062	−	0.389926i
$466$		0			0
$467$		−	729.350i		−	1.56178i	−0.624670	−	0.780889i	$-0.714766\pi$
$467$		−	729.350i		−	1.56178i	0.624670	−	0.780889i	$-0.285234\pi$
$468$		0			0
$469$	−311.839	−	667.269i	−0.664901	−	1.42275i
$470$		0			0
$471$	125.555			0.266570
$472$		0			0
$473$		−	222.132i		−	0.469623i
$474$		0			0
$475$	−79.6072	+	44.7393i	−0.167594	+	0.0941879i
$476$		0			0
$477$	−379.687			−0.795990
$478$		0			0
$479$		−	878.089i		−	1.83317i	−0.399840	−	0.916585i	$-0.630934\pi$
$479$		−	878.089i		−	1.83317i	0.399840	−	0.916585i	$-0.369066\pi$
$480$		0			0
$481$		−	124.670i		−	0.259190i
$482$		0			0
$483$	5.25204	+	11.2383i	0.0108738	+	0.0232676i
$484$		0			0
$485$	148.625	+	567.815i	0.306443	+	1.17075i
$486$		0			0
$487$		−	220.513i		−	0.452799i	−0.974035	−	0.226399i	$-0.927305\pi$
$487$		−	220.513i		−	0.452799i	0.974035	−	0.226399i	$-0.0726954\pi$
$488$		0			0
$489$			132.183i			0.270313i
$490$		0			0
$491$			143.245i			0.291741i	0.989304	+	0.145871i	$0.0465983\pi$
$491$			143.245i			0.291741i	−0.989304	+	0.145871i	$0.953402\pi$
$492$		0			0
$493$			249.892i			0.506880i
$494$		0			0
$495$	232.151	−	60.7652i	0.468992	−	0.122758i
$496$		0			0
$497$	−193.508	−	414.066i	−0.389352	−	0.833131i
$498$		0			0
$499$		−	302.198i		−	0.605607i	−0.953053	−	0.302804i	$-0.902077\pi$
$499$		−	302.198i		−	0.605607i	0.953053	−	0.302804i	$-0.0979227\pi$
$500$		0			0
$501$			138.016i			0.275482i
$502$		0			0
$503$	−635.540			−1.26350			−0.631750	−	0.775172i	$-0.717663\pi$
$503$	−635.540			−1.26350			−0.631750	+	0.775172i	$0.717663\pi$
$504$		0			0
$505$	105.388	+	402.631i	0.208689	+	0.797289i
$506$		0			0
$507$		−	45.1369i		−	0.0890274i
$508$		0			0
$509$	616.524			1.21125			0.605623	−	0.795752i	$-0.292924\pi$
$509$	616.524			1.21125			0.605623	+	0.795752i	$0.292924\pi$
$510$		0			0
$511$	288.909	+	618.205i	0.565380	+	1.20979i
$512$		0			0
$513$		−	72.6672i		−	0.141652i
$514$		0			0
$515$	199.427	+	761.903i	0.387236	+	1.47942i
$516$		0			0
$517$		−	347.186i		−	0.671539i
$518$		0			0
$519$	97.7742			0.188390
$520$		0			0
$521$		−	754.179i		−	1.44756i	−0.690030	−	0.723780i	$-0.742403\pi$
$521$		−	754.179i		−	1.44756i	0.690030	−	0.723780i	$-0.257597\pi$
$522$		0			0
$523$			245.714i			0.469816i	0.972018	+	0.234908i	$0.0754789\pi$
$523$			245.714i			0.469816i	−0.972018	+	0.234908i	$0.924521\pi$
$524$		0			0
$525$	−122.457	+	170.948i	−0.233252	+	0.325615i
$526$		0			0
$527$		−	557.484i		−	1.05784i
$528$		0			0
$529$	526.825			0.995888
$530$		0			0
$531$	−369.002			−0.694919
$532$		0			0
$533$	−696.442			−1.30665
$534$		0			0
$535$	−113.896	−	435.134i	−0.212889	−	0.813335i
$536$		0			0
$537$	−239.170			−0.445382
$538$		0			0
$539$	238.757	+	199.655i	0.442963	+	0.370417i
$540$		0			0
$541$			347.839i			0.642955i	0.946917	+	0.321478i	$0.104180\pi$
$541$			347.839i			0.642955i	−0.946917	+	0.321478i	$0.895820\pi$
$542$		0			0
$543$			326.236i			0.600803i
$544$		0			0
$545$	186.226	−	48.7442i	0.341698	−	0.0894389i
$546$		0			0
$547$	−88.7131			−0.162181			−0.0810906	−	0.996707i	$-0.525840\pi$
$547$	−88.7131			−0.162181			−0.0810906	+	0.996707i	$0.525840\pi$
$548$		0			0
$549$	−100.684			−0.183395
$550$		0			0
$551$			51.0768i			0.0926983i
$552$		0			0
$553$	38.1429	+	81.6177i	0.0689744	+	0.147591i
$554$		0			0
$555$	−63.2049	+	16.5438i	−0.113883	+	0.0298086i
$556$		0			0
$557$	−244.783			−0.439467			−0.219733	−	0.975560i	$-0.570519\pi$
$557$	−244.783			−0.439467			−0.219733	+	0.975560i	$0.570519\pi$
$558$		0			0
$559$			400.937i			0.717240i
$560$		0			0
$561$	−136.396			−0.243129
$562$		0			0
$563$		−	330.152i		−	0.586416i	−0.956049	−	0.293208i	$-0.905277\pi$
$563$		−	330.152i		−	0.586416i	0.956049	−	0.293208i	$-0.0947228\pi$
$564$		0			0
$565$	−710.568	+	185.990i	−1.25764	+	0.329186i
$566$		0			0
$567$	130.698	+	279.667i	0.230509	+	0.493240i
$568$		0			0
$569$	−65.2889			−0.114743			−0.0573716	−	0.998353i	$-0.518272\pi$
$569$	−65.2889			−0.114743			−0.0573716	+	0.998353i	$0.518272\pi$
$570$		0			0
$571$			408.808i			0.715950i	0.933731	+	0.357975i	$0.116533\pi$
$571$			408.808i			0.715950i	−0.933731	+	0.357975i	$0.883467\pi$
$572$		0			0
$573$			182.973i			0.319325i
$574$		0			0
$575$	−18.0636	−	32.1417i	−0.0314150	−	0.0558985i
$576$		0			0
$577$	848.550			1.47062			0.735312	−	0.677729i	$-0.237036\pi$
$577$	848.550			1.47062			0.735312	+	0.677729i	$0.237036\pi$
$578$		0			0
$579$	214.946			0.371236
$580$		0			0
$581$	−889.688	+	415.783i	−1.53130	+	0.715633i
$582$		0			0
$583$			319.168i			0.547457i
$584$		0			0
$585$	−419.022	+	109.678i	−0.716277	+	0.187484i
$586$		0			0
$587$		−	224.450i		−	0.382369i	−0.981554	−	0.191184i	$-0.938767\pi$
$587$		−	224.450i		−	0.382369i	0.981554	−	0.191184i	$-0.0612328\pi$
$588$		0			0
$589$		−	113.947i		−	0.193459i
$590$		0			0
$591$		−	326.600i		−	0.552622i
$592$		0			0
$593$	−278.865			−0.470261			−0.235130	−	0.971964i	$-0.575552\pi$
$593$	−278.865			−0.470261			−0.235130	+	0.971964i	$0.575552\pi$
$594$		0			0
$595$	−481.127	+	399.672i	−0.808617	+	0.671717i
$596$		0			0
$597$	47.6877			0.0798789
$598$		0			0
$599$	−434.423			−0.725247			−0.362623	−	0.931936i	$-0.618119\pi$
$599$	−434.423			−0.725247			−0.362623	+	0.931936i	$0.618119\pi$
$600$		0			0
$601$		−	324.120i		−	0.539301i	−0.962958	−	0.269651i	$-0.913092\pi$
$601$		−	324.120i		−	0.539301i	0.962958	−	0.269651i	$-0.0869082\pi$
$602$		0			0
$603$	−795.055			−1.31850
$604$		0			0
$605$	102.117	+	390.135i	0.168788	+	0.644851i
$606$		0			0
$607$	674.884			1.11184			0.555918	−	0.831237i	$-0.312367\pi$
$607$	674.884			1.11184			0.555918	+	0.831237i	$0.312367\pi$
$608$		0			0
$609$	49.7973	+	106.556i	0.0817690	+	0.174968i
$610$		0			0
$611$			626.654i			1.02562i
$612$		0			0
$613$	1087.67			1.77434			0.887172	−	0.461440i	$-0.152667\pi$
$613$	1087.67			1.77434			0.887172	+	0.461440i	$0.152667\pi$
$614$		0			0
$615$	92.4180	+	353.080i	0.150273	+	0.574113i
$616$		0			0
$617$		−	124.521i		−	0.201817i	−0.994896	−	0.100909i	$-0.967825\pi$
$617$		−	124.521i		−	0.201817i	0.994896	−	0.100909i	$-0.0321750\pi$
$618$		0			0
$619$	1059.42			1.71150			0.855752	−	0.517387i	$-0.173095\pi$
$619$	1059.42			1.71150			0.855752	+	0.517387i	$0.173095\pi$
$620$		0			0
$621$	29.3396			0.0472458
$622$		0			0
$623$	161.708	−	75.5719i	0.259563	−	0.121303i
$624$		0			0
$625$	324.960	−	533.878i	0.519937	−	0.854205i
$626$		0			0
$627$	−27.8787			−0.0444636
$628$		0			0
$629$	−194.334			−0.308956
$630$		0			0
$631$	−893.784			−1.41646			−0.708228	−	0.705983i	$-0.750505\pi$
$631$	−893.784			−1.41646			−0.708228	+	0.705983i	$0.750505\pi$
$632$		0			0
$633$	499.772			0.789529
$634$		0			0
$635$	−783.895	+	205.183i	−1.23448	+	0.323123i
$636$		0			0
$637$	−430.945	−	360.368i	−0.676523	−	0.565727i
$638$		0			0
$639$	−493.362			−0.772084
$640$		0			0
$641$	−395.495			−0.616997			−0.308499	−	0.951225i	$-0.599827\pi$
$641$	−395.495			−0.616997			−0.308499	+	0.951225i	$0.599827\pi$
$642$		0			0
$643$			733.874i			1.14133i	0.821183	+	0.570664i	$0.193314\pi$
$643$			733.874i			1.14133i	−0.821183	+	0.570664i	$0.806686\pi$
$644$		0			0
$645$	203.266	−	53.2045i	0.315141	−	0.0824875i
$646$		0			0
$647$	196.080			0.303060			0.151530	−	0.988453i	$-0.451580\pi$
$647$	196.080			0.303060			0.151530	+	0.988453i	$0.451580\pi$
$648$		0			0
$649$			310.186i			0.477944i
$650$		0			0
$651$	−111.093	−	237.715i	−0.170650	−	0.365154i
$652$		0			0
$653$	−183.968			−0.281728			−0.140864	−	0.990029i	$-0.544988\pi$
$653$	−183.968			−0.281728			−0.140864	+	0.990029i	$0.544988\pi$
$654$		0			0
$655$	21.5938	+	82.4985i	0.0329677	+	0.125952i
$656$		0			0
$657$	736.595			1.12115
$658$		0			0
$659$		−	846.874i		−	1.28509i	−0.766248	−	0.642544i	$-0.777879\pi$
$659$		−	846.874i		−	1.28509i	0.766248	−	0.642544i	$-0.222121\pi$
$660$		0			0
$661$	−498.387			−0.753990			−0.376995	−	0.926215i	$-0.623043\pi$
$661$	−498.387			−0.753990			−0.376995	+	0.926215i	$0.623043\pi$
$662$		0			0
$663$	246.188			0.371324
$664$		0			0
$665$	−98.3403	+	81.6911i	−0.147880	+	0.122844i
$666$		0			0
$667$	−20.6224			−0.0309182
$668$		0			0
$669$			96.5948i			0.144387i
$670$		0			0
$671$			84.6357i			0.126134i
$672$		0			0
$673$			26.2761i			0.0390432i	0.999809	+	0.0195216i	$0.00621431\pi$
$673$			26.2761i			0.0390432i	−0.999809	+	0.0195216i	$0.993786\pi$
$674$		0			0
$675$	243.668	+	433.573i	0.360990	+	0.642330i
$676$		0			0
$677$		−	273.868i		−	0.404532i	−0.979331	−	0.202266i	$-0.935169\pi$
$677$		−	273.868i		−	0.404532i	0.979331	−	0.202266i	$-0.0648305\pi$
$678$		0			0
$679$	347.903	+	744.439i	0.512376	+	1.09638i
$680$		0			0
$681$	−139.225			−0.204442
$682$		0			0
$683$	163.569			0.239486			0.119743	−	0.992805i	$-0.461793\pi$
$683$	163.569			0.239486			0.119743	+	0.992805i	$0.461793\pi$
$684$		0			0
$685$	594.716	−	155.666i	0.868199	−	0.227250i
$686$		0			0
$687$			443.168i			0.645077i
$688$		0			0
$689$		−	576.083i		−	0.836114i
$690$		0			0
$691$	−217.130			−0.314226			−0.157113	−	0.987581i	$-0.550219\pi$
$691$	−217.130			−0.314226			−0.157113	+	0.987581i	$0.550219\pi$
$692$		0			0
$693$	304.364	−	142.240i	0.439198	−	0.205253i
$694$		0			0
$695$	295.562	+	1129.18i	0.425269	+	1.62473i
$696$		0			0
$697$			1085.60i			1.55753i
$698$		0			0
$699$	−359.105			−0.513741
$700$		0			0
$701$			187.844i			0.267966i	0.990984	+	0.133983i	$0.0427767\pi$
$701$			187.844i			0.267966i	−0.990984	+	0.133983i	$0.957223\pi$
$702$		0			0
$703$	−39.7209			−0.0565020
$704$		0			0
$705$	317.699	−	83.1571i	0.450637	−	0.117953i
$706$		0			0
$707$	246.694	+	527.873i	0.348931	+	0.746638i
$708$		0			0
$709$		−	1273.19i		−	1.79575i	−0.440252	−	0.897874i	$-0.645111\pi$
$709$		−	1273.19i		−	1.79575i	0.440252	−	0.897874i	$-0.354889\pi$
$710$		0			0
$711$	97.2480			0.136776
$712$		0			0
$713$	46.0065			0.0645253
$714$		0			0
$715$	92.1964	+	352.233i	0.128946	+	0.492634i
$716$		0			0
$717$		−	63.0324i		−	0.0879113i
$718$		0			0
$719$			1161.36i			1.61524i	0.589704	+	0.807620i	$0.299245\pi$
$719$			1161.36i			1.61524i	−0.589704	+	0.807620i	$0.700755\pi$
$720$		0			0
$721$	466.821	+	998.900i	0.647464	+	1.38544i
$722$		0			0
$723$	−467.920			−0.647192
$724$		0			0
$725$	−171.271	−	304.752i	−0.236236	−	0.420348i
$726$		0			0
$727$	952.100			1.30963			0.654815	−	0.755790i	$-0.272747\pi$
$727$	952.100			1.30963			0.654815	+	0.755790i	$0.272747\pi$
$728$		0			0
$729$	118.080			0.161975
$730$		0			0
$731$	624.973			0.854957
$732$		0			0
$733$		−	67.7576i		−	0.0924387i	−0.998931	−	0.0462194i	$-0.985283\pi$
$733$		−	67.7576i		−	0.0924387i	0.998931	−	0.0462194i	$-0.0147173\pi$
$734$		0			0
$735$	−125.512	+	266.300i	−0.170764	+	0.362313i
$736$		0			0
$737$			668.329i			0.906823i
$738$		0			0
$739$		−	866.575i		−	1.17263i	−0.810082	−	0.586316i	$-0.800578\pi$
$739$		−	866.575i		−	1.17263i	0.810082	−	0.586316i	$-0.199422\pi$
$740$		0			0
$741$	50.3197			0.0679078
$742$		0			0
$743$		−	977.257i		−	1.31529i	−0.753330	−	0.657643i	$-0.771554\pi$
$743$		−	977.257i		−	1.31529i	0.753330	−	0.657643i	$-0.228446\pi$
$744$		0			0
$745$	1031.36	−	269.956i	1.38437	−	0.362357i
$746$		0			0
$747$			1060.07i			1.41910i
$748$		0			0
$749$	−266.609	−	570.487i	−0.355953	−	0.761665i
$750$		0			0
$751$	−310.507			−0.413459			−0.206729	−	0.978398i	$-0.566282\pi$
$751$	−310.507			−0.413459			−0.206729	+	0.978398i	$0.566282\pi$
$752$		0			0
$753$			25.9018i			0.0343981i
$754$		0			0
$755$	−111.227	−	424.938i	−0.147320	−	0.562832i
$756$		0			0
$757$	−309.239			−0.408506			−0.204253	−	0.978918i	$-0.565477\pi$
$757$	−309.239			−0.408506			−0.204253	+	0.978918i	$0.565477\pi$
$758$		0			0
$759$		−	11.2561i		−	0.0148302i
$760$		0			0
$761$			83.9096i			0.110262i	0.998479	+	0.0551312i	$0.0175577\pi$
$761$			83.9096i			0.110262i	−0.998479	+	0.0551312i	$0.982442\pi$
$762$		0			0
$763$	244.153	−	114.101i	0.319991	−	0.149543i
$764$		0			0
$765$	170.964	+	653.164i	0.223483	+	0.853809i
$766$		0			0
$767$		−	559.871i		−	0.729949i
$768$		0			0
$769$		−	676.111i		−	0.879208i	−0.898192	−	0.439604i	$-0.855119\pi$
$769$		−	676.111i		−	0.879208i	0.898192	−	0.439604i	$-0.144881\pi$
$770$		0			0
$771$			345.657i			0.448322i
$772$		0			0
$773$		−	105.163i		−	0.136046i	−0.997684	−	0.0680229i	$-0.978331\pi$
$773$		−	105.163i		−	0.136046i	0.997684	−	0.0680229i	$-0.0216691\pi$
$774$		0			0
$775$	382.088	+	679.872i	0.493017	+	0.877254i
$776$		0			0
$777$	−82.8654	+	38.7259i	−0.106648	+	0.0498403i
$778$		0			0
$779$			221.892i			0.284842i
$780$		0			0
$781$			414.724i			0.531016i
$782$		0			0
$783$	278.185			0.355280
$784$		0			0
$785$	−505.414	+	132.291i	−0.643840	+	0.168524i
$786$		0			0
$787$			1346.17i			1.71051i	0.518211	+	0.855253i	$0.326598\pi$
$787$			1346.17i			1.71051i	−0.518211	+	0.855253i	$0.673402\pi$
$788$		0			0
$789$	−267.942			−0.339596
$790$		0			0
$791$	−931.597	+	435.368i	−1.17775	+	0.550402i
$792$		0			0
$793$		−	152.763i		−	0.192640i
$794$		0			0
$795$	−292.060	+	76.4463i	−0.367372	+	0.0961589i
$796$		0			0
$797$			1053.41i			1.32172i	0.750509	+	0.660860i	$0.229809\pi$
$797$			1053.41i			1.32172i	−0.750509	+	0.660860i	$0.770191\pi$
$798$		0			0
$799$	976.816			1.22255
$800$		0			0
$801$		−	192.676i		−	0.240544i
$802$		0			0
$803$		−	619.187i		−	0.771092i
$804$		0			0
$805$	−32.9831	−	39.7052i	−0.0409727	−	0.0493232i
$806$		0			0
$807$		−	568.626i		−	0.704617i
$808$		0			0
$809$	784.349			0.969529			0.484764	−	0.874645i	$-0.338905\pi$
$809$	784.349			0.969529			0.484764	+	0.874645i	$0.338905\pi$
$810$		0			0
$811$	−309.590			−0.381738			−0.190869	−	0.981615i	$-0.561131\pi$
$811$	−309.590			−0.381738			−0.190869	+	0.981615i	$0.561131\pi$
$812$		0			0
$813$	348.282			0.428391
$814$		0			0
$815$	−139.275	−	532.096i	−0.170890	−	0.652878i
$816$		0			0
$817$	127.742			0.156355
$818$		0			0
$819$	−549.363	+	256.737i	−0.670773	+	0.313476i
$820$		0			0
$821$		−	567.809i		−	0.691607i	−0.938307	−	0.345804i	$-0.887606\pi$
$821$		−	567.809i		−	0.691607i	0.938307	−	0.345804i	$-0.112394\pi$
$822$		0			0
$823$			1545.86i			1.87833i	0.343471	+	0.939163i	$0.388397\pi$
$823$			1545.86i			1.87833i	−0.343471	+	0.939163i	$0.611603\pi$
$824$		0			0
$825$	166.339	−	93.4828i	0.201624	−	0.113313i
$826$		0			0
$827$	706.209			0.853940			0.426970	−	0.904266i	$-0.359581\pi$
$827$	706.209			0.853940			0.426970	+	0.904266i	$0.359581\pi$
$828$		0			0
$829$	798.772			0.963537			0.481769	−	0.876298i	$-0.339995\pi$
$829$	798.772			0.963537			0.481769	+	0.876298i	$0.339995\pi$
$830$		0			0
$831$			91.4048i			0.109994i
$832$		0			0
$833$	−561.735	+	671.749i	−0.674351	+	0.806422i
$834$		0			0
$835$	−145.422	−	555.579i	−0.174158	−	0.665364i
$836$		0			0
$837$	−620.602			−0.741460
$838$		0			0
$839$		−	809.432i		−	0.964759i	−0.875962	−	0.482379i	$-0.839773\pi$
$839$		−	809.432i		−	0.964759i	0.875962	−	0.482379i	$-0.160227\pi$
$840$		0			0
$841$	645.468			0.767501
$842$		0			0
$843$			46.1508i			0.0547459i
$844$		0			0
$845$	47.5587	+	181.696i	0.0562825	+	0.215025i
$846$		0			0
$847$	239.037	+	511.490i	0.282216	+	0.603884i
$848$		0			0
$849$	44.4577			0.0523648
$850$		0			0
$851$		−	16.0375i		−	0.0188454i
$852$		0			0
$853$			661.922i			0.775992i	0.921661	+	0.387996i	$0.126833\pi$
$853$			661.922i			0.775992i	−0.921661	+	0.387996i	$0.873167\pi$
$854$		0			0
$855$	34.9444	+	133.504i	0.0408706	+	0.156145i
$856$		0			0
$857$	−293.366			−0.342317			−0.171159	−	0.985243i	$-0.554751\pi$
$857$	−293.366			−0.342317			−0.171159	+	0.985243i	$0.554751\pi$
$858$		0			0
$859$	187.895			0.218737			0.109369	−	0.994001i	$-0.465117\pi$
$859$	187.895			0.218737			0.109369	+	0.994001i	$0.465117\pi$
$860$		0			0
$861$	216.334	+	462.909i	0.251259	+	0.537641i
$862$		0			0
$863$		−	165.554i		−	0.191835i	−0.995389	−	0.0959176i	$-0.969421\pi$
$863$		−	165.554i		−	0.191835i	0.995389	−	0.0959176i	$-0.0305785\pi$
$864$		0			0
$865$	−393.586	+	103.020i	−0.455012	+	0.119099i
$866$		0			0
$867$		−	36.4858i		−	0.0420829i
$868$		0			0
$869$		−	81.7473i		−	0.0940706i
$870$		0			0
$871$		−	1206.30i		−	1.38496i
$872$		0			0
$873$	887.004			1.01604
$874$		0			0
$875$	312.825	−	817.169i	0.357515	−	0.933908i
$876$		0			0
$877$	−742.892			−0.847084			−0.423542	−	0.905877i	$-0.639213\pi$
$877$	−742.892			−0.847084			−0.423542	+	0.905877i	$0.639213\pi$
$878$		0			0
$879$	−533.521			−0.606964
$880$		0			0
$881$			190.302i			0.216007i	0.994151	+	0.108003i	$0.0344457\pi$
$881$			190.302i			0.216007i	−0.994151	+	0.108003i	$0.965554\pi$
$882$		0			0
$883$	−976.283			−1.10564			−0.552821	−	0.833300i	$-0.686449\pi$
$883$	−976.283			−1.10564			−0.552821	+	0.833300i	$0.686449\pi$
$884$		0			0
$885$	−283.841	+	74.2950i	−0.320725	+	0.0839491i
$886$		0			0
$887$	−1136.32			−1.28108			−0.640542	−	0.767923i	$-0.721290\pi$
$887$	−1136.32			−1.28108			−0.640542	+	0.767923i	$0.721290\pi$
$888$		0			0
$889$	−1027.73	+	480.296i	−1.15606	+	0.540266i
$890$		0			0
$891$		−	280.111i		−	0.314379i
$892$		0			0
$893$	199.657			0.223580
$894$		0			0
$895$	962.769	−	252.003i	1.07572	−	0.281568i
$896$		0			0
$897$			20.3167i			0.0226497i
$898$		0			0
$899$	436.212			0.485220
$900$		0			0
$901$	−897.987			−0.996656
$902$		0			0
$903$	266.494	−	124.542i	0.295120	−	0.137920i
$904$		0			0
$905$	−343.740	−	1313.25i	−0.379823	−	1.45110i
$906$		0			0
$907$	1526.57			1.68310			0.841549	−	0.540181i	$-0.181644\pi$
$907$	1526.57			1.68310			0.841549	+	0.540181i	$0.181644\pi$
$908$		0			0
$909$	628.964			0.691929
$910$		0			0
$911$	479.174			0.525986			0.262993	−	0.964798i	$-0.415290\pi$
$911$	479.174			0.525986			0.262993	+	0.964798i	$0.415290\pi$
$912$		0			0
$913$	891.101			0.976014
$914$		0			0
$915$	−77.4475	+	20.2717i	−0.0846421	+	0.0221549i
$916$		0			0
$917$	50.5472	+	108.160i	0.0551224	+	0.117950i
$918$		0			0
$919$	1645.66			1.79071			0.895354	−	0.445355i	$-0.146923\pi$
$919$	1645.66			1.79071			0.895354	+	0.445355i	$0.146923\pi$
$920$		0			0
$921$	262.535			0.285054
$922$		0			0
$923$		−	748.557i		−	0.811004i
$924$		0			0
$925$	236.997	−	133.192i	0.256213	−	0.143992i
$926$		0			0
$927$	1190.19			1.28392
$928$		0			0
$929$			1749.94i			1.88368i	0.336061	+	0.941840i	$0.390905\pi$
$929$			1749.94i			1.88368i	−0.336061	+	0.941840i	$0.609095\pi$
$930$		0			0
$931$	−114.816	+	137.303i	−0.123326	+	0.147479i
$932$		0			0
$933$	−634.181			−0.679722
$934$		0			0
$935$	549.054	−	143.714i	0.587224	−	0.153705i
$936$		0			0
$937$	−571.445			−0.609867			−0.304934	−	0.952374i	$-0.598634\pi$
$937$	−571.445			−0.609867			−0.304934	+	0.952374i	$0.598634\pi$
$938$		0			0
$939$			487.003i			0.518641i
$940$		0			0
$941$	1275.74			1.35572			0.677862	−	0.735189i	$-0.262907\pi$
$941$	1275.74			1.35572			0.677862	+	0.735189i	$0.262907\pi$
$942$		0			0
$943$	−89.5896			−0.0950049
$944$		0			0
$945$	444.923	+	535.601i	0.470818	+	0.566773i
$946$		0			0
$947$	−107.958			−0.114000			−0.0570001	−	0.998374i	$-0.518154\pi$
$947$	−107.958			−0.114000			−0.0570001	+	0.998374i	$0.518154\pi$
$948$		0			0
$949$			1117.60i			1.17766i
$950$		0			0
$951$		−	659.275i		−	0.693244i
$952$		0			0
$953$		−	582.319i		−	0.611037i	−0.952186	−	0.305519i	$-0.901170\pi$
$953$		−	582.319i		−	0.611037i	0.952186	−	0.305519i	$-0.0988299\pi$
$954$		0			0
$955$	−192.791	−	736.551i	−0.201875	−	0.771257i
$956$		0			0
$957$		−	106.725i		−	0.111520i
$958$		0			0
$959$	779.708	−	364.386i	0.813043	−	0.379964i
$960$		0			0
$961$	−12.1464			−0.0126393
$962$		0			0
$963$	−679.738			−0.705855
$964$		0			0
$965$	−865.254	+	226.479i	−0.896637	+	0.234693i
$966$		0			0
$967$			856.298i			0.885520i	0.896640	+	0.442760i	$0.146001\pi$
$967$			856.298i			0.885520i	−0.896640	+	0.442760i	$0.853999\pi$
$968$		0			0
$969$		−	78.4374i		−	0.0809467i
$970$		0			0
$971$	−888.179			−0.914706			−0.457353	−	0.889285i	$-0.651202\pi$
$971$	−888.179			−0.914706			−0.457353	+	0.889285i	$0.651202\pi$
$972$		0			0
$973$	691.857	+	1480.43i	0.711055	+	1.52151i
$974$		0			0
$975$	−300.235	+	168.732i	−0.307933	+	0.173059i
$976$		0			0
$977$			195.521i			0.200124i	0.994981	+	0.100062i	$0.0319041\pi$
$977$			195.521i			0.200124i	−0.994981	+	0.100062i	$0.968096\pi$
$978$		0			0
$979$	−161.965			−0.165439
$980$		0			0
$981$		−	290.909i		−	0.296544i
$982$		0			0
$983$	281.780			0.286653			0.143327	−	0.989675i	$-0.454220\pi$
$983$	281.780			0.286653			0.143327	+	0.989675i	$0.454220\pi$
$984$		0			0
$985$	344.123	+	1314.71i	0.349364	+	1.33473i
$986$		0			0
$987$	416.522	−	194.656i	0.422008	−	0.197220i
$988$		0			0
$989$			51.5762i			0.0521498i
$990$		0			0
$991$	−1196.50			−1.20737			−0.603685	−	0.797223i	$-0.706302\pi$
$991$	−1196.50			−1.20737			−0.603685	+	0.797223i	$0.706302\pi$
$992$		0			0
$993$	92.5342			0.0931865
$994$		0			0
$995$	−191.965	+	50.2464i	−0.192929	+	0.0504989i
$996$		0			0
$997$			209.026i			0.209655i	0.994490	+	0.104828i	$0.0334291\pi$
$997$			209.026i			0.209655i	−0.994490	+	0.104828i	$0.966571\pi$
$998$		0			0
$999$			216.336i			0.216553i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.3.c.g.209.53		80
4.3	odd	2		280.3.c.g.69.12	yes	80
5.4	even	2	inner	1120.3.c.g.209.32		80
7.6	odd	2	inner	1120.3.c.g.209.46		80
8.3	odd	2		280.3.c.g.69.71	yes	80
8.5	even	2	inner	1120.3.c.g.209.48		80
20.19	odd	2		280.3.c.g.69.69	yes	80
28.27	even	2		280.3.c.g.69.11	yes	80
35.34	odd	2	inner	1120.3.c.g.209.47		80
40.19	odd	2		280.3.c.g.69.10	yes	80
40.29	even	2	inner	1120.3.c.g.209.45		80
56.13	odd	2	inner	1120.3.c.g.209.31		80
56.27	even	2		280.3.c.g.69.72	yes	80
140.139	even	2		280.3.c.g.69.70	yes	80
280.69	odd	2	inner	1120.3.c.g.209.54		80
280.139	even	2		280.3.c.g.69.9	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.3.c.g.69.9	✓	80	280.139	even	2
280.3.c.g.69.10	yes	80	40.19	odd	2
280.3.c.g.69.11	yes	80	28.27	even	2
280.3.c.g.69.12	yes	80	4.3	odd	2
280.3.c.g.69.69	yes	80	20.19	odd	2
280.3.c.g.69.70	yes	80	140.139	even	2
280.3.c.g.69.71	yes	80	8.3	odd	2
280.3.c.g.69.72	yes	80	56.27	even	2
1120.3.c.g.209.31		80	56.13	odd	2	inner
1120.3.c.g.209.32		80	5.4	even	2	inner
1120.3.c.g.209.45		80	40.29	even	2	inner
1120.3.c.g.209.46		80	7.6	odd	2	inner
1120.3.c.g.209.47		80	35.34	odd	2	inner
1120.3.c.g.209.48		80	8.5	even	2	inner
1120.3.c.g.209.53		80	1.1	even	1	trivial
1120.3.c.g.209.54		80	280.69	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)

\(f(q)\)	\(=\)	\(q-1.20162i q^{3} +(1.26609 + 4.83705i) q^{5} +(2.96368 + 6.34166i) q^{7} +7.55612 q^{9} +O(q^{10})\)	\(q-1.20162i q^{3} +(1.26609 + 4.83705i) q^{5} +(2.96368 + 6.34166i) q^{7} +7.55612 q^{9} -6.35173i q^{11} +11.4646i q^{13} +(5.81227 - 1.52135i) q^{15} +17.8708 q^{17} +3.65270 q^{19} +(7.62023 - 3.56121i) q^{21} +1.47479i q^{23} +(-21.7940 + 12.2483i) q^{25} -19.8941i q^{27} +13.9833i q^{29} -31.1953i q^{31} -7.63234 q^{33} +(-26.9226 + 22.3646i) q^{35} -10.8744 q^{37} +13.7760 q^{39} +60.7473i q^{41} +34.9718 q^{43} +(9.56671 + 36.5493i) q^{45} +54.6600 q^{47} +(-31.4332 + 37.5893i) q^{49} -21.4738i q^{51} -50.2489 q^{53} +(30.7236 - 8.04185i) q^{55} -4.38915i q^{57} -48.8348 q^{59} -13.3248 q^{61} +(22.3939 + 47.9183i) q^{63} +(-55.4547 + 14.5152i) q^{65} -105.220 q^{67} +1.77213 q^{69} -65.2930 q^{71} +97.4832 q^{73} +(14.7177 + 26.1881i) q^{75} +(40.2805 - 18.8245i) q^{77} +12.8701 q^{79} +44.1000 q^{81} +140.293i q^{83} +(22.6260 + 86.4417i) q^{85} +16.8025 q^{87} -25.4993i q^{89} +(-72.7044 + 33.9774i) q^{91} -37.4848 q^{93} +(4.62465 + 17.6683i) q^{95} +117.389 q^{97} -47.9944i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 224 q^{9}+O(q^{10}) \) 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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