LMFDB - Embedded newform 880.4.b.h.529.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,4,Mod(529,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("880.529");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	880.b (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:	$51.9216808051$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 $ x^8 + 151x^6 + 7935x^4 + 171721*x^2 + 1308736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.7
Root			$4.82127i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.4.b.h.529.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.82127i q^{3} +(-8.37442 + 7.40737i) q^{5} -13.6360i q^{7} -19.5297 q^{9} +11.0000 q^{11} +66.9953i q^{13} +(-50.5277 - 57.1242i) q^{15} +136.870i q^{17} +126.061 q^{19} +93.0149 q^{21} +62.5641i q^{23} +(15.2617 - 124.065i) q^{25} +50.9568i q^{27} -144.857 q^{29} -229.288 q^{31} +75.0340i q^{33} +(101.007 + 114.194i) q^{35} +6.56930i q^{37} -456.993 q^{39} -276.684 q^{41} -233.339i q^{43} +(163.550 - 144.664i) q^{45} +381.735i q^{47} +157.059 q^{49} -933.627 q^{51} +182.225i q^{53} +(-92.1186 + 81.4811i) q^{55} +859.898i q^{57} +111.466 q^{59} +116.977 q^{61} +266.307i q^{63} +(-496.259 - 561.047i) q^{65} -480.882i q^{67} -426.766 q^{69} +602.696 q^{71} -622.178i q^{73} +(846.280 + 104.104i) q^{75} -149.996i q^{77} -768.900 q^{79} -874.892 q^{81} -953.137i q^{83} +(-1013.85 - 1146.21i) q^{85} -988.109i q^{87} +757.114 q^{89} +913.549 q^{91} -1564.03i q^{93} +(-1055.69 + 933.783i) q^{95} +1118.29i q^{97} -214.827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 110 q^{9} + 88 q^{11} - 8 q^{15} + 302 q^{19} + 230 q^{21} - 162 q^{25} - 58 q^{29} - 1022 q^{31} + 1058 q^{35} + 320 q^{39} + 452 q^{41} - 622 q^{45} + 222 q^{49} - 834 q^{51} + 176 q^{55}+ \cdots - 1210 q^{99}+O(q^{100}) $ 8 * q + 16 * q^5 - 110 * q^9 + 88 * q^11 - 8 * q^15 + 302 * q^19 + 230 * q^21 - 162 * q^25 - 58 * q^29 - 1022 * q^31 + 1058 * q^35 + 320 * q^39 + 452 * q^41 - 622 * q^45 + 222 * q^49 - 834 * q^51 + 176 * q^55 - 268 * q^59 - 490 * q^61 - 1408 * q^65 + 2820 * q^69 - 2034 * q^71 + 2110 * q^75 - 272 * q^79 - 2488 * q^81 + 330 * q^85 + 5782 * q^89 + 3520 * q^91 - 1268 * q^95 - 1210 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			6.82127i			1.31275i	0.754433	+	0.656377i	$0.227912\pi$
$3$			6.82127i			1.31275i	−0.754433	+	0.656377i	$0.772088\pi$
$4$		0			0
$5$	−8.37442	+	7.40737i	−0.749031	+	0.662535i
$5$	−8.37442	+	7.40737i	−0.749031	+	0.662535i
$6$		0			0
$7$		−	13.6360i		−	0.736275i	−0.929771	−	0.368138i	$-0.879995\pi$
$7$		−	13.6360i		−	0.736275i	0.929771	−	0.368138i	$-0.120005\pi$
$8$		0			0
$9$	−19.5297			−0.723323
$10$		0			0
$11$	11.0000			0.301511
$11$	11.0000			0.301511
$12$		0			0
$13$			66.9953i			1.42932i	0.699472	+	0.714660i	$0.253419\pi$
$13$			66.9953i			1.42932i	−0.699472	+	0.714660i	$0.746581\pi$
$14$		0			0
$15$	−50.5277	−	57.1242i	−0.869746	−	0.983293i
$16$		0			0
$17$			136.870i			1.95270i	0.216201	+	0.976349i	$0.430633\pi$
$17$			136.870i			1.95270i	−0.216201	+	0.976349i	$0.569367\pi$
$18$		0			0
$19$	126.061			1.52213			0.761064	−	0.648676i	$-0.224677\pi$
$19$	126.061			1.52213			0.761064	+	0.648676i	$0.224677\pi$
$20$		0			0
$21$	93.0149			0.966548
$22$		0			0
$23$			62.5641i			0.567196i	0.958943	+	0.283598i	$0.0915281\pi$
$23$			62.5641i			0.567196i	−0.958943	+	0.283598i	$0.908472\pi$
$24$		0			0
$25$	15.2617	−	124.065i	0.122094	−	0.992519i
$26$		0			0
$27$			50.9568i			0.363209i
$28$		0			0
$29$	−144.857			−0.927562			−0.463781	−	0.885950i	$-0.653507\pi$
$29$	−144.857			−0.927562			−0.463781	+	0.885950i	$0.653507\pi$
$30$		0			0
$31$	−229.288			−1.32843			−0.664214	−	0.747543i	$-0.731234\pi$
$31$	−229.288			−1.32843			−0.664214	+	0.747543i	$0.731234\pi$
$32$		0			0
$33$			75.0340i			0.395810i
$34$		0			0
$35$	101.007	+	114.194i	0.487809	+	0.551493i
$36$		0			0
$37$			6.56930i			0.0291888i	0.999893	+	0.0145944i	$0.00464571\pi$
$37$			6.56930i			0.0291888i	−0.999893	+	0.0145944i	$0.995354\pi$
$38$		0			0
$39$	−456.993			−1.87635
$40$		0			0
$41$	−276.684			−1.05392			−0.526961	−	0.849889i	$-0.676669\pi$
$41$	−276.684			−1.05392			−0.526961	+	0.849889i	$0.676669\pi$
$42$		0			0
$43$		−	233.339i		−	0.827531i	−0.910384	−	0.413765i	$-0.864213\pi$
$43$		−	233.339i		−	0.827531i	0.910384	−	0.413765i	$-0.135787\pi$
$44$		0			0
$45$	163.550	−	144.664i	0.541791	−	0.479227i
$46$		0			0
$47$			381.735i			1.18472i	0.805674	+	0.592359i	$0.201803\pi$
$47$			381.735i			1.18472i	−0.805674	+	0.592359i	$0.798197\pi$
$48$		0			0
$49$	157.059			0.457898
$50$		0			0
$51$	−933.627			−2.56341
$52$		0			0
$53$			182.225i			0.472275i	0.971720	+	0.236137i	$0.0758815\pi$
$53$			182.225i			0.472275i	−0.971720	+	0.236137i	$0.924118\pi$
$54$		0			0
$55$	−92.1186	+	81.4811i	−0.225841	+	0.199762i
$56$		0			0
$57$			859.898i			1.99818i
$58$		0			0
$59$	111.466			0.245961			0.122980	−	0.992409i	$-0.460755\pi$
$59$	111.466			0.245961			0.122980	+	0.992409i	$0.460755\pi$
$60$		0			0
$61$	116.977			0.245530			0.122765	−	0.992436i	$-0.460824\pi$
$61$	116.977			0.245530			0.122765	+	0.992436i	$0.460824\pi$
$62$		0			0
$63$			266.307i			0.532565i
$64$		0			0
$65$	−496.259	−	561.047i	−0.946975	−	1.07060i
$66$		0			0
$67$		−	480.882i		−	0.876853i	−0.898767	−	0.438426i	$-0.855536\pi$
$67$		−	480.882i		−	0.876853i	0.898767	−	0.438426i	$-0.144464\pi$
$68$		0			0
$69$	−426.766			−0.744589
$70$		0			0
$71$	602.696			1.00742			0.503710	−	0.863873i	$-0.331968\pi$
$71$	602.696			1.00742			0.503710	+	0.863873i	$0.331968\pi$
$72$		0			0
$73$		−	622.178i		−	0.997540i	−0.866734	−	0.498770i	$-0.833785\pi$
$73$		−	622.178i		−	0.997540i	0.866734	−	0.498770i	$-0.166215\pi$
$74$		0			0
$75$	846.280	+	104.104i	1.30293	+	0.160279i
$76$		0			0
$77$		−	149.996i		−	0.221995i
$78$		0			0
$79$	−768.900			−1.09504			−0.547519	−	0.836793i	$-0.684428\pi$
$79$	−768.900			−1.09504			−0.547519	+	0.836793i	$0.684428\pi$
$80$		0			0
$81$	−874.892			−1.20013
$82$		0			0
$83$		−	953.137i		−	1.26049i	−0.776398	−	0.630243i	$-0.782955\pi$
$83$		−	953.137i		−	1.26049i	0.776398	−	0.630243i	$-0.217045\pi$
$84$		0			0
$85$	−1013.85	−	1146.21i	−1.29373	−	1.46263i
$86$		0			0
$87$		−	988.109i		−	1.21766i
$88$		0			0
$89$	757.114			0.901730			0.450865	−	0.892592i	$-0.351116\pi$
$89$	757.114			0.901730			0.450865	+	0.892592i	$0.351116\pi$
$90$		0			0
$91$	913.549			1.05237
$92$		0			0
$93$		−	1564.03i		−	1.74390i
$94$		0			0
$95$	−1055.69	+	933.783i	−1.14012	+	1.00846i
$96$		0			0
$97$			1118.29i			1.17057i	0.810827	+	0.585285i	$0.199018\pi$
$97$			1118.29i			1.17057i	−0.810827	+	0.585285i	$0.800982\pi$
$98$		0			0
$99$	−214.827			−0.218090
$100$		0			0
$101$	−1053.05			−1.03745			−0.518723	−	0.854942i	$-0.673592\pi$
$101$	−1053.05			−1.03745			−0.518723	+	0.854942i	$0.673592\pi$
$102$		0			0
$103$		−	728.192i		−	0.696610i	−0.937381	−	0.348305i	$-0.886757\pi$
$103$		−	728.192i		−	0.696610i	0.937381	−	0.348305i	$-0.113243\pi$
$104$		0			0
$105$	−778.946	+	688.996i	−0.723974	+	0.640373i
$106$		0			0
$107$		−	988.919i		−	0.893480i	−0.894664	−	0.446740i	$-0.852585\pi$
$107$		−	988.919i		−	0.893480i	0.894664	−	0.446740i	$-0.147415\pi$
$108$		0			0
$109$	1736.51			1.52594			0.762970	−	0.646434i	$-0.223740\pi$
$109$	1736.51			1.52594			0.762970	+	0.646434i	$0.223740\pi$
$110$		0			0
$111$	−44.8110			−0.0383178
$112$		0			0
$113$		−	1115.15i		−	0.928360i	−0.885741	−	0.464180i	$-0.846349\pi$
$113$		−	1115.15i		−	0.928360i	0.885741	−	0.464180i	$-0.153651\pi$
$114$		0			0
$115$	−463.435	−	523.937i	−0.375787	−	0.424847i
$116$		0			0
$117$		−	1308.40i		−	1.03386i
$118$		0			0
$119$	1866.36			1.43772
$120$		0			0
$121$	121.000			0.0909091
$122$		0			0
$123$		−	1887.34i		−	1.38354i
$124$		0			0
$125$	791.186	+	1152.02i	0.566127	+	0.824318i
$126$		0			0
$127$			1558.13i			1.08868i	0.838866	+	0.544338i	$0.183219\pi$
$127$			1558.13i			1.08868i	−0.838866	+	0.544338i	$0.816781\pi$
$128$		0			0
$129$	1591.67			1.08634
$130$		0			0
$131$	620.634			0.413932			0.206966	−	0.978348i	$-0.433641\pi$
$131$	620.634			0.413932			0.206966	+	0.978348i	$0.433641\pi$
$132$		0			0
$133$		−	1718.97i		−	1.12071i
$134$		0			0
$135$	−377.456	−	426.734i	−0.240639	−	0.272055i
$136$		0			0
$137$			1138.81i			0.710185i	0.934831	+	0.355093i	$0.115551\pi$
$137$			1138.81i			0.710185i	−0.934831	+	0.355093i	$0.884449\pi$
$138$		0			0
$139$	−1363.18			−0.831826			−0.415913	−	0.909404i	$-0.636538\pi$
$139$	−1363.18			−0.831826			−0.415913	+	0.909404i	$0.636538\pi$
$140$		0			0
$141$	−2603.91			−1.55524
$142$		0			0
$143$			736.949i			0.430956i
$144$		0			0
$145$	1213.09	−	1073.01i	0.694772	−	0.614542i
$146$		0			0
$147$			1071.34i			0.601108i
$148$		0			0
$149$	−2823.58			−1.55246			−0.776232	−	0.630448i	$-0.782871\pi$
$149$	−2823.58			−1.55246			−0.776232	+	0.630448i	$0.782871\pi$
$150$		0			0
$151$	−975.281			−0.525611			−0.262806	−	0.964849i	$-0.584648\pi$
$151$	−975.281			−0.525611			−0.262806	+	0.964849i	$0.584648\pi$
$152$		0			0
$153$		−	2673.03i		−	1.41243i
$154$		0			0
$155$	1920.15	−	1698.42i	0.995033	−	0.880130i
$156$		0			0
$157$		−	842.402i		−	0.428223i	−0.976809	−	0.214111i	$-0.931314\pi$
$157$		−	842.402i		−	0.428223i	0.976809	−	0.214111i	$-0.0686856\pi$
$158$		0			0
$159$	−1243.01			−0.619980
$160$		0			0
$161$	853.124			0.417612
$162$		0			0
$163$			376.648i			0.180990i	0.995897	+	0.0904950i	$0.0288449\pi$
$163$			376.648i			0.180990i	−0.995897	+	0.0904950i	$0.971155\pi$
$164$		0			0
$165$	−555.804	−	628.366i	−0.262238	−	0.296474i
$166$		0			0
$167$		−	1298.93i		−	0.601882i	−0.953643	−	0.300941i	$-0.902699\pi$
$167$		−	1298.93i		−	0.601882i	0.953643	−	0.300941i	$-0.0973008\pi$
$168$		0			0
$169$	−2291.37			−1.04296
$170$		0			0
$171$	−2461.94			−1.10099
$172$		0			0
$173$			2066.43i			0.908138i	0.890967	+	0.454069i	$0.150028\pi$
$173$			2066.43i			0.908138i	−0.890967	+	0.454069i	$0.849972\pi$
$174$		0			0
$175$	−1691.75	−	208.109i	−0.730767	−	0.0898945i
$176$		0			0
$177$			760.342i			0.322886i
$178$		0			0
$179$	−2046.17			−0.854400			−0.427200	−	0.904157i	$-0.640500\pi$
$179$	−2046.17			−0.854400			−0.427200	+	0.904157i	$0.640500\pi$
$180$		0			0
$181$	2561.44			1.05188			0.525941	−	0.850521i	$-0.323713\pi$
$181$	2561.44			1.05188			0.525941	+	0.850521i	$0.323713\pi$
$182$		0			0
$183$			797.931i			0.322321i
$184$		0			0
$185$	−48.6613	−	55.0141i	−0.0193386	−	0.0218633i
$186$		0			0
$187$			1505.57i			0.588761i
$188$		0			0
$189$	694.848			0.267422
$190$		0			0
$191$	−3474.08			−1.31610			−0.658051	−	0.752973i	$-0.728619\pi$
$191$	−3474.08			−1.31610			−0.658051	+	0.752973i	$0.728619\pi$
$192$		0			0
$193$			1741.67i			0.649577i	0.945787	+	0.324788i	$0.105293\pi$
$193$			1741.67i			0.649577i	−0.945787	+	0.324788i	$0.894707\pi$
$194$		0			0
$195$	3827.05	−	3385.12i	1.40544	−	1.24315i
$196$		0			0
$197$			112.400i			0.0406507i	0.999793	+	0.0203253i	$0.00647020\pi$
$197$			112.400i			0.0406507i	−0.999793	+	0.0203253i	$0.993530\pi$
$198$		0			0
$199$	−4526.79			−1.61254			−0.806270	−	0.591548i	$-0.798517\pi$
$199$	−4526.79			−1.61254			−0.806270	+	0.591548i	$0.798517\pi$
$200$		0			0
$201$	3280.23			1.15109
$202$		0			0
$203$			1975.27i			0.682941i
$204$		0			0
$205$	2317.07	−	2049.50i	0.789420	−	0.698261i
$206$		0			0
$207$		−	1221.86i		−	0.410266i
$208$		0			0
$209$	1386.67			0.458939
$210$		0			0
$211$	3743.57			1.22141			0.610707	−	0.791857i	$-0.290885\pi$
$211$	3743.57			1.22141			0.610707	+	0.791857i	$0.290885\pi$
$212$		0			0
$213$			4111.15i			1.32249i
$214$		0			0
$215$	1728.43	+	1954.08i	0.548269	+	0.619846i
$216$		0			0
$217$			3126.57i			0.978089i
$218$		0			0
$219$	4244.04			1.30952
$220$		0			0
$221$	−9169.65			−2.79103
$222$		0			0
$223$		−	2773.89i		−	0.832975i	−0.909141	−	0.416487i	$-0.863261\pi$
$223$		−	2773.89i		−	0.832975i	0.909141	−	0.416487i	$-0.136739\pi$
$224$		0			0
$225$	−298.057	+	2422.95i	−0.0883131	+	0.717911i
$226$		0			0
$227$		−	175.125i		−	0.0512046i	−0.999672	−	0.0256023i	$-0.991850\pi$
$227$		−	175.125i		−	0.0512046i	0.999672	−	0.0256023i	$-0.00815036\pi$
$228$		0			0
$229$	−2789.77			−0.805035			−0.402518	−	0.915412i	$-0.631865\pi$
$229$	−2789.77			−0.805035			−0.402518	+	0.915412i	$0.631865\pi$
$230$		0			0
$231$	1023.16			0.291425
$232$		0			0
$233$		−	3107.83i		−	0.873822i	−0.899505	−	0.436911i	$-0.856072\pi$
$233$		−	3107.83i		−	0.873822i	0.899505	−	0.436911i	$-0.143928\pi$
$234$		0			0
$235$	−2827.65	−	3196.80i	−0.784917	−	0.887389i
$236$		0			0
$237$		−	5244.88i		−	1.43752i
$238$		0			0
$239$	−5263.47			−1.42454			−0.712271	−	0.701904i	$-0.752333\pi$
$239$	−5263.47			−1.42454			−0.712271	+	0.701904i	$0.752333\pi$
$240$		0			0
$241$	−2063.46			−0.551533			−0.275767	−	0.961225i	$-0.588932\pi$
$241$	−2063.46			−0.551533			−0.275767	+	0.961225i	$0.588932\pi$
$242$		0			0
$243$		−	4592.04i		−	1.21226i
$244$		0			0
$245$	−1315.28	+	1163.40i	−0.342980	+	0.303374i
$246$		0			0
$247$			8445.52i			2.17561i
$248$		0			0
$249$	6501.60			1.65471
$250$		0			0
$251$	−2087.66			−0.524988			−0.262494	−	0.964934i	$-0.584545\pi$
$251$	−2087.66			−0.524988			−0.262494	+	0.964934i	$0.584545\pi$
$252$		0			0
$253$			688.205i			0.171016i
$254$		0			0
$255$	7818.58	−	6915.72i	1.92007	−	1.69835i
$256$		0			0
$257$			4901.68i			1.18972i	0.803829	+	0.594861i	$0.202793\pi$
$257$			4901.68i			1.18972i	−0.803829	+	0.594861i	$0.797207\pi$
$258$		0			0
$259$	89.5791			0.0214910
$260$		0			0
$261$	2829.02			0.670927
$262$		0			0
$263$			5075.49i			1.18999i	0.803728	+	0.594997i	$0.202847\pi$
$263$			5075.49i			1.18999i	−0.803728	+	0.594997i	$0.797153\pi$
$264$		0			0
$265$	−1349.81	−	1526.03i	−0.312899	−	0.353748i
$266$		0			0
$267$			5164.48i			1.18375i
$268$		0			0
$269$	3906.32			0.885400			0.442700	−	0.896670i	$-0.354021\pi$
$269$	3906.32			0.885400			0.442700	+	0.896670i	$0.354021\pi$
$270$		0			0
$271$	−2849.44			−0.638714			−0.319357	−	0.947635i	$-0.603467\pi$
$271$	−2849.44			−0.638714			−0.319357	+	0.947635i	$0.603467\pi$
$272$		0			0
$273$			6231.56i			1.38151i
$274$		0			0
$275$	167.879	−	1364.71i	0.0368126	−	0.299256i
$276$		0			0
$277$		−	5076.18i		−	1.10108i	−0.834810	−	0.550539i	$-0.814422\pi$
$277$		−	5076.18i		−	1.10108i	0.834810	−	0.550539i	$-0.185578\pi$
$278$		0			0
$279$	4477.92			0.960882
$280$		0			0
$281$	−2815.17			−0.597647			−0.298823	−	0.954308i	$-0.596594\pi$
$281$	−2815.17			−0.597647			−0.298823	+	0.954308i	$0.596594\pi$
$282$		0			0
$283$		−	2420.95i		−	0.508519i	−0.967136	−	0.254259i	$-0.918168\pi$
$283$		−	2420.95i		−	0.508519i	0.967136	−	0.254259i	$-0.0818317\pi$
$284$		0			0
$285$	−6369.58	−	7201.14i	−1.32387	−	1.49670i
$286$		0			0
$287$			3772.87i			0.775978i
$288$		0			0
$289$	−13820.4			−2.81303
$290$		0			0
$291$	−7628.17			−1.53667
$292$		0			0
$293$			2429.28i			0.484370i	0.970230	+	0.242185i	$0.0778640\pi$
$293$			2429.28i			0.484370i	−0.970230	+	0.242185i	$0.922136\pi$
$294$		0			0
$295$	−933.465	+	825.672i	−0.184232	+	0.162958i
$296$		0			0
$297$			560.525i			0.109512i
$298$		0			0
$299$	−4191.50			−0.810705
$300$		0			0
$301$	−3181.81			−0.609291
$302$		0			0
$303$		−	7183.12i		−	1.36191i
$304$		0			0
$305$	−979.613	+	866.491i	−0.183910	+	0.162673i
$306$		0			0
$307$			162.937i			0.0302910i	0.999885	+	0.0151455i	$0.00482114\pi$
$307$			162.937i			0.0302910i	−0.999885	+	0.0151455i	$0.995179\pi$
$308$		0			0
$309$	4967.19			0.914478
$310$		0			0
$311$	6155.28			1.12230			0.561148	−	0.827715i	$-0.310360\pi$
$311$	6155.28			1.12230			0.561148	+	0.827715i	$0.310360\pi$
$312$		0			0
$313$		−	6165.59i		−	1.11342i	−0.830708	−	0.556709i	$-0.812064\pi$
$313$		−	6165.59i		−	1.11342i	0.830708	−	0.556709i	$-0.187936\pi$
$314$		0			0
$315$	−1972.64	−	2230.17i	−0.352843	−	0.398907i
$316$		0			0
$317$			3099.89i			0.549234i	0.961554	+	0.274617i	$0.0885510\pi$
$317$			3099.89i			0.549234i	−0.961554	+	0.274617i	$0.911449\pi$
$318$		0			0
$319$	−1593.43			−0.279670
$320$		0			0
$321$	6745.68			1.17292
$322$		0			0
$323$			17254.0i			2.97226i
$324$		0			0
$325$	8311.76	+	1022.46i	1.41863	+	0.174511i
$326$		0			0
$327$			11845.2i			2.00318i
$328$		0			0
$329$	5205.34			0.872278
$330$		0			0
$331$	7423.91			1.23279			0.616397	−	0.787436i	$-0.288592\pi$
$331$	7423.91			1.23279			0.616397	+	0.787436i	$0.288592\pi$
$332$		0			0
$333$		−	128.297i		−	0.0211130i
$334$		0			0
$335$	3562.07	+	4027.11i	0.580946	+	0.656789i
$336$		0			0
$337$			1808.87i			0.292391i	0.989256	+	0.146195i	$0.0467028\pi$
$337$			1808.87i			0.292391i	−0.989256	+	0.146195i	$0.953297\pi$
$338$		0			0
$339$	7606.76			1.21871
$340$		0			0
$341$	−2522.16			−0.400536
$342$		0			0
$343$		−	6818.81i		−	1.07341i
$344$		0			0
$345$	3573.92	−	3161.22i	0.557720	−	0.493316i
$346$		0			0
$347$			5850.60i			0.905120i	0.891734	+	0.452560i	$0.149489\pi$
$347$			5850.60i			0.905120i	−0.891734	+	0.452560i	$0.850511\pi$
$348$		0			0
$349$	10231.6			1.56930			0.784648	−	0.619941i	$-0.212844\pi$
$349$	10231.6			1.56930			0.784648	+	0.619941i	$0.212844\pi$
$350$		0			0
$351$	−3413.87			−0.519142
$352$		0			0
$353$			438.848i			0.0661687i	0.999453	+	0.0330843i	$0.0105330\pi$
$353$			438.848i			0.0661687i	−0.999453	+	0.0330843i	$0.989467\pi$
$354$		0			0
$355$	−5047.22	+	4464.39i	−0.754588	+	0.667451i
$356$		0			0
$357$			12731.0i			1.88738i
$358$		0			0
$359$	9743.06			1.43236			0.716182	−	0.697913i	$-0.245888\pi$
$359$	9743.06			1.43236			0.716182	+	0.697913i	$0.245888\pi$
$360$		0			0
$361$	9032.45			1.31688
$362$		0			0
$363$			825.374i			0.119341i
$364$		0			0
$365$	4608.70	+	5210.38i	0.660905	+	0.747188i
$366$		0			0
$367$		−	11674.8i		−	1.66054i	−0.557363	−	0.830269i	$-0.688187\pi$
$367$		−	11674.8i		−	1.66054i	0.557363	−	0.830269i	$-0.311813\pi$
$368$		0			0
$369$	5403.57			0.762327
$370$		0			0
$371$	2484.82			0.347724
$372$		0			0
$373$		−	537.719i		−	0.0746435i	−0.999303	−	0.0373218i	$-0.988117\pi$
$373$		−	537.719i		−	0.0746435i	0.999303	−	0.0373218i	$-0.0118826\pi$
$374$		0			0
$375$	−7858.24	+	5396.90i	−1.08213	+	0.743185i
$376$		0			0
$377$		−	9704.75i		−	1.32578i
$378$		0			0
$379$	9268.43			1.25617			0.628084	−	0.778146i	$-0.283840\pi$
$379$	9268.43			1.25617			0.628084	+	0.778146i	$0.283840\pi$
$380$		0			0
$381$	−10628.4			−1.42916
$382$		0			0
$383$			5038.62i			0.672223i	0.941822	+	0.336111i	$0.109112\pi$
$383$			5038.62i			0.672223i	−0.941822	+	0.336111i	$0.890888\pi$
$384$		0			0
$385$	1111.08	+	1256.13i	0.147080	+	0.166281i
$386$		0			0
$387$			4557.04i			0.598572i
$388$		0			0
$389$	−12545.6			−1.63518			−0.817590	−	0.575800i	$-0.804691\pi$
$389$	−12545.6			−1.63518			−0.817590	+	0.575800i	$0.804691\pi$
$390$		0			0
$391$	−8563.14			−1.10756
$392$		0			0
$393$			4233.51i			0.543390i
$394$		0			0
$395$	6439.09	−	5695.53i	0.820217	−	0.725502i
$396$		0			0
$397$			9116.35i			1.15248i	0.817279	+	0.576242i	$0.195482\pi$
$397$			9116.35i			1.15248i	−0.817279	+	0.576242i	$0.804518\pi$
$398$		0			0
$399$	11725.6			1.47121
$400$		0			0
$401$	−3052.85			−0.380179			−0.190090	−	0.981767i	$-0.560878\pi$
$401$	−3052.85			−0.380179			−0.190090	+	0.981767i	$0.560878\pi$
$402$		0			0
$403$		−	15361.2i		−	1.89875i
$404$		0			0
$405$	7326.71	−	6480.65i	0.898932	−	0.795127i
$406$		0			0
$407$			72.2623i			0.00880077i
$408$		0			0
$409$	4417.02			0.534004			0.267002	−	0.963696i	$-0.413967\pi$
$409$	4417.02			0.534004			0.267002	+	0.963696i	$0.413967\pi$
$410$		0			0
$411$	−7768.15			−0.932298
$412$		0			0
$413$		−	1519.96i		−	0.181095i
$414$		0			0
$415$	7060.24	+	7981.96i	0.835117	+	0.944143i
$416$		0			0
$417$		−	9298.65i		−	1.09198i
$418$		0			0
$419$	6184.20			0.721045			0.360522	−	0.932751i	$-0.382598\pi$
$419$	6184.20			0.721045			0.360522	+	0.932751i	$0.382598\pi$
$420$		0			0
$421$	−14153.4			−1.63846			−0.819231	−	0.573464i	$-0.805599\pi$
$421$	−14153.4			−1.63846			−0.819231	+	0.573464i	$0.805599\pi$
$422$		0			0
$423$		−	7455.17i		−	0.856933i
$424$		0			0
$425$	16980.8	+	2088.87i	1.93809	+	0.238412i
$426$		0			0
$427$		−	1595.10i		−	0.180778i
$428$		0			0
$429$	−5026.93			−0.565739
$430$		0			0
$431$	−539.740			−0.0603210			−0.0301605	−	0.999545i	$-0.509602\pi$
$431$	−539.740			−0.0603210			−0.0301605	+	0.999545i	$0.509602\pi$
$432$		0			0
$433$			13447.3i			1.49246i	0.665687	+	0.746231i	$0.268139\pi$
$433$			13447.3i			1.49246i	−0.665687	+	0.746231i	$0.731861\pi$
$434$		0			0
$435$	7319.29	+	8274.84i	0.806743	+	0.912065i
$436$		0			0
$437$			7886.91i			0.863345i
$438$		0			0
$439$	9970.49			1.08398			0.541988	−	0.840386i	$-0.317672\pi$
$439$	9970.49			1.08398			0.541988	+	0.840386i	$0.317672\pi$
$440$		0			0
$441$	−3067.32			−0.331208
$442$		0			0
$443$			11600.7i			1.24416i	0.782952	+	0.622082i	$0.213713\pi$
$443$			11600.7i			1.24416i	−0.782952	+	0.622082i	$0.786287\pi$
$444$		0			0
$445$	−6340.39	+	5608.23i	−0.675423	+	0.597428i
$446$		0			0
$447$		−	19260.4i		−	2.03800i
$448$		0			0
$449$	12957.9			1.36196			0.680982	−	0.732300i	$-0.261553\pi$
$449$	12957.9			1.36196			0.680982	+	0.732300i	$0.261553\pi$
$450$		0			0
$451$	−3043.53			−0.317770
$452$		0			0
$453$		−	6652.66i		−	0.689998i
$454$		0			0
$455$	−7650.44	+	6767.00i	−0.788260	+	0.697235i
$456$		0			0
$457$			16667.8i			1.70610i	0.521831	+	0.853049i	$0.325249\pi$
$457$			16667.8i			1.70610i	−0.521831	+	0.853049i	$0.674751\pi$
$458$		0			0
$459$	−6974.46			−0.709237
$460$		0			0
$461$	5625.88			0.568380			0.284190	−	0.958768i	$-0.408275\pi$
$461$	5625.88			0.568380			0.284190	+	0.958768i	$0.408275\pi$
$462$		0			0
$463$		−	3395.81i		−	0.340857i	−0.985370	−	0.170429i	$-0.945485\pi$
$463$		−	3395.81i		−	0.340857i	0.985370	−	0.170429i	$-0.0545152\pi$
$464$		0			0
$465$	11585.4	+	13097.9i	1.15539	+	1.30623i
$466$		0			0
$467$		−	12556.6i		−	1.24422i	−0.782930	−	0.622110i	$-0.786276\pi$
$467$		−	12556.6i		−	1.24422i	0.782930	−	0.622110i	$-0.213724\pi$
$468$		0			0
$469$	−6557.32			−0.645605
$470$		0			0
$471$	5746.25			0.562151
$472$		0			0
$473$		−	2566.73i		−	0.249510i
$474$		0			0
$475$	1923.91	−	15639.8i	0.185842	−	1.51074i
$476$		0			0
$477$		−	3558.81i		−	0.341607i
$478$		0			0
$479$	7841.71			0.748010			0.374005	−	0.927427i	$-0.377984\pi$
$479$	7841.71			0.748010			0.374005	+	0.927427i	$0.377984\pi$
$480$		0			0
$481$	−440.113			−0.0417202
$482$		0			0
$483$			5819.39i			0.548222i
$484$		0			0
$485$	−8283.61	−	9365.05i	−0.775545	−	0.876793i
$486$		0			0
$487$		−	1205.34i		−	0.112154i	−0.998426	−	0.0560772i	$-0.982141\pi$
$487$		−	1205.34i		−	0.112154i	0.998426	−	0.0560772i	$-0.0178593\pi$
$488$		0			0
$489$	−2569.22			−0.237595
$490$		0			0
$491$	2708.46			0.248944			0.124472	−	0.992223i	$-0.460276\pi$
$491$	2708.46			0.248944			0.124472	+	0.992223i	$0.460276\pi$
$492$		0			0
$493$		−	19826.6i		−	1.81125i
$494$		0			0
$495$	1799.05	−	1591.30i	0.163356	−	0.144492i
$496$		0			0
$497$		−	8218.36i		−	0.741739i
$498$		0			0
$499$	−17160.4			−1.53949			−0.769745	−	0.638351i	$-0.779617\pi$
$499$	−17160.4			−1.53949			−0.769745	+	0.638351i	$0.779617\pi$
$500$		0			0
$501$	8860.36			0.790124
$502$		0			0
$503$		−	263.269i		−	0.0233371i	−0.999932	−	0.0116686i	$-0.996286\pi$
$503$		−	263.269i		−	0.0233371i	0.999932	−	0.0116686i	$-0.00371431\pi$
$504$		0			0
$505$	8818.65	−	7800.31i	0.777079	−	0.687345i
$506$		0			0
$507$		−	15630.1i		−	1.36914i
$508$		0			0
$509$	21083.7			1.83599			0.917997	−	0.396588i	$-0.129806\pi$
$509$	21083.7			1.83599			0.917997	+	0.396588i	$0.129806\pi$
$510$		0			0
$511$	−8484.02			−0.734464
$512$		0			0
$513$			6423.68i			0.552851i
$514$		0			0
$515$	5393.99	+	6098.18i	0.461529	+	0.521782i
$516$		0			0
$517$			4199.08i			0.357206i
$518$		0			0
$519$	−14095.7			−1.19216
$520$		0			0
$521$	20745.1			1.74445			0.872226	−	0.489102i	$-0.162676\pi$
$521$	20745.1			1.74445			0.872226	+	0.489102i	$0.162676\pi$
$522$		0			0
$523$			19060.9i			1.59364i	0.604214	+	0.796822i	$0.293487\pi$
$523$			19060.9i			1.59364i	−0.604214	+	0.796822i	$0.706513\pi$
$524$		0			0
$525$	1419.57	−	11539.9i	0.118009	−	0.959317i
$526$		0			0
$527$		−	31382.6i		−	2.59402i
$528$		0			0
$529$	8252.74			0.678289
$530$		0			0
$531$	−2176.91			−0.177909
$532$		0			0
$533$		−	18536.6i		−	1.50639i
$534$		0			0
$535$	7325.29	+	8281.62i	0.591962	+	0.669244i
$536$		0			0
$537$		−	13957.5i		−	1.12162i
$538$		0			0
$539$	1727.65			0.138062
$540$		0			0
$541$	−3953.81			−0.314210			−0.157105	−	0.987582i	$-0.550216\pi$
$541$	−3953.81			−0.314210			−0.157105	+	0.987582i	$0.550216\pi$
$542$		0			0
$543$			17472.3i			1.38086i
$544$		0			0
$545$	−14542.3	+	12863.0i	−1.14298	+	1.01099i
$546$		0			0
$547$			10226.2i			0.799340i	0.916659	+	0.399670i	$0.130875\pi$
$547$			10226.2i			0.799340i	−0.916659	+	0.399670i	$0.869125\pi$
$548$		0			0
$549$	−2284.53			−0.177598
$550$		0			0
$551$	−18260.9			−1.41187
$552$		0			0
$553$			10484.7i			0.806250i
$554$		0			0
$555$	375.266	−	331.932i	0.0287012	−	0.0253869i
$556$		0			0
$557$		−	17378.7i		−	1.32201i	−0.750381	−	0.661005i	$-0.770130\pi$
$557$		−	17378.7i		−	1.32201i	0.750381	−	0.661005i	$-0.229870\pi$
$558$		0			0
$559$	15632.6			1.18281
$560$		0			0
$561$	−10269.9			−0.772898
$562$		0			0
$563$			12073.4i			0.903791i	0.892071	+	0.451895i	$0.149252\pi$
$563$			12073.4i			0.903791i	−0.892071	+	0.451895i	$0.850748\pi$
$564$		0			0
$565$	8260.35	+	9338.75i	0.615072	+	0.695370i
$566$		0			0
$567$			11930.0i			0.883624i
$568$		0			0
$569$	−11995.8			−0.883811			−0.441906	−	0.897062i	$-0.645697\pi$
$569$	−11995.8			−0.883811			−0.441906	+	0.897062i	$0.645697\pi$
$570$		0			0
$571$	6422.26			0.470688			0.235344	−	0.971912i	$-0.424378\pi$
$571$	6422.26			0.470688			0.235344	+	0.971912i	$0.424378\pi$
$572$		0			0
$573$		−	23697.6i		−	1.72772i
$574$		0			0
$575$	7762.00	+	954.834i	0.562953	+	0.0692510i
$576$		0			0
$577$			21036.3i			1.51777i	0.651227	+	0.758883i	$0.274255\pi$
$577$			21036.3i			1.51777i	−0.651227	+	0.758883i	$0.725745\pi$
$578$		0			0
$579$	−11880.4			−0.852735
$580$		0			0
$581$	−12997.0			−0.928065
$582$		0			0
$583$			2004.48i			0.142396i
$584$		0			0
$585$	9691.80	+	10957.1i	0.684969	+	0.774393i
$586$		0			0
$587$		−	5970.28i		−	0.419795i	−0.977723	−	0.209898i	$-0.932687\pi$
$587$		−	5970.28i		−	0.419795i	0.977723	−	0.209898i	$-0.0673131\pi$
$588$		0			0
$589$	−28904.3			−2.02204
$590$		0			0
$591$	−766.712			−0.0533643
$592$		0			0
$593$			25442.7i			1.76190i	0.473209	+	0.880950i	$0.343096\pi$
$593$			25442.7i			1.76190i	−0.473209	+	0.880950i	$0.656904\pi$
$594$		0			0
$595$	−15629.7	+	13824.8i	−1.07690	+	0.952543i
$596$		0			0
$597$		−	30878.4i		−	2.11687i
$598$		0			0
$599$	15519.2			1.05860			0.529298	−	0.848436i	$-0.322456\pi$
$599$	15519.2			1.05860			0.529298	+	0.848436i	$0.322456\pi$
$600$		0			0
$601$	−12438.2			−0.844203			−0.422102	−	0.906549i	$-0.638707\pi$
$601$	−12438.2			−0.844203			−0.422102	+	0.906549i	$0.638707\pi$
$602$		0			0
$603$			9391.50i			0.634248i
$604$		0			0
$605$	−1013.30	+	896.292i	−0.0680937	+	0.0602305i
$606$		0			0
$607$			16809.7i			1.12403i	0.827128	+	0.562014i	$0.189973\pi$
$607$			16809.7i			1.12403i	−0.827128	+	0.562014i	$0.810027\pi$
$608$		0			0
$609$	−13473.9			−0.896533
$610$		0			0
$611$	−25574.4			−1.69334
$612$		0			0
$613$			18549.3i			1.22218i	0.791560	+	0.611092i	$0.209269\pi$
$613$			18549.3i			1.22218i	−0.791560	+	0.611092i	$0.790731\pi$
$614$		0			0
$615$	13980.2	+	15805.4i	0.916645	+	1.03631i
$616$		0			0
$617$			21973.9i			1.43377i	0.697191	+	0.716885i	$0.254433\pi$
$617$			21973.9i			1.43377i	−0.697191	+	0.716885i	$0.745567\pi$
$618$		0			0
$619$	3457.29			0.224492			0.112246	−	0.993680i	$-0.464196\pi$
$619$	3457.29			0.224492			0.112246	+	0.993680i	$0.464196\pi$
$620$		0			0
$621$	−3188.06			−0.206011
$622$		0			0
$623$		−	10324.0i		−	0.663922i
$624$		0			0
$625$	−15159.2	−	3786.88i	−0.970186	−	0.242360i
$626$		0			0
$627$			9458.88i			0.602474i
$628$		0			0
$629$	−899.141			−0.0569970
$630$		0			0
$631$	−4266.16			−0.269149			−0.134575	−	0.990903i	$-0.542967\pi$
$631$	−4266.16			−0.269149			−0.134575	+	0.990903i	$0.542967\pi$
$632$		0			0
$633$			25535.9i			1.60342i
$634$		0			0
$635$	−11541.7	−	13048.4i	−0.721286	−	0.815451i
$636$		0			0
$637$			10522.2i			0.654484i
$638$		0			0
$639$	−11770.5			−0.728690
$640$		0			0
$641$	−4556.59			−0.280771			−0.140386	−	0.990097i	$-0.544834\pi$
$641$	−4556.59			−0.280771			−0.140386	+	0.990097i	$0.544834\pi$
$642$		0			0
$643$			4616.92i			0.283162i	0.989927	+	0.141581i	$0.0452186\pi$
$643$			4616.92i			0.283162i	−0.989927	+	0.141581i	$0.954781\pi$
$644$		0			0
$645$	−13329.3	+	11790.1i	−0.813705	+	0.719742i
$646$		0			0
$647$			1201.79i			0.0730252i	0.999333	+	0.0365126i	$0.0116249\pi$
$647$			1201.79i			0.0730252i	−0.999333	+	0.0365126i	$0.988375\pi$
$648$		0			0
$649$	1226.13			0.0741599
$650$		0			0
$651$	−21327.2			−1.28399
$652$		0			0
$653$			8658.24i			0.518871i	0.965760	+	0.259436i	$0.0835366\pi$
$653$			8658.24i			0.518871i	−0.965760	+	0.259436i	$0.916463\pi$
$654$		0			0
$655$	−5197.45	+	4597.27i	−0.310047	+	0.274244i
$656$		0			0
$657$			12151.0i			0.721543i
$658$		0			0
$659$	15311.7			0.905096			0.452548	−	0.891740i	$-0.350515\pi$
$659$	15311.7			0.905096			0.452548	+	0.891740i	$0.350515\pi$
$660$		0			0
$661$	−24981.1			−1.46997			−0.734987	−	0.678081i	$-0.762812\pi$
$661$	−24981.1			−1.46997			−0.734987	+	0.678081i	$0.762812\pi$
$662$		0			0
$663$		−	62548.7i		−	3.66394i
$664$		0			0
$665$	12733.1	+	14395.4i	0.742507	+	0.839443i
$666$		0			0
$667$		−	9062.85i		−	0.526109i
$668$		0			0
$669$	18921.5			1.09349
$670$		0			0
$671$	1286.75			0.0740302
$672$		0			0
$673$		−	7248.66i		−	0.415179i	−0.978216	−	0.207589i	$-0.933438\pi$
$673$		−	7248.66i		−	0.415179i	0.978216	−	0.207589i	$-0.0665618\pi$
$674$		0			0
$675$	6321.95	+	777.688i	0.360492	+	0.0443455i
$676$		0			0
$677$		−	10382.1i		−	0.589388i	−0.955592	−	0.294694i	$-0.904782\pi$
$677$		−	10382.1i		−	0.589388i	0.955592	−	0.294694i	$-0.0952178\pi$
$678$		0			0
$679$	15249.0			0.861863
$680$		0			0
$681$	1194.57			0.0672191
$682$		0			0
$683$			32874.9i			1.84176i	0.389844	+	0.920881i	$0.372529\pi$
$683$			32874.9i			1.84176i	−0.389844	+	0.920881i	$0.627471\pi$
$684$		0			0
$685$	−8435.61	−	9536.89i	−0.470523	−	0.531950i
$686$		0			0
$687$		−	19029.8i		−	1.05681i
$688$		0			0
$689$	−12208.2			−0.675032
$690$		0			0
$691$	−20337.8			−1.11966			−0.559830	−	0.828607i	$-0.689134\pi$
$691$	−20337.8			−1.11966			−0.559830	+	0.828607i	$0.689134\pi$
$692$		0			0
$693$			2929.38i			0.160574i
$694$		0			0
$695$	11415.9	−	10097.6i	0.623063	−	0.551114i
$696$		0			0
$697$		−	37869.8i		−	2.05799i
$698$		0			0
$699$	21199.3			1.14711
$700$		0			0
$701$	12930.7			0.696699			0.348349	−	0.937365i	$-0.386742\pi$
$701$	12930.7			0.696699			0.348349	+	0.937365i	$0.386742\pi$
$702$		0			0
$703$			828.135i			0.0444292i
$704$		0			0
$705$	21806.3	−	19288.2i	1.16492	−	1.03040i
$706$		0			0
$707$			14359.4i			0.763846i
$708$		0			0
$709$	−7078.40			−0.374944			−0.187472	−	0.982270i	$-0.560029\pi$
$709$	−7078.40			−0.374944			−0.187472	+	0.982270i	$0.560029\pi$
$710$		0			0
$711$	15016.4			0.792066
$712$		0			0
$713$		−	14345.2i		−	0.753479i
$714$		0			0
$715$	−5458.85	−	6171.51i	−0.285524	−	0.322799i
$716$		0			0
$717$		−	35903.6i		−	1.87007i
$718$		0			0
$719$	−9201.25			−0.477258			−0.238629	−	0.971111i	$-0.576698\pi$
$719$	−9201.25			−0.477258			−0.238629	+	0.971111i	$0.576698\pi$
$720$		0			0
$721$	−9929.63			−0.512897
$722$		0			0
$723$		−	14075.5i		−	0.724027i
$724$		0			0
$725$	−2210.77	+	17971.7i	−0.113249	+	0.920622i
$726$		0			0
$727$		−	1522.05i		−	0.0776477i	−0.999246	−	0.0388239i	$-0.987639\pi$
$727$		−	1522.05i		−	0.0776477i	0.999246	−	0.0388239i	$-0.0123611\pi$
$728$		0			0
$729$	7701.47			0.391275
$730$		0			0
$731$	31937.1			1.61592
$732$		0			0
$733$		−	21264.1i		−	1.07150i	−0.844377	−	0.535749i	$-0.820029\pi$
$733$		−	21264.1i		−	1.07150i	0.844377	−	0.535749i	$-0.179971\pi$
$734$		0			0
$735$	−7935.84	−	8971.87i	−0.398255	−	0.450248i
$736$		0			0
$737$		−	5289.71i		−	0.264381i
$738$		0			0
$739$	21555.5			1.07298			0.536490	−	0.843907i	$-0.319750\pi$
$739$	21555.5			1.07298			0.536490	+	0.843907i	$0.319750\pi$
$740$		0			0
$741$	−57609.2			−2.85604
$742$		0			0
$743$		−	18974.7i		−	0.936895i	−0.883491	−	0.468447i	$-0.844814\pi$
$743$		−	18974.7i		−	0.936895i	0.883491	−	0.468447i	$-0.155186\pi$
$744$		0			0
$745$	23645.9	−	20915.3i	1.16284	−	1.02856i
$746$		0			0
$747$			18614.5i			0.911738i
$748$		0			0
$749$	−13484.9			−0.657848
$750$		0			0
$751$	−24260.6			−1.17880			−0.589402	−	0.807840i	$-0.700637\pi$
$751$	−24260.6			−1.17880			−0.589402	+	0.807840i	$0.700637\pi$
$752$		0			0
$753$		−	14240.5i		−	0.689180i
$754$		0			0
$755$	8167.41	−	7224.27i	0.393699	−	0.348236i
$756$		0			0
$757$			14128.2i			0.678333i	0.940726	+	0.339166i	$0.110145\pi$
$757$			14128.2i			0.678333i	−0.940726	+	0.339166i	$0.889855\pi$
$758$		0			0
$759$	−4694.43			−0.224502
$760$		0			0
$761$	−917.035			−0.0436826			−0.0218413	−	0.999761i	$-0.506953\pi$
$761$	−917.035			−0.0436826			−0.0218413	+	0.999761i	$0.506953\pi$
$762$		0			0
$763$		−	23679.1i		−	1.12351i
$764$		0			0
$765$	19800.1	+	22385.1i	0.935786	+	1.05795i
$766$		0			0
$767$			7467.72i			0.351556i
$768$		0			0
$769$	21609.1			1.01332			0.506661	−	0.862145i	$-0.330880\pi$
$769$	21609.1			1.01332			0.506661	+	0.862145i	$0.330880\pi$
$770$		0			0
$771$	−33435.7			−1.56181
$772$		0			0
$773$		−	16205.3i		−	0.754029i	−0.926207	−	0.377014i	$-0.876951\pi$
$773$		−	16205.3i		−	0.754029i	0.926207	−	0.377014i	$-0.123049\pi$
$774$		0			0
$775$	−3499.32	+	28446.5i	−0.162193	+	1.31849i
$776$		0			0
$777$			611.043i			0.0282124i
$778$		0			0
$779$	−34879.2			−1.60421
$780$		0			0
$781$	6629.65			0.303749
$782$		0			0
$783$		−	7381.45i		−	0.336899i
$784$		0			0
$785$	6239.98	+	7054.63i	0.283713	+	0.320752i
$786$		0			0
$787$			6106.91i			0.276605i	0.990390	+	0.138302i	$0.0441645\pi$
$787$			6106.91i			0.276605i	−0.990390	+	0.138302i	$0.955835\pi$
$788$		0			0
$789$	−34621.3			−1.56217
$790$		0			0
$791$	−15206.2			−0.683529
$792$		0			0
$793$			7836.91i			0.350942i
$794$		0			0
$795$	10409.5	−	9207.41i	0.464384	−	0.410759i
$796$		0			0
$797$			1702.01i			0.0756442i	0.999284	+	0.0378221i	$0.0120420\pi$
$797$			1702.01i			0.0756442i	−0.999284	+	0.0378221i	$0.987958\pi$
$798$		0			0
$799$	−52248.0			−2.31339
$800$		0			0
$801$	−14786.2			−0.652242
$802$		0			0
$803$		−	6843.96i		−	0.300770i
$804$		0			0
$805$	−7144.42	+	6319.41i	−0.312804	+	0.276683i
$806$		0			0
$807$			26646.1i			1.16231i
$808$		0			0
$809$	22987.7			0.999016			0.499508	−	0.866309i	$-0.333514\pi$
$809$	22987.7			0.999016			0.499508	+	0.866309i	$0.333514\pi$
$810$		0			0
$811$	−24705.7			−1.06971			−0.534855	−	0.844944i	$-0.679634\pi$
$811$	−24705.7			−1.06971			−0.534855	+	0.844944i	$0.679634\pi$
$812$		0			0
$813$		−	19436.8i		−	0.838474i
$814$		0			0
$815$	−2789.97	−	3154.21i	−0.119912	−	0.135567i
$816$		0			0
$817$		−	29415.0i		−	1.25961i
$818$		0			0
$819$	−17841.4			−0.761206
$820$		0			0
$821$	37165.6			1.57989			0.789945	−	0.613178i	$-0.210109\pi$
$821$	37165.6			1.57989			0.789945	+	0.613178i	$0.210109\pi$
$822$		0			0
$823$		−	1333.29i		−	0.0564707i	−0.999601	−	0.0282354i	$-0.991011\pi$
$823$		−	1333.29i		−	0.0564707i	0.999601	−	0.0282354i	$-0.00898879\pi$
$824$		0			0
$825$	9309.08	+	1145.15i	0.392849	+	0.0483259i
$826$		0			0
$827$		−	19441.3i		−	0.817459i	−0.912656	−	0.408729i	$-0.865972\pi$
$827$		−	19441.3i		−	0.817459i	0.912656	−	0.408729i	$-0.134028\pi$
$828$		0			0
$829$	30978.1			1.29785			0.648923	−	0.760854i	$-0.275220\pi$
$829$	30978.1			1.29785			0.648923	+	0.760854i	$0.275220\pi$
$830$		0			0
$831$	34626.0			1.44544
$832$		0			0
$833$			21496.7i			0.894137i
$834$		0			0
$835$	9621.67	+	10877.8i	0.398768	+	0.450828i
$836$		0			0
$837$		−	11683.8i		−	0.482497i
$838$		0			0
$839$	−19548.7			−0.804406			−0.402203	−	0.915551i	$-0.631755\pi$
$839$	−19548.7			−0.804406			−0.402203	+	0.915551i	$0.631755\pi$
$840$		0			0
$841$	−3405.42			−0.139629
$842$		0			0
$843$		−	19203.0i		−	0.784563i
$844$		0			0
$845$	19188.9	−	16973.1i	0.781206	−	0.690995i
$846$		0			0
$847$		−	1649.96i		−	0.0669341i
$848$		0			0
$849$	16514.0			0.667560
$850$		0			0
$851$	−411.002			−0.0165558
$852$		0			0
$853$			27242.9i			1.09353i	0.837287	+	0.546764i	$0.184140\pi$
$853$			27242.9i			1.09353i	−0.837287	+	0.546764i	$0.815860\pi$
$854$		0			0
$855$	20617.3	−	18236.5i	0.824676	−	0.729445i
$856$		0			0
$857$			16567.2i			0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$857$			16567.2i			0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$858$		0			0
$859$	−2689.42			−0.106824			−0.0534121	−	0.998573i	$-0.517010\pi$
$859$	−2689.42			−0.106824			−0.0534121	+	0.998573i	$0.517010\pi$
$860$		0			0
$861$	−25735.8			−1.01867
$862$		0			0
$863$			17196.1i			0.678289i	0.940734	+	0.339144i	$0.110138\pi$
$863$			17196.1i			0.678289i	−0.940734	+	0.339144i	$0.889862\pi$
$864$		0			0
$865$	−15306.8	−	17305.2i	−0.601673	−	0.680223i
$866$		0			0
$867$		−	94272.7i		−	3.69281i
$868$		0			0
$869$	−8457.90			−0.330167
$870$		0			0
$871$	32216.9			1.25330
$872$		0			0
$873$		−	21839.9i		−	0.846701i
$874$		0			0
$875$	15709.0	−	10788.6i	0.606925	−	0.416825i
$876$		0			0
$877$		−	43591.8i		−	1.67844i	−0.543794	−	0.839219i	$-0.683013\pi$
$877$		−	43591.8i		−	1.67844i	0.543794	−	0.839219i	$-0.316987\pi$
$878$		0			0
$879$	−16570.8			−0.635858
$880$		0			0
$881$	−28746.0			−1.09929			−0.549646	−	0.835398i	$-0.685237\pi$
$881$	−28746.0			−1.09929			−0.549646	+	0.835398i	$0.685237\pi$
$882$		0			0
$883$		−	2883.18i		−	0.109883i	−0.998490	−	0.0549415i	$-0.982503\pi$
$883$		−	2883.18i		−	0.109883i	0.998490	−	0.0549415i	$-0.0174972\pi$
$884$		0			0
$885$	−5632.13	−	6367.42i	−0.213923	−	0.241851i
$886$		0			0
$887$			10797.1i			0.408718i	0.978896	+	0.204359i	$0.0655109\pi$
$887$			10797.1i			0.408718i	−0.978896	+	0.204359i	$0.934489\pi$
$888$		0			0
$889$	21246.7			0.801565
$890$		0			0
$891$	−9623.82			−0.361852
$892$		0			0
$893$			48122.0i			1.80329i
$894$		0			0
$895$	17135.5	−	15156.7i	0.639972	−	0.566071i
$896$		0			0
$897$		−	28591.3i		−	1.06426i
$898$		0			0
$899$	33213.9			1.23220
$900$		0			0
$901$	−24941.2			−0.922209
$902$		0			0
$903$		−	21704.0i		−	0.799849i
$904$		0			0
$905$	−21450.6	+	18973.6i	−0.787891	+	0.696909i
$906$		0			0
$907$			6158.05i			0.225441i	0.993627	+	0.112720i	$0.0359564\pi$
$907$			6158.05i			0.225441i	−0.993627	+	0.112720i	$0.964044\pi$
$908$		0			0
$909$	20565.7			0.750409
$910$		0			0
$911$	−8119.84			−0.295304			−0.147652	−	0.989039i	$-0.547172\pi$
$911$	−8119.84			−0.295304			−0.147652	+	0.989039i	$0.547172\pi$
$912$		0			0
$913$		−	10484.5i		−	0.380051i
$914$		0			0
$915$	−5910.57	−	6682.21i	−0.213549	−	0.241428i
$916$		0			0
$917$		−	8462.97i		−	0.304768i
$918$		0			0
$919$	−20945.0			−0.751807			−0.375904	−	0.926659i	$-0.622668\pi$
$919$	−20945.0			−0.751807			−0.375904	+	0.926659i	$0.622668\pi$
$920$		0			0
$921$	−1111.44			−0.0397646
$922$		0			0
$923$			40377.8i			1.43993i
$924$		0			0
$925$	815.020	+	100.259i	0.0289705	+	0.00356377i
$926$		0			0
$927$			14221.4i			0.503874i
$928$		0			0
$929$	34175.1			1.20694			0.603470	−	0.797386i	$-0.293784\pi$
$929$	34175.1			1.20694			0.603470	+	0.797386i	$0.293784\pi$
$930$		0			0
$931$	19799.1			0.696980
$932$		0			0
$933$			41986.8i			1.47330i
$934$		0			0
$935$	−11152.3	−	12608.3i	−0.390075	−	0.441000i
$936$		0			0
$937$		−	41106.5i		−	1.43318i	−0.697494	−	0.716591i	$-0.745702\pi$
$937$		−	41106.5i		−	1.43318i	0.697494	−	0.716591i	$-0.254298\pi$
$938$		0			0
$939$	42057.1			1.46164
$940$		0			0
$941$	32287.7			1.11854			0.559271	−	0.828985i	$-0.311081\pi$
$941$	32287.7			1.11854			0.559271	+	0.828985i	$0.311081\pi$
$942$		0			0
$943$		−	17310.5i		−	0.597781i
$944$		0			0
$945$	−5818.94	+	5146.99i	−0.200307	+	0.177176i
$946$		0			0
$947$		−	57062.9i		−	1.95807i	−0.203683	−	0.979037i	$-0.565291\pi$
$947$		−	57062.9i		−	1.95807i	0.203683	−	0.979037i	$-0.434709\pi$
$948$		0			0
$949$	41683.0			1.42580
$950$		0			0
$951$	−21145.2			−0.721009
$952$		0			0
$953$		−	51251.6i		−	1.74208i	−0.491211	−	0.871040i	$-0.663446\pi$
$953$		−	51251.6i		−	1.74208i	0.491211	−	0.871040i	$-0.336554\pi$
$954$		0			0
$955$	29093.4	−	25733.8i	0.985801	−	0.871964i
$956$		0			0
$957$		−	10869.2i		−	0.367138i
$958$		0			0
$959$	15528.9			0.522892
$960$		0			0
$961$	22781.8			0.764720
$962$		0			0
$963$			19313.3i			0.646275i
$964$		0			0
$965$	−12901.2	−	14585.5i	−0.430368	−	0.486553i
$966$		0			0
$967$		−	23103.2i		−	0.768303i	−0.923270	−	0.384151i	$-0.874494\pi$
$967$		−	23103.2i		−	0.768303i	0.923270	−	0.384151i	$-0.125506\pi$
$968$		0			0
$969$	−117694.			−3.90184
$970$		0			0
$971$	−27006.6			−0.892566			−0.446283	−	0.894892i	$-0.647253\pi$
$971$	−27006.6			−0.892566			−0.446283	+	0.894892i	$0.647253\pi$
$972$		0			0
$973$			18588.4i			0.612453i
$974$		0			0
$975$	−6974.49	+	56696.8i	−0.229090	+	1.86231i
$976$		0			0
$977$			50695.2i			1.66006i	0.557715	+	0.830032i	$0.311678\pi$
$977$			50695.2i			1.66006i	−0.557715	+	0.830032i	$0.688322\pi$
$978$		0			0
$979$	8328.26			0.271882
$980$		0			0
$981$	−33913.5			−1.10375
$982$		0			0
$983$			34483.1i			1.11886i	0.828877	+	0.559431i	$0.188980\pi$
$983$			34483.1i			1.11886i	−0.828877	+	0.559431i	$0.811020\pi$
$984$		0			0
$985$	−832.590	−	941.286i	−0.0269325	−	0.0304486i
$986$		0			0
$987$			35507.0i			1.14509i
$988$		0			0
$989$	14598.6			0.469372
$990$		0			0
$991$	−22631.4			−0.725439			−0.362719	−	0.931898i	$-0.618152\pi$
$991$	−22631.4			−0.725439			−0.362719	+	0.931898i	$0.618152\pi$
$992$		0			0
$993$			50640.5i			1.61835i
$994$		0			0
$995$	37909.2	−	33531.6i	1.20784	−	1.06836i
$996$		0			0
$997$		−	14507.6i		−	0.460844i	−0.973091	−	0.230422i	$-0.925989\pi$
$997$		−	14507.6i		−	0.460844i	0.973091	−	0.230422i	$-0.0740107\pi$
$998$		0			0
$999$	−334.751			−0.0106016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.4.b.h.529.7		8
4.3	odd	2		110.4.b.c.89.6	yes	8
5.4	even	2	inner	880.4.b.h.529.2		8
12.11	even	2		990.4.c.i.199.4		8
20.3	even	4		550.4.a.bb.1.3		4
20.7	even	4		550.4.a.ba.1.2		4
20.19	odd	2		110.4.b.c.89.3	✓	8
60.59	even	2		990.4.c.i.199.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.4.b.c.89.3	✓	8	20.19	odd	2
110.4.b.c.89.6	yes	8	4.3	odd	2
550.4.a.ba.1.2		4	20.7	even	4
550.4.a.bb.1.3		4	20.3	even	4
880.4.b.h.529.2		8	5.4	even	2	inner
880.4.b.h.529.7		8	1.1	even	1	trivial
990.4.c.i.199.4		8	12.11	even	2
990.4.c.i.199.8		8	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.82127i q^{3} +(-8.37442 + 7.40737i) q^{5} -13.6360i q^{7} -19.5297 q^{9} +11.0000 q^{11} +66.9953i q^{13} +(-50.5277 - 57.1242i) q^{15} +136.870i q^{17} +126.061 q^{19} +93.0149 q^{21} +62.5641i q^{23} +(15.2617 - 124.065i) q^{25} +50.9568i q^{27} -144.857 q^{29} -229.288 q^{31} +75.0340i q^{33} +(101.007 + 114.194i) q^{35} +6.56930i q^{37} -456.993 q^{39} -276.684 q^{41} -233.339i q^{43} +(163.550 - 144.664i) q^{45} +381.735i q^{47} +157.059 q^{49} -933.627 q^{51} +182.225i q^{53} +(-92.1186 + 81.4811i) q^{55} +859.898i q^{57} +111.466 q^{59} +116.977 q^{61} +266.307i q^{63} +(-496.259 - 561.047i) q^{65} -480.882i q^{67} -426.766 q^{69} +602.696 q^{71} -622.178i q^{73} +(846.280 + 104.104i) q^{75} -149.996i q^{77} -768.900 q^{79} -874.892 q^{81} -953.137i q^{83} +(-1013.85 - 1146.21i) q^{85} -988.109i q^{87} +757.114 q^{89} +913.549 q^{91} -1564.03i q^{93} +(-1055.69 + 933.783i) q^{95} +1118.29i q^{97} -214.827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 110 q^{9} + 88 q^{11} - 8 q^{15} + 302 q^{19} + 230 q^{21} - 162 q^{25} - 58 q^{29} - 1022 q^{31} + 1058 q^{35} + 320 q^{39} + 452 q^{41} - 622 q^{45} + 222 q^{49} - 834 q^{51} + 176 q^{55}+ \cdots - 1210 q^{99}+O(q^{100}) \) 8 * q + 16 * q^5 - 110 * q^9 + 88 * q^11 - 8 * q^15 + 302 * q^19 + 230 * q^21 - 162 * q^25 - 58 * q^29 - 1022 * q^31 + 1058 * q^35 + 320 * q^39 + 452 * q^41 - 622 * q^45 + 222 * q^49 - 834 * q^51 + 176 * q^55 - 268 * q^59 - 490 * q^61 - 1408 * q^65 + 2820 * q^69 - 2034 * q^71 + 2110 * q^75 - 272 * q^79 - 2488 * q^81 + 330 * q^85 + 5782 * q^89 + 3520 * q^91 - 1268 * q^95 - 1210 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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