LMFDB - Embedded newform 880.3.i.d.769.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,3,Mod(769,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	880.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.9782632637$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{-21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 41x^{2} - 40x + 505 $ x^4 - 2x^3 + 41x^2 - 40*x + 505
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$-0.618034 + 4.58258i$ of defining polynomial
Character	$\chi$	$=$	880.769
Dual form			880.3.i.d.769.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.58258i q^{3} +(-2.00000 - 4.58258i) q^{5} -11.1803 q^{7} -12.0000 q^{9} +(4.00000 + 10.2470i) q^{11} +8.94427 q^{13} +(-21.0000 + 9.16515i) q^{15} -15.6525 q^{17} +10.2470i q^{19} +51.2348i q^{21} -27.4955i q^{23} +(-17.0000 + 18.3303i) q^{25} +13.7477i q^{27} +10.2470i q^{29} +3.00000 q^{31} +(46.9574 - 18.3303i) q^{33} +(22.3607 + 51.2348i) q^{35} +4.58258i q^{37} -40.9878i q^{39} +20.4939i q^{41} -22.3607 q^{43} +(24.0000 + 54.9909i) q^{45} +64.1561i q^{47} +76.0000 q^{49} +71.7287i q^{51} +4.58258i q^{53} +(38.9574 - 38.8242i) q^{55} +46.9574 q^{57} +18.0000 q^{59} -71.7287i q^{61} +134.164 q^{63} +(-17.8885 - 40.9878i) q^{65} -27.4955i q^{67} -126.000 q^{69} -27.0000 q^{71} +58.1378 q^{73} +(84.0000 + 77.9038i) q^{75} +(-44.7214 - 114.564i) q^{77} -61.4817i q^{79} -45.0000 q^{81} +71.5542 q^{83} +(31.3050 + 71.7287i) q^{85} +46.9574 q^{87} +37.0000 q^{89} -100.000 q^{91} -13.7477i q^{93} +(46.9574 - 20.4939i) q^{95} +119.147i q^{97} +(-48.0000 - 122.963i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} - 48 q^{9} + 16 q^{11} - 84 q^{15} - 68 q^{25} + 12 q^{31} + 96 q^{45} + 304 q^{49} - 32 q^{55} + 72 q^{59} - 504 q^{69} - 108 q^{71} + 336 q^{75} - 180 q^{81} + 148 q^{89} - 400 q^{91} - 192 q^{99}+O(q^{100}) $ 4 * q - 8 * q^5 - 48 * q^9 + 16 * q^11 - 84 * q^15 - 68 * q^25 + 12 * q^31 + 96 * q^45 + 304 * q^49 - 32 * q^55 + 72 * q^59 - 504 * q^69 - 108 * q^71 + 336 * q^75 - 180 * q^81 + 148 * q^89 - 400 * q^91 - 192 * 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
$3$		−	4.58258i		−	1.52753i	−0.645497	−	0.763763i	$-0.723350\pi$
$3$		−	4.58258i		−	1.52753i	0.645497	−	0.763763i	$-0.276650\pi$
$4$		0			0
$5$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$5$	−2.00000	−	4.58258i	−0.400000	−	0.916515i
$6$		0			0
$7$	−11.1803			−1.59719			−0.798596	−	0.601868i	$-0.794423\pi$
$7$	−11.1803			−1.59719			−0.798596	+	0.601868i	$0.794423\pi$
$8$		0			0
$9$	−12.0000			−1.33333
$10$		0			0
$11$	4.00000	+	10.2470i	0.363636	+	0.931541i
$11$	4.00000	+	10.2470i	0.363636	+	0.931541i
$12$		0			0
$13$	8.94427			0.688021			0.344010	−	0.938966i	$-0.388214\pi$
$13$	8.94427			0.688021			0.344010	+	0.938966i	$0.388214\pi$
$14$		0			0
$15$	−21.0000	+	9.16515i	−1.40000	+	0.611010i
$16$		0			0
$17$	−15.6525			−0.920734			−0.460367	−	0.887729i	$-0.652282\pi$
$17$	−15.6525			−0.920734			−0.460367	+	0.887729i	$0.652282\pi$
$18$		0			0
$19$			10.2470i			0.539313i	0.962957	+	0.269657i	$0.0869102\pi$
$19$			10.2470i			0.539313i	−0.962957	+	0.269657i	$0.913090\pi$
$20$		0			0
$21$			51.2348i			2.43975i
$22$		0			0
$23$		−	27.4955i		−	1.19545i	−0.801700	−	0.597727i	$-0.796071\pi$
$23$		−	27.4955i		−	1.19545i	0.801700	−	0.597727i	$-0.203929\pi$
$24$		0			0
$25$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$26$		0			0
$27$			13.7477i			0.509175i
$28$		0			0
$29$			10.2470i			0.353343i	0.984270	+	0.176672i	$0.0565330\pi$
$29$			10.2470i			0.353343i	−0.984270	+	0.176672i	$0.943467\pi$
$30$		0			0
$31$	3.00000			0.0967742			0.0483871	−	0.998829i	$-0.484592\pi$
$31$	3.00000			0.0967742			0.0483871	+	0.998829i	$0.484592\pi$
$32$		0			0
$33$	46.9574	−	18.3303i	1.42295	−	0.555464i
$34$		0			0
$35$	22.3607	+	51.2348i	0.638877	+	1.46385i
$36$		0			0
$37$			4.58258i			0.123853i	0.998081	+	0.0619267i	$0.0197245\pi$
$37$			4.58258i			0.123853i	−0.998081	+	0.0619267i	$0.980275\pi$
$38$		0			0
$39$		−	40.9878i		−	1.05097i
$40$		0			0
$41$			20.4939i			0.499851i	0.968265	+	0.249926i	$0.0804062\pi$
$41$			20.4939i			0.499851i	−0.968265	+	0.249926i	$0.919594\pi$
$42$		0			0
$43$	−22.3607			−0.520016			−0.260008	−	0.965606i	$-0.583725\pi$
$43$	−22.3607			−0.520016			−0.260008	+	0.965606i	$0.583725\pi$
$44$		0			0
$45$	24.0000	+	54.9909i	0.533333	+	1.22202i
$46$		0			0
$47$			64.1561i			1.36502i	0.730875	+	0.682511i	$0.239112\pi$
$47$			64.1561i			1.36502i	−0.730875	+	0.682511i	$0.760888\pi$
$48$		0			0
$49$	76.0000			1.55102
$50$		0			0
$51$			71.7287i			1.40644i
$52$		0			0
$53$			4.58258i			0.0864637i	0.999065	+	0.0432318i	$0.0137654\pi$
$53$			4.58258i			0.0864637i	−0.999065	+	0.0432318i	$0.986235\pi$
$54$		0			0
$55$	38.9574	−	38.8242i	0.708317	−	0.705895i
$56$		0			0
$57$	46.9574			0.823815
$58$		0			0
$59$	18.0000			0.305085			0.152542	−	0.988297i	$-0.451254\pi$
$59$	18.0000			0.305085			0.152542	+	0.988297i	$0.451254\pi$
$60$		0			0
$61$		−	71.7287i		−	1.17588i	−0.808905	−	0.587940i	$-0.799939\pi$
$61$		−	71.7287i		−	1.17588i	0.808905	−	0.587940i	$-0.200061\pi$
$62$		0			0
$63$	134.164			2.12959
$64$		0			0
$65$	−17.8885	−	40.9878i	−0.275208	−	0.630582i
$66$		0			0
$67$		−	27.4955i		−	0.410380i	−0.978722	−	0.205190i	$-0.934219\pi$
$67$		−	27.4955i		−	0.410380i	0.978722	−	0.205190i	$-0.0657812\pi$
$68$		0			0
$69$	−126.000			−1.82609
$70$		0			0
$71$	−27.0000			−0.380282			−0.190141	−	0.981757i	$-0.560894\pi$
$71$	−27.0000			−0.380282			−0.190141	+	0.981757i	$0.560894\pi$
$72$		0			0
$73$	58.1378			0.796408			0.398204	−	0.917297i	$-0.369634\pi$
$73$	58.1378			0.796408			0.398204	+	0.917297i	$0.369634\pi$
$74$		0			0
$75$	84.0000	+	77.9038i	1.12000	+	1.03872i
$76$		0			0
$77$	−44.7214	−	114.564i	−0.580797	−	1.48785i
$78$		0			0
$79$		−	61.4817i		−	0.778249i	−0.921185	−	0.389125i	$-0.872778\pi$
$79$		−	61.4817i		−	0.778249i	0.921185	−	0.389125i	$-0.127222\pi$
$80$		0			0
$81$	−45.0000			−0.555556
$82$		0			0
$83$	71.5542			0.862098			0.431049	−	0.902328i	$-0.358144\pi$
$83$	71.5542			0.862098			0.431049	+	0.902328i	$0.358144\pi$
$84$		0			0
$85$	31.3050	+	71.7287i	0.368294	+	0.843867i
$86$		0			0
$87$	46.9574			0.539741
$88$		0			0
$89$	37.0000			0.415730			0.207865	−	0.978157i	$-0.433348\pi$
$89$	37.0000			0.415730			0.207865	+	0.978157i	$0.433348\pi$
$90$		0			0
$91$	−100.000			−1.09890
$92$		0			0
$93$		−	13.7477i		−	0.147825i
$94$		0			0
$95$	46.9574	−	20.4939i	0.494289	−	0.215725i
$96$		0			0
$97$			119.147i			1.22832i	0.789182	+	0.614160i	$0.210505\pi$
$97$			119.147i			1.22832i	−0.789182	+	0.614160i	$0.789495\pi$
$98$		0			0
$99$	−48.0000	−	122.963i	−0.484848	−	1.24205i
$100$		0			0
$101$		−	102.470i		−	1.01455i	−0.861784	−	0.507275i	$-0.830653\pi$
$101$		−	102.470i		−	1.01455i	0.861784	−	0.507275i	$-0.169347\pi$
$102$		0			0
$103$			18.3303i			0.177964i	0.996033	+	0.0889821i	$0.0283614\pi$
$103$			18.3303i			0.177964i	−0.996033	+	0.0889821i	$0.971639\pi$
$104$		0			0
$105$	234.787	−	102.470i	2.23607	−	0.975900i
$106$		0			0
$107$	−156.525			−1.46285			−0.731424	−	0.681923i	$-0.761144\pi$
$107$	−156.525			−1.46285			−0.731424	+	0.681923i	$0.761144\pi$
$108$		0			0
$109$			20.4939i			0.188017i	0.995571	+	0.0940087i	$0.0299682\pi$
$109$			20.4939i			0.188017i	−0.995571	+	0.0940087i	$0.970032\pi$
$110$		0			0
$111$	21.0000			0.189189
$112$		0			0
$113$			164.973i			1.45994i	0.683482	+	0.729968i	$0.260465\pi$
$113$			164.973i			1.45994i	−0.683482	+	0.729968i	$0.739535\pi$
$114$		0			0
$115$	−126.000	+	54.9909i	−1.09565	+	0.478182i
$116$		0			0
$117$	−107.331			−0.917361
$118$		0			0
$119$	175.000			1.47059
$120$		0			0
$121$	−89.0000	+	81.9756i	−0.735537	+	0.677484i
$122$		0			0
$123$	93.9149			0.763535
$124$		0			0
$125$	118.000	+	41.2432i	0.944000	+	0.329945i
$126$		0			0
$127$	31.3050			0.246496			0.123248	−	0.992376i	$-0.460669\pi$
$127$	31.3050			0.246496			0.123248	+	0.992376i	$0.460669\pi$
$128$		0			0
$129$			102.470i			0.794337i
$130$		0			0
$131$			215.186i			1.64264i	0.570467	+	0.821320i	$0.306762\pi$
$131$			215.186i			1.64264i	−0.570467	+	0.821320i	$0.693238\pi$
$132$		0			0
$133$		−	114.564i		−	0.861386i
$134$		0			0
$135$	63.0000	−	27.4955i	0.466667	−	0.203670i
$136$		0			0
$137$		−	64.1561i		−	0.468292i	−0.972201	−	0.234146i	$-0.924771\pi$
$137$		−	64.1561i		−	0.468292i	0.972201	−	0.234146i	$-0.0752294\pi$
$138$		0			0
$139$			225.433i			1.62182i	0.585171	+	0.810910i	$0.301027\pi$
$139$			225.433i			1.62182i	−0.585171	+	0.810910i	$0.698973\pi$
$140$		0			0
$141$	294.000			2.08511
$142$		0			0
$143$	35.7771	+	91.6515i	0.250189	+	0.640920i
$144$		0			0
$145$	46.9574	−	20.4939i	0.323844	−	0.141337i
$146$		0			0
$147$		−	348.276i		−	2.36922i
$148$		0			0
$149$			153.704i			1.03157i	0.856717	+	0.515786i	$0.172500\pi$
$149$			153.704i			1.03157i	−0.856717	+	0.515786i	$0.827500\pi$
$150$		0			0
$151$		−	102.470i		−	0.678606i	−0.940677	−	0.339303i	$-0.889809\pi$
$151$		−	102.470i		−	0.678606i	0.940677	−	0.339303i	$-0.110191\pi$
$152$		0			0
$153$	187.830			1.22765
$154$		0			0
$155$	−6.00000	−	13.7477i	−0.0387097	−	0.0886950i
$156$		0			0
$157$			50.4083i			0.321072i	0.987030	+	0.160536i	$0.0513223\pi$
$157$			50.4083i			0.321072i	−0.987030	+	0.160536i	$0.948678\pi$
$158$		0			0
$159$	21.0000			0.132075
$160$		0			0
$161$			307.409i			1.90937i
$162$		0			0
$163$			270.372i			1.65872i	0.558712	+	0.829362i	$0.311296\pi$
$163$			270.372i			1.65872i	−0.558712	+	0.829362i	$0.688704\pi$
$164$		0			0
$165$	−177.915	−	178.525i	−1.07827	−	1.08197i
$166$		0			0
$167$	−297.397			−1.78082			−0.890410	−	0.455159i	$-0.849583\pi$
$167$	−297.397			−1.78082			−0.890410	+	0.455159i	$0.849583\pi$
$168$		0			0
$169$	−89.0000			−0.526627
$170$		0			0
$171$		−	122.963i		−	0.719084i
$172$		0			0
$173$	−143.108			−0.827216			−0.413608	−	0.910455i	$-0.635732\pi$
$173$	−143.108			−0.827216			−0.413608	+	0.910455i	$0.635732\pi$
$174$		0			0
$175$	190.066	−	204.939i	1.08609	−	1.17108i
$176$		0			0
$177$		−	82.4864i		−	0.466025i
$178$		0			0
$179$	−242.000			−1.35196			−0.675978	−	0.736922i	$-0.736278\pi$
$179$	−242.000			−1.35196			−0.675978	+	0.736922i	$0.736278\pi$
$180$		0			0
$181$	−258.000			−1.42541			−0.712707	−	0.701462i	$-0.752531\pi$
$181$	−258.000			−1.42541			−0.712707	+	0.701462i	$0.752531\pi$
$182$		0			0
$183$	−328.702			−1.79619
$184$		0			0
$185$	21.0000	−	9.16515i	0.113514	−	0.0495414i
$186$		0			0
$187$	−62.6099	−	160.390i	−0.334812	−	0.857701i
$188$		0			0
$189$		−	153.704i		−	0.813250i
$190$		0			0
$191$	278.000			1.45550			0.727749	−	0.685844i	$-0.240567\pi$
$191$	278.000			1.45550			0.727749	+	0.685844i	$0.240567\pi$
$192$		0			0
$193$	−78.2624			−0.405505			−0.202752	−	0.979230i	$-0.564989\pi$
$193$	−78.2624			−0.405505			−0.202752	+	0.979230i	$0.564989\pi$
$194$		0			0
$195$	−187.830	+	81.9756i	−0.963229	+	0.420388i
$196$		0			0
$197$	−196.774			−0.998853			−0.499426	−	0.866356i	$-0.666456\pi$
$197$	−196.774			−0.998853			−0.499426	+	0.866356i	$0.666456\pi$
$198$		0			0
$199$	53.0000			0.266332			0.133166	−	0.991094i	$-0.457486\pi$
$199$	53.0000			0.266332			0.133166	+	0.991094i	$0.457486\pi$
$200$		0			0
$201$	−126.000			−0.626866
$202$		0			0
$203$		−	114.564i		−	0.564357i
$204$		0			0
$205$	93.9149	−	40.9878i	0.458121	−	0.199941i
$206$		0			0
$207$			329.945i			1.59394i
$208$		0			0
$209$	−105.000	+	40.9878i	−0.502392	+	0.196114i
$210$		0			0
$211$			174.198i			0.825584i	0.910825	+	0.412792i	$0.135446\pi$
$211$			174.198i			0.825584i	−0.910825	+	0.412792i	$0.864554\pi$
$212$		0			0
$213$			123.730i			0.580890i
$214$		0			0
$215$	44.7214	+	102.470i	0.208006	+	0.476602i
$216$		0			0
$217$	−33.5410			−0.154567
$218$		0			0
$219$		−	266.421i		−	1.21653i
$220$		0			0
$221$	−140.000			−0.633484
$222$		0			0
$223$		−	394.102i		−	1.76727i	−0.468175	−	0.883636i	$-0.655088\pi$
$223$		−	394.102i		−	1.76727i	0.468175	−	0.883636i	$-0.344912\pi$
$224$		0			0
$225$	204.000	−	219.964i	0.906667	−	0.977616i
$226$		0			0
$227$	277.272			1.22146			0.610732	−	0.791837i	$-0.290875\pi$
$227$	277.272			1.22146			0.610732	+	0.791837i	$0.290875\pi$
$228$		0			0
$229$	292.000			1.27511			0.637555	−	0.770405i	$-0.279946\pi$
$229$	292.000			1.27511			0.637555	+	0.770405i	$0.279946\pi$
$230$		0			0
$231$	−525.000	+	204.939i	−2.27273	+	0.887182i
$232$		0			0
$233$	−172.177			−0.738958			−0.369479	−	0.929239i	$-0.620464\pi$
$233$	−172.177			−0.738958			−0.369479	+	0.929239i	$0.620464\pi$
$234$		0			0
$235$	294.000	−	128.312i	1.25106	−	0.546009i
$236$		0			0
$237$	−281.745			−1.18880
$238$		0			0
$239$			225.433i			0.943234i	0.881804	+	0.471617i	$0.156329\pi$
$239$			225.433i			0.943234i	−0.881804	+	0.471617i	$0.843671\pi$
$240$		0			0
$241$			327.902i			1.36059i	0.732938	+	0.680295i	$0.238149\pi$
$241$			327.902i			1.36059i	−0.732938	+	0.680295i	$0.761851\pi$
$242$		0			0
$243$			329.945i			1.35780i
$244$		0			0
$245$	−152.000	−	348.276i	−0.620408	−	1.42153i
$246$		0			0
$247$			91.6515i			0.371059i
$248$		0			0
$249$		−	327.902i		−	1.31688i
$250$		0			0
$251$	−282.000			−1.12351			−0.561753	−	0.827305i	$-0.689873\pi$
$251$	−282.000			−1.12351			−0.561753	+	0.827305i	$0.689873\pi$
$252$		0			0
$253$	281.745	−	109.982i	1.11361	−	0.434711i
$254$		0			0
$255$	328.702	−	143.457i	1.28903	−	0.562578i
$256$		0			0
$257$		−	64.1561i		−	0.249634i	−0.992180	−	0.124817i	$-0.960166\pi$
$257$		−	64.1561i		−	0.249634i	0.992180	−	0.124817i	$-0.0398345\pi$
$258$		0			0
$259$		−	51.2348i		−	0.197818i
$260$		0			0
$261$		−	122.963i		−	0.471124i
$262$		0			0
$263$	−199.010			−0.756692			−0.378346	−	0.925664i	$-0.623507\pi$
$263$	−199.010			−0.756692			−0.378346	+	0.925664i	$0.623507\pi$
$264$		0			0
$265$	21.0000	−	9.16515i	0.0792453	−	0.0345855i
$266$		0			0
$267$		−	169.555i		−	0.635039i
$268$		0			0
$269$	162.000			0.602230			0.301115	−	0.953588i	$-0.402641\pi$
$269$	162.000			0.602230			0.301115	+	0.953588i	$0.402641\pi$
$270$		0			0
$271$			245.927i			0.907479i	0.891134	+	0.453740i	$0.149910\pi$
$271$			245.927i			0.907479i	−0.891134	+	0.453740i	$0.850090\pi$
$272$		0			0
$273$			458.258i			1.67860i
$274$		0			0
$275$	−255.830	−	100.877i	−0.930290	−	0.366825i
$276$		0			0
$277$	58.1378			0.209884			0.104942	−	0.994478i	$-0.466534\pi$
$277$	58.1378			0.209884			0.104942	+	0.994478i	$0.466534\pi$
$278$		0			0
$279$	−36.0000			−0.129032
$280$		0			0
$281$		−	163.951i		−	0.583456i	−0.956501	−	0.291728i	$-0.905770\pi$
$281$		−	163.951i		−	0.583456i	0.956501	−	0.291728i	$-0.0942303\pi$
$282$		0			0
$283$	250.440			0.884946			0.442473	−	0.896782i	$-0.354101\pi$
$283$	250.440			0.884946			0.442473	+	0.896782i	$0.354101\pi$
$284$		0			0
$285$	−93.9149	−	215.186i	−0.329526	−	0.755038i
$286$		0			0
$287$		−	229.129i		−	0.798358i
$288$		0			0
$289$	−44.0000			−0.152249
$290$		0			0
$291$	546.000			1.87629
$292$		0			0
$293$	−447.214			−1.52633			−0.763163	−	0.646206i	$-0.776355\pi$
$293$	−447.214			−1.52633			−0.763163	+	0.646206i	$0.776355\pi$
$294$		0			0
$295$	−36.0000	−	82.4864i	−0.122034	−	0.279615i
$296$		0			0
$297$	−140.872	+	54.9909i	−0.474317	+	0.185155i
$298$		0			0
$299$		−	245.927i		−	0.822498i
$300$		0			0
$301$	250.000			0.830565
$302$		0			0
$303$	−469.574			−1.54975
$304$		0			0
$305$	−328.702	+	143.457i	−1.07771	+	0.470352i
$306$		0			0
$307$	−447.214			−1.45672			−0.728361	−	0.685194i	$-0.759718\pi$
$307$	−447.214			−1.45672			−0.728361	+	0.685194i	$0.759718\pi$
$308$		0			0
$309$	84.0000			0.271845
$310$		0			0
$311$	−87.0000			−0.279743			−0.139871	−	0.990170i	$-0.544669\pi$
$311$	−87.0000			−0.279743			−0.139871	+	0.990170i	$0.544669\pi$
$312$		0			0
$313$		−	293.285i		−	0.937012i	−0.883460	−	0.468506i	$-0.844792\pi$
$313$		−	293.285i		−	0.937012i	0.883460	−	0.468506i	$-0.155208\pi$
$314$		0			0
$315$	−268.328	−	614.817i	−0.851835	−	1.95180i
$316$		0			0
$317$			50.4083i			0.159017i	0.996834	+	0.0795084i	$0.0253351\pi$
$317$			50.4083i			0.159017i	−0.996834	+	0.0795084i	$0.974665\pi$
$318$		0			0
$319$	−105.000	+	40.9878i	−0.329154	+	0.128488i
$320$		0			0
$321$			717.287i			2.23454i
$322$		0			0
$323$		−	160.390i		−	0.496564i
$324$		0			0
$325$	−152.053	+	163.951i	−0.467854	+	0.504465i
$326$		0			0
$327$	93.9149			0.287201
$328$		0			0
$329$		−	717.287i		−	2.18020i
$330$		0			0
$331$	138.000			0.416918			0.208459	−	0.978031i	$-0.433155\pi$
$331$	138.000			0.416918			0.208459	+	0.978031i	$0.433155\pi$
$332$		0			0
$333$		−	54.9909i		−	0.165138i
$334$		0			0
$335$	−126.000	+	54.9909i	−0.376119	+	0.164152i
$336$		0			0
$337$	243.731			0.723239			0.361619	−	0.932326i	$-0.382224\pi$
$337$	243.731			0.723239			0.361619	+	0.932326i	$0.382224\pi$
$338$		0			0
$339$	756.000			2.23009
$340$		0			0
$341$	12.0000	+	30.7409i	0.0351906	+	0.0901491i
$342$		0			0
$343$	−301.869			−0.880085
$344$		0			0
$345$	252.000	+	577.405i	0.730435	+	1.67364i
$346$		0			0
$347$	−4.47214			−0.0128880			−0.00644400	−	0.999979i	$-0.502051\pi$
$347$	−4.47214			−0.0128880			−0.00644400	+	0.999979i	$0.502051\pi$
$348$		0			0
$349$		−	512.348i		−	1.46804i	−0.679125	−	0.734022i	$-0.737641\pi$
$349$		−	512.348i		−	1.46804i	0.679125	−	0.734022i	$-0.262359\pi$
$350$		0			0
$351$			122.963i			0.350323i
$352$		0			0
$353$		−	293.285i		−	0.830835i	−0.909631	−	0.415418i	$-0.863635\pi$
$353$		−	293.285i		−	0.830835i	0.909631	−	0.415418i	$-0.136365\pi$
$354$		0			0
$355$	54.0000	+	123.730i	0.152113	+	0.348534i
$356$		0			0
$357$		−	801.951i		−	2.24636i
$358$		0			0
$359$		−	635.311i		−	1.76967i	−0.465906	−	0.884834i	$-0.654272\pi$
$359$		−	635.311i		−	1.76967i	0.465906	−	0.884834i	$-0.345728\pi$
$360$		0			0
$361$	256.000			0.709141
$362$		0			0
$363$	375.659	+	407.849i	1.03487	+	1.12355i
$364$		0			0
$365$	−116.276	−	266.421i	−0.318563	−	0.729920i
$366$		0			0
$367$			430.762i			1.17374i	0.809682	+	0.586869i	$0.199640\pi$
$367$			430.762i			1.17374i	−0.809682	+	0.586869i	$0.800360\pi$
$368$		0			0
$369$		−	245.927i		−	0.666468i
$370$		0			0
$371$		−	51.2348i		−	0.138099i
$372$		0			0
$373$	−545.601			−1.46274			−0.731368	−	0.681983i	$-0.761118\pi$
$373$	−545.601			−1.46274			−0.731368	+	0.681983i	$0.761118\pi$
$374$		0			0
$375$	189.000	−	540.744i	0.504000	−	1.44198i
$376$		0			0
$377$			91.6515i			0.243107i
$378$		0			0
$379$	428.000			1.12929			0.564644	−	0.825335i	$-0.309014\pi$
$379$	428.000			1.12929			0.564644	+	0.825335i	$0.309014\pi$
$380$		0			0
$381$		−	143.457i		−	0.376528i
$382$		0			0
$383$		−	164.973i		−	0.430738i	−0.976533	−	0.215369i	$-0.930905\pi$
$383$		−	164.973i		−	0.430738i	0.976533	−	0.215369i	$-0.0690955\pi$
$384$		0			0
$385$	−435.557	+	434.068i	−1.13132	+	1.12745i
$386$		0			0
$387$	268.328			0.693354
$388$		0			0
$389$	−228.000			−0.586118			−0.293059	−	0.956094i	$-0.594673\pi$
$389$	−228.000			−0.586118			−0.293059	+	0.956094i	$0.594673\pi$
$390$		0			0
$391$			430.372i			1.10070i
$392$		0			0
$393$	986.106			2.50918
$394$		0			0
$395$	−281.745	+	122.963i	−0.713277	+	0.311300i
$396$		0			0
$397$			669.056i			1.68528i	0.538478	+	0.842640i	$0.319000\pi$
$397$			669.056i			1.68528i	−0.538478	+	0.842640i	$0.681000\pi$
$398$		0			0
$399$	−525.000			−1.31579
$400$		0			0
$401$	107.000			0.266833			0.133416	−	0.991060i	$-0.457405\pi$
$401$	107.000			0.266833			0.133416	+	0.991060i	$0.457405\pi$
$402$		0			0
$403$	26.8328			0.0665827
$404$		0			0
$405$	90.0000	+	206.216i	0.222222	+	0.509175i
$406$		0			0
$407$	−46.9574	+	18.3303i	−0.115375	+	0.0450376i
$408$		0			0
$409$			327.902i			0.801717i	0.916140	+	0.400859i	$0.131288\pi$
$409$			327.902i			0.801717i	−0.916140	+	0.400859i	$0.868712\pi$
$410$		0			0
$411$	−294.000			−0.715328
$412$		0			0
$413$	−201.246			−0.487279
$414$		0			0
$415$	−143.108	−	327.902i	−0.344839	−	0.790126i
$416$		0			0
$417$	1033.06			2.47737
$418$		0			0
$419$	−722.000			−1.72315			−0.861575	−	0.507630i	$-0.830522\pi$
$419$	−722.000			−1.72315			−0.861575	+	0.507630i	$0.830522\pi$
$420$		0			0
$421$	−408.000			−0.969121			−0.484561	−	0.874758i	$-0.661020\pi$
$421$	−408.000			−0.969121			−0.484561	+	0.874758i	$0.661020\pi$
$422$		0			0
$423$		−	769.873i		−	1.82003i
$424$		0			0
$425$	266.092	−	286.915i	0.626099	−	0.675093i
$426$		0			0
$427$			801.951i			1.87810i
$428$		0			0
$429$	420.000	−	163.951i	0.979021	−	0.382171i
$430$		0			0
$431$			61.4817i			0.142649i	0.997453	+	0.0713245i	$0.0227226\pi$
$431$			61.4817i			0.142649i	−0.997453	+	0.0713245i	$0.977277\pi$
$432$		0			0
$433$		−	247.459i		−	0.571499i	−0.958304	−	0.285750i	$-0.907757\pi$
$433$		−	247.459i		−	0.571499i	0.958304	−	0.285750i	$-0.0922425\pi$
$434$		0			0
$435$	−93.9149	−	215.186i	−0.215896	−	0.494680i
$436$		0			0
$437$	281.745			0.644724
$438$		0			0
$439$			430.372i			0.980346i	0.871625	+	0.490173i	$0.163066\pi$
$439$			430.372i			0.980346i	−0.871625	+	0.490173i	$0.836934\pi$
$440$		0			0
$441$	−912.000			−2.06803
$442$		0			0
$443$		−	302.450i		−	0.682731i	−0.939931	−	0.341366i	$-0.889111\pi$
$443$		−	302.450i		−	0.682731i	0.939931	−	0.341366i	$-0.110889\pi$
$444$		0			0
$445$	−74.0000	−	169.555i	−0.166292	−	0.381023i
$446$		0			0
$447$	704.361			1.57575
$448$		0			0
$449$	22.0000			0.0489978			0.0244989	−	0.999700i	$-0.492201\pi$
$449$	22.0000			0.0489978			0.0244989	+	0.999700i	$0.492201\pi$
$450$		0			0
$451$	−210.000	+	81.9756i	−0.465632	+	0.181764i
$452$		0			0
$453$	−469.574			−1.03659
$454$		0			0
$455$	200.000	+	458.258i	0.439560	+	1.00716i
$456$		0			0
$457$	346.591			0.758404			0.379202	−	0.925314i	$-0.376199\pi$
$457$	346.591			0.758404			0.379202	+	0.925314i	$0.376199\pi$
$458$		0			0
$459$		−	215.186i		−	0.468815i
$460$		0			0
$461$			338.149i			0.733513i	0.930317	+	0.366756i	$0.119532\pi$
$461$			338.149i			0.733513i	−0.930317	+	0.366756i	$0.880468\pi$
$462$		0			0
$463$		−	623.230i		−	1.34607i	−0.739611	−	0.673035i	$-0.764990\pi$
$463$		−	623.230i		−	1.34607i	0.739611	−	0.673035i	$-0.235010\pi$
$464$		0			0
$465$	−63.0000	+	27.4955i	−0.135484	+	0.0591300i
$466$		0			0
$467$		−	554.492i		−	1.18735i	−0.804706	−	0.593674i	$-0.797677\pi$
$467$		−	554.492i		−	1.18735i	0.804706	−	0.593674i	$-0.202323\pi$
$468$		0			0
$469$			307.409i			0.655455i
$470$		0			0
$471$	231.000			0.490446
$472$		0			0
$473$	−89.4427	−	229.129i	−0.189097	−	0.484416i
$474$		0			0
$475$	−187.830	−	174.198i	−0.395431	−	0.366733i
$476$		0			0
$477$		−	54.9909i		−	0.115285i
$478$		0			0
$479$			40.9878i			0.0855695i	0.999084	+	0.0427848i	$0.0136230\pi$
$479$			40.9878i			0.0855695i	−0.999084	+	0.0427848i	$0.986377\pi$
$480$		0			0
$481$			40.9878i			0.0852137i
$482$		0			0
$483$	1408.72			2.91661
$484$		0			0
$485$	546.000	−	238.294i	1.12577	−	0.491328i
$486$		0			0
$487$			339.111i			0.696326i	0.937434	+	0.348163i	$0.113194\pi$
$487$			339.111i			0.696326i	−0.937434	+	0.348163i	$0.886806\pi$
$488$		0			0
$489$	1239.00			2.53374
$490$		0			0
$491$		−	584.076i		−	1.18956i	−0.803887	−	0.594782i	$-0.797238\pi$
$491$		−	584.076i		−	1.18956i	0.803887	−	0.594782i	$-0.202762\pi$
$492$		0			0
$493$		−	160.390i		−	0.325335i
$494$		0			0
$495$	−467.489	+	465.890i	−0.944422	+	0.941193i
$496$		0			0
$497$	301.869			0.607383
$498$		0			0
$499$	268.000			0.537074			0.268537	−	0.963269i	$-0.413460\pi$
$499$	268.000			0.537074			0.268537	+	0.963269i	$0.413460\pi$
$500$		0			0
$501$			1362.84i			2.72025i
$502$		0			0
$503$	−541.128			−1.07580			−0.537901	−	0.843008i	$-0.680783\pi$
$503$	−541.128			−1.07580			−0.537901	+	0.843008i	$0.680783\pi$
$504$		0			0
$505$	−469.574	+	204.939i	−0.929850	+	0.405820i
$506$		0			0
$507$			407.849i			0.804436i
$508$		0			0
$509$	−828.000			−1.62672			−0.813360	−	0.581761i	$-0.802364\pi$
$509$	−828.000			−1.62672			−0.813360	+	0.581761i	$0.802364\pi$
$510$		0			0
$511$	−650.000			−1.27202
$512$		0			0
$513$	−140.872			−0.274605
$514$		0			0
$515$	84.0000	−	36.6606i	0.163107	−	0.0711856i
$516$		0			0
$517$	−657.404	+	256.624i	−1.27157	+	0.496372i
$518$		0			0
$519$			655.805i			1.26359i
$520$		0			0
$521$	242.000			0.464491			0.232246	−	0.972657i	$-0.425393\pi$
$521$	242.000			0.464491			0.232246	+	0.972657i	$0.425393\pi$
$522$		0			0
$523$	143.108			0.273630			0.136815	−	0.990597i	$-0.456313\pi$
$523$	143.108			0.273630			0.136815	+	0.990597i	$0.456313\pi$
$524$		0			0
$525$	−939.149	−	870.991i	−1.78885	−	1.65903i
$526$		0			0
$527$	−46.9574			−0.0891033
$528$		0			0
$529$	−227.000			−0.429112
$530$		0			0
$531$	−216.000			−0.406780
$532$		0			0
$533$			183.303i			0.343908i
$534$		0			0
$535$	313.050	+	717.287i	0.585139	+	1.34072i
$536$		0			0
$537$			1108.98i			2.06515i
$538$		0			0
$539$	304.000	+	778.768i	0.564007	+	1.44484i
$540$		0			0
$541$		−	440.619i		−	0.814453i	−0.913327	−	0.407226i	$-0.866496\pi$
$541$		−	440.619i		−	0.814453i	0.913327	−	0.407226i	$-0.133504\pi$
$542$		0			0
$543$			1182.30i			2.17736i
$544$		0			0
$545$	93.9149	−	40.9878i	0.172321	−	0.0752070i
$546$		0			0
$547$	398.020			0.727642			0.363821	−	0.931469i	$-0.381472\pi$
$547$	398.020			0.727642			0.363821	+	0.931469i	$0.381472\pi$
$548$		0			0
$549$			860.744i			1.56784i
$550$		0			0
$551$	−105.000			−0.190563
$552$		0			0
$553$			687.386i			1.24301i
$554$		0			0
$555$	−42.0000	−	96.2341i	−0.0756757	−	0.173395i
$556$		0			0
$557$	156.525			0.281014			0.140507	−	0.990080i	$-0.455127\pi$
$557$	156.525			0.281014			0.140507	+	0.990080i	$0.455127\pi$
$558$		0			0
$559$	−200.000			−0.357782
$560$		0			0
$561$	−735.000	+	286.915i	−1.31016	+	0.511434i
$562$		0			0
$563$	−881.011			−1.56485			−0.782425	−	0.622745i	$-0.786018\pi$
$563$	−881.011			−1.56485			−0.782425	+	0.622745i	$0.786018\pi$
$564$		0			0
$565$	756.000	−	329.945i	1.33805	−	0.583974i
$566$		0			0
$567$	503.115			0.887329
$568$		0			0
$569$			143.457i			0.252122i	0.992023	+	0.126061i	$0.0402335\pi$
$569$			143.457i			0.252122i	−0.992023	+	0.126061i	$0.959767\pi$
$570$		0			0
$571$			604.570i			1.05879i	0.848375	+	0.529396i	$0.177581\pi$
$571$			604.570i			1.05879i	−0.848375	+	0.529396i	$0.822419\pi$
$572$		0			0
$573$		−	1273.96i		−	2.22331i
$574$		0			0
$575$	504.000	+	467.423i	0.876522	+	0.812909i
$576$		0			0
$577$			164.973i			0.285915i	0.989729	+	0.142957i	$0.0456612\pi$
$577$			164.973i			0.285915i	−0.989729	+	0.142957i	$0.954339\pi$
$578$		0			0
$579$			358.643i			0.619418i
$580$		0			0
$581$	−800.000			−1.37694
$582$		0			0
$583$	−46.9574	+	18.3303i	−0.0805445	+	0.0314413i
$584$		0			0
$585$	214.663	+	491.854i	0.366944	+	0.840775i
$586$		0			0
$587$		−	875.272i		−	1.49109i	−0.666453	−	0.745547i	$-0.732188\pi$
$587$		−	875.272i		−	1.49109i	0.666453	−	0.745547i	$-0.267812\pi$
$588$		0			0
$589$			30.7409i			0.0521916i
$590$		0			0
$591$			901.732i			1.52577i
$592$		0			0
$593$	−111.803			−0.188539			−0.0942693	−	0.995547i	$-0.530051\pi$
$593$	−111.803			−0.188539			−0.0942693	+	0.995547i	$0.530051\pi$
$594$		0			0
$595$	−350.000	−	801.951i	−0.588235	−	1.34782i
$596$		0			0
$597$		−	242.877i		−	0.406828i
$598$		0			0
$599$	493.000			0.823038			0.411519	−	0.911401i	$-0.364998\pi$
$599$	493.000			0.823038			0.411519	+	0.911401i	$0.364998\pi$
$600$		0			0
$601$			102.470i			0.170498i	0.996360	+	0.0852492i	$0.0271686\pi$
$601$			102.470i			0.170498i	−0.996360	+	0.0852492i	$0.972831\pi$
$602$		0			0
$603$			329.945i			0.547173i
$604$		0			0
$605$	553.659	+	243.898i	0.915140	+	0.403137i
$606$		0			0
$607$	816.165			1.34459			0.672294	−	0.740284i	$-0.265309\pi$
$607$	816.165			1.34459			0.672294	+	0.740284i	$0.265309\pi$
$608$		0			0
$609$	−525.000			−0.862069
$610$		0			0
$611$			573.829i			0.939164i
$612$		0			0
$613$	881.011			1.43721			0.718606	−	0.695418i	$-0.244781\pi$
$613$	881.011			1.43721			0.718606	+	0.695418i	$0.244781\pi$
$614$		0			0
$615$	−187.830	−	430.372i	−0.305414	−	0.699792i
$616$		0			0
$617$		−	293.285i		−	0.475340i	−0.971346	−	0.237670i	$-0.923616\pi$
$617$		−	293.285i		−	0.475340i	0.971346	−	0.237670i	$-0.0763837\pi$
$618$		0			0
$619$	−972.000			−1.57027			−0.785137	−	0.619322i	$-0.787408\pi$
$619$	−972.000			−1.57027			−0.785137	+	0.619322i	$0.787408\pi$
$620$		0			0
$621$	378.000			0.608696
$622$		0			0
$623$	−413.673			−0.664001
$624$		0			0
$625$	−47.0000	−	623.230i	−0.0752000	−	0.997168i
$626$		0			0
$627$	187.830	+	481.170i	0.299569	+	0.767417i
$628$		0			0
$629$		−	71.7287i		−	0.114036i
$630$		0			0
$631$	113.000			0.179081			0.0895404	−	0.995983i	$-0.471460\pi$
$631$	113.000			0.179081			0.0895404	+	0.995983i	$0.471460\pi$
$632$		0			0
$633$	798.276			1.26110
$634$		0			0
$635$	−62.6099	−	143.457i	−0.0985983	−	0.225917i
$636$		0			0
$637$	679.765			1.06713
$638$		0			0
$639$	324.000			0.507042
$640$		0			0
$641$	1007.00			1.57098			0.785491	−	0.618873i	$-0.212410\pi$
$641$	1007.00			1.57098			0.785491	+	0.618873i	$0.212410\pi$
$642$		0			0
$643$			270.372i			0.420485i	0.977649	+	0.210243i	$0.0674254\pi$
$643$			270.372i			0.420485i	−0.977649	+	0.210243i	$0.932575\pi$
$644$		0			0
$645$	469.574	−	204.939i	0.728022	−	0.317735i
$646$		0			0
$647$			18.3303i			0.0283312i	0.999900	+	0.0141656i	$0.00450921\pi$
$647$			18.3303i			0.0283312i	−0.999900	+	0.0141656i	$0.995491\pi$
$648$		0			0
$649$	72.0000	+	184.445i	0.110940	+	0.284199i
$650$		0			0
$651$			153.704i			0.236105i
$652$		0			0
$653$			691.969i			1.05968i	0.848099	+	0.529838i	$0.177747\pi$
$653$			691.969i			1.05968i	−0.848099	+	0.529838i	$0.822253\pi$
$654$		0			0
$655$	986.106	−	430.372i	1.50551	−	0.657056i
$656$		0			0
$657$	−697.653			−1.06188
$658$		0			0
$659$		−	1096.42i		−	1.66377i	−0.554949	−	0.831884i	$-0.687262\pi$
$659$		−	1096.42i		−	1.66377i	0.554949	−	0.831884i	$-0.312738\pi$
$660$		0			0
$661$	862.000			1.30408			0.652042	−	0.758183i	$-0.273912\pi$
$661$	862.000			1.30408			0.652042	+	0.758183i	$0.273912\pi$
$662$		0			0
$663$			641.561i			0.967663i
$664$		0			0
$665$	−525.000	+	229.129i	−0.789474	+	0.344555i
$666$		0			0
$667$	281.745			0.422406
$668$		0			0
$669$	−1806.00			−2.69955
$670$		0			0
$671$	735.000	−	286.915i	1.09538	−	0.427593i
$672$		0			0
$673$	−713.306			−1.05989			−0.529945	−	0.848032i	$-0.677787\pi$
$673$	−713.306			−1.05989			−0.529945	+	0.848032i	$0.677787\pi$
$674$		0			0
$675$	−252.000	−	233.711i	−0.373333	−	0.346239i
$676$		0			0
$677$	−31.3050			−0.0462407			−0.0231203	−	0.999733i	$-0.507360\pi$
$677$	−31.3050			−0.0462407			−0.0231203	+	0.999733i	$0.507360\pi$
$678$		0			0
$679$		−	1332.10i		−	1.96186i
$680$		0			0
$681$		−	1270.62i		−	1.86582i
$682$		0			0
$683$		−	600.317i		−	0.878942i	−0.898257	−	0.439471i	$-0.855166\pi$
$683$		−	600.317i		−	0.878942i	0.898257	−	0.439471i	$-0.144834\pi$
$684$		0			0
$685$	−294.000	+	128.312i	−0.429197	+	0.187317i
$686$		0			0
$687$		−	1338.11i		−	1.94776i
$688$		0			0
$689$			40.9878i			0.0594888i
$690$		0			0
$691$	−872.000			−1.26194			−0.630970	−	0.775808i	$-0.717343\pi$
$691$	−872.000			−1.26194			−0.630970	+	0.775808i	$0.717343\pi$
$692$		0			0
$693$	536.656	+	1374.77i	0.774396	+	1.98380i
$694$		0			0
$695$	1033.06	−	450.866i	1.48642	−	0.648728i
$696$		0			0
$697$		−	320.780i		−	0.460230i
$698$		0			0
$699$			789.015i			1.12878i
$700$		0			0
$701$			870.991i			1.24250i	0.783613	+	0.621249i	$0.213374\pi$
$701$			870.991i			1.24250i	−0.783613	+	0.621249i	$0.786626\pi$
$702$		0			0
$703$	−46.9574			−0.0667958
$704$		0			0
$705$	−588.000	−	1347.28i	−0.834043	−	1.91103i
$706$		0			0
$707$			1145.64i			1.62043i
$708$		0			0
$709$	−768.000			−1.08322			−0.541608	−	0.840631i	$-0.682184\pi$
$709$	−768.000			−1.08322			−0.541608	+	0.840631i	$0.682184\pi$
$710$		0			0
$711$			737.780i			1.03767i
$712$		0			0
$713$		−	82.4864i		−	0.115689i
$714$		0			0
$715$	348.446	−	347.254i	0.487337	−	0.485670i
$716$		0			0
$717$	1033.06			1.44081
$718$		0			0
$719$	263.000			0.365786			0.182893	−	0.983133i	$-0.441454\pi$
$719$	263.000			0.365786			0.182893	+	0.983133i	$0.441454\pi$
$720$		0			0
$721$		−	204.939i		−	0.284243i
$722$		0			0
$723$	1502.64			2.07834
$724$		0			0
$725$	−187.830	−	174.198i	−0.259075	−	0.240273i
$726$		0			0
$727$			293.285i			0.403418i	0.979446	+	0.201709i	$0.0646495\pi$
$727$			293.285i			0.403418i	−0.979446	+	0.201709i	$0.935350\pi$
$728$		0			0
$729$	1107.00			1.51852
$730$		0			0
$731$	350.000			0.478796
$732$		0			0
$733$	219.135			0.298956			0.149478	−	0.988765i	$-0.452241\pi$
$733$	219.135			0.298956			0.149478	+	0.988765i	$0.452241\pi$
$734$		0			0
$735$	−1596.00	+	696.552i	−2.17143	+	0.947689i
$736$		0			0
$737$	281.745	−	109.982i	0.382286	−	0.149229i
$738$		0			0
$739$		−	389.384i		−	0.526907i	−0.964672	−	0.263453i	$-0.915138\pi$
$739$		−	389.384i		−	0.526907i	0.964672	−	0.263453i	$-0.0848615\pi$
$740$		0			0
$741$	420.000			0.566802
$742$		0			0
$743$	−682.001			−0.917901			−0.458951	−	0.888462i	$-0.651775\pi$
$743$	−682.001			−0.917901			−0.458951	+	0.888462i	$0.651775\pi$
$744$		0			0
$745$	704.361	−	307.409i	0.945452	−	0.412629i
$746$		0			0
$747$	−858.650			−1.14946
$748$		0			0
$749$	1750.00			2.33645
$750$		0			0
$751$	1003.00			1.33555			0.667776	−	0.744362i	$-0.267246\pi$
$751$	1003.00			1.33555			0.667776	+	0.744362i	$0.267246\pi$
$752$		0			0
$753$			1292.29i			1.71618i
$754$		0			0
$755$	−469.574	+	204.939i	−0.621953	+	0.271442i
$756$		0			0
$757$			1218.97i			1.61026i	0.593100	+	0.805129i	$0.297904\pi$
$757$			1218.97i			1.61026i	−0.593100	+	0.805129i	$0.702096\pi$
$758$		0			0
$759$	−504.000	−	1291.12i	−0.664032	−	1.70107i
$760$		0			0
$761$			184.445i			0.242372i	0.992630	+	0.121186i	$0.0386698\pi$
$761$			184.445i			0.242372i	−0.992630	+	0.121186i	$0.961330\pi$
$762$		0			0
$763$		−	229.129i		−	0.300300i
$764$		0			0
$765$	−375.659	−	860.744i	−0.491058	−	1.12516i
$766$		0			0
$767$	160.997			0.209905
$768$		0			0
$769$			245.927i			0.319801i	0.987133	+	0.159900i	$0.0511173\pi$
$769$			245.927i			0.319801i	−0.987133	+	0.159900i	$0.948883\pi$
$770$		0			0
$771$	−294.000			−0.381323
$772$		0			0
$773$			279.537i			0.361626i	0.983517	+	0.180813i	$0.0578729\pi$
$773$			279.537i			0.361626i	−0.983517	+	0.180813i	$0.942127\pi$
$774$		0			0
$775$	−51.0000	+	54.9909i	−0.0658065	+	0.0709560i
$776$		0			0
$777$	−234.787			−0.302171
$778$		0			0
$779$	−210.000			−0.269576
$780$		0			0
$781$	−108.000	−	276.668i	−0.138284	−	0.354248i
$782$		0			0
$783$	−140.872			−0.179914
$784$		0			0
$785$	231.000	−	100.817i	0.294268	−	0.128429i
$786$		0			0
$787$	−746.847			−0.948979			−0.474490	−	0.880261i	$-0.657367\pi$
$787$	−746.847			−0.948979			−0.474490	+	0.880261i	$0.657367\pi$
$788$		0			0
$789$			911.979i			1.15587i
$790$		0			0
$791$		−	1844.45i		−	2.33180i
$792$		0			0
$793$		−	641.561i		−	0.809030i
$794$		0			0
$795$	−42.0000	−	96.2341i	−0.0528302	−	0.121049i
$796$		0			0
$797$		−	705.717i		−	0.885466i	−0.896653	−	0.442733i	$-0.854009\pi$
$797$		−	705.717i		−	0.885466i	0.896653	−	0.442733i	$-0.145991\pi$
$798$		0			0
$799$		−	1004.20i		−	1.25682i
$800$		0			0
$801$	−444.000			−0.554307
$802$		0			0
$803$	232.551	+	595.735i	0.289603	+	0.741886i
$804$		0			0
$805$	1408.72	−	614.817i	1.74997	−	0.763748i
$806$		0			0
$807$		−	742.377i		−	0.919922i
$808$		0			0
$809$		−	1209.14i		−	1.49461i	−0.664481	−	0.747305i	$-0.731347\pi$
$809$		−	1209.14i		−	1.49461i	0.664481	−	0.747305i	$-0.268653\pi$
$810$		0			0
$811$			1096.42i			1.35194i	0.736929	+	0.675970i	$0.236275\pi$
$811$			1096.42i			1.35194i	−0.736929	+	0.675970i	$0.763725\pi$
$812$		0			0
$813$	1126.98			1.38620
$814$		0			0
$815$	1239.00	−	540.744i	1.52025	−	0.663489i
$816$		0			0
$817$		−	229.129i		−	0.280451i
$818$		0			0
$819$	1200.00			1.46520
$820$		0			0
$821$		−	758.274i		−	0.923598i	−0.886984	−	0.461799i	$-0.847204\pi$
$821$		−	758.274i		−	0.923598i	0.886984	−	0.461799i	$-0.152796\pi$
$822$		0			0
$823$			293.285i			0.356361i	0.983998	+	0.178180i	$0.0570210\pi$
$823$			293.285i			0.356361i	−0.983998	+	0.178180i	$0.942979\pi$
$824$		0			0
$825$	−462.276	+	1172.36i	−0.560335	+	1.42104i
$826$		0			0
$827$	−594.794			−0.719219			−0.359609	−	0.933103i	$-0.617090\pi$
$827$	−594.794			−0.719219			−0.359609	+	0.933103i	$0.617090\pi$
$828$		0			0
$829$	102.000			0.123040			0.0615199	−	0.998106i	$-0.480405\pi$
$829$	102.000			0.123040			0.0615199	+	0.998106i	$0.480405\pi$
$830$		0			0
$831$		−	266.421i		−	0.320603i
$832$		0			0
$833$	−1189.59			−1.42808
$834$		0			0
$835$	594.794	+	1362.84i	0.712328	+	1.63215i
$836$		0			0
$837$			41.2432i			0.0492750i
$838$		0			0
$839$	138.000			0.164482			0.0822408	−	0.996612i	$-0.473792\pi$
$839$	138.000			0.164482			0.0822408	+	0.996612i	$0.473792\pi$
$840$		0			0
$841$	736.000			0.875149
$842$		0			0
$843$	−751.319			−0.891244
$844$		0			0
$845$	178.000	+	407.849i	0.210651	+	0.482662i
$846$		0			0
$847$	995.050	−	916.515i	1.17479	−	1.08207i
$848$		0			0
$849$		−	1147.66i		−	1.35178i
$850$		0			0
$851$	126.000			0.148061
$852$		0			0
$853$	−532.184			−0.623897			−0.311949	−	0.950099i	$-0.600982\pi$
$853$	−532.184			−0.623897			−0.311949	+	0.950099i	$0.600982\pi$
$854$		0			0
$855$	−563.489	+	245.927i	−0.659052	+	0.287634i
$856$		0			0
$857$	−950.329			−1.10890			−0.554451	−	0.832216i	$-0.687072\pi$
$857$	−950.329			−1.10890			−0.554451	+	0.832216i	$0.687072\pi$
$858$		0			0
$859$	1068.00			1.24331			0.621653	−	0.783293i	$-0.286461\pi$
$859$	1068.00			1.24331			0.621653	+	0.783293i	$0.286461\pi$
$860$		0			0
$861$	−1050.00			−1.21951
$862$		0			0
$863$		−	73.3212i		−	0.0849608i	−0.999097	−	0.0424804i	$-0.986474\pi$
$863$		−	73.3212i		−	0.0849608i	0.999097	−	0.0424804i	$-0.0135260\pi$
$864$		0			0
$865$	286.217	+	655.805i	0.330886	+	0.758156i
$866$		0			0
$867$			201.633i			0.232564i
$868$		0			0
$869$	630.000	−	245.927i	0.724971	−	0.283000i
$870$		0			0
$871$		−	245.927i		−	0.282350i
$872$		0			0
$873$		−	1429.76i		−	1.63776i
$874$		0			0
$875$	−1319.28	−	461.113i	−1.50775	−	0.526986i
$876$		0			0
$877$	169.941			0.193776			0.0968878	−	0.995295i	$-0.469111\pi$
$877$	169.941			0.193776			0.0968878	+	0.995295i	$0.469111\pi$
$878$		0			0
$879$			2049.39i			2.33150i
$880$		0			0
$881$	−78.0000			−0.0885358			−0.0442679	−	0.999020i	$-0.514096\pi$
$881$	−78.0000			−0.0885358			−0.0442679	+	0.999020i	$0.514096\pi$
$882$		0			0
$883$			1553.49i			1.75934i	0.475589	+	0.879668i	$0.342235\pi$
$883$			1553.49i			1.75934i	−0.475589	+	0.879668i	$0.657765\pi$
$884$		0			0
$885$	−378.000	+	164.973i	−0.427119	+	0.186410i
$886$		0			0
$887$	−299.633			−0.337805			−0.168903	−	0.985633i	$-0.554022\pi$
$887$	−299.633			−0.337805			−0.168903	+	0.985633i	$0.554022\pi$
$888$		0			0
$889$	−350.000			−0.393701
$890$		0			0
$891$	−180.000	−	461.113i	−0.202020	−	0.517523i
$892$		0			0
$893$	−657.404			−0.736175
$894$		0			0
$895$	484.000	+	1108.98i	0.540782	+	1.23909i
$896$		0			0
$897$	−1126.98			−1.25639
$898$		0			0
$899$			30.7409i			0.0341945i
$900$		0			0
$901$		−	71.7287i		−	0.0796101i
$902$		0			0
$903$		−	1145.64i		−	1.26871i
$904$		0			0
$905$	516.000	+	1182.30i	0.570166	+	1.30641i
$906$		0			0
$907$		−	691.969i		−	0.762921i	−0.924385	−	0.381460i	$-0.875421\pi$
$907$		−	691.969i		−	0.762921i	0.924385	−	0.381460i	$-0.124579\pi$
$908$		0			0
$909$			1229.63i			1.35273i
$910$		0			0
$911$	−1037.00			−1.13831			−0.569155	−	0.822230i	$-0.692729\pi$
$911$	−1037.00			−1.13831			−0.569155	+	0.822230i	$0.692729\pi$
$912$		0			0
$913$	286.217	+	733.212i	0.313490	+	0.803080i
$914$		0			0
$915$	657.404	+	1506.30i	0.718474	+	1.64623i
$916$		0			0
$917$		−	2405.85i		−	2.62361i
$918$		0			0
$919$		−	1782.97i		−	1.94012i	−0.242867	−	0.970060i	$-0.578088\pi$
$919$		−	1782.97i		−	1.94012i	0.242867	−	0.970060i	$-0.421912\pi$
$920$		0			0
$921$			2049.39i			2.22518i
$922$		0			0
$923$	−241.495			−0.261642
$924$		0			0
$925$	−84.0000	−	77.9038i	−0.0908108	−	0.0842203i
$926$		0			0
$927$		−	219.964i		−	0.237285i
$928$		0			0
$929$	−633.000			−0.681378			−0.340689	−	0.940176i	$-0.610660\pi$
$929$	−633.000			−0.681378			−0.340689	+	0.940176i	$0.610660\pi$
$930$		0			0
$931$			778.768i			0.836486i
$932$		0			0
$933$			398.684i			0.427314i
$934$		0			0
$935$	−609.780	+	607.695i	−0.652171	+	0.649941i
$936$		0			0
$937$	−728.958			−0.777970			−0.388985	−	0.921244i	$-0.627174\pi$
$937$	−728.958			−0.777970			−0.388985	+	0.921244i	$0.627174\pi$
$938$		0			0
$939$	−1344.00			−1.43131
$940$		0			0
$941$		−	850.497i		−	0.903822i	−0.892063	−	0.451911i	$-0.850742\pi$
$941$		−	850.497i		−	0.903822i	0.892063	−	0.451911i	$-0.149258\pi$
$942$		0			0
$943$	563.489			0.597549
$944$		0			0
$945$	−704.361	+	307.409i	−0.745356	+	0.325300i
$946$		0			0
$947$			911.933i			0.962970i	0.876454	+	0.481485i	$0.159902\pi$
$947$			911.933i			0.962970i	−0.876454	+	0.481485i	$0.840098\pi$
$948$		0			0
$949$	520.000			0.547945
$950$		0			0
$951$	231.000			0.242902
$952$		0			0
$953$	−1236.55			−1.29753			−0.648765	−	0.760989i	$-0.724714\pi$
$953$	−1236.55			−1.29753			−0.648765	+	0.760989i	$0.724714\pi$
$954$		0			0
$955$	−556.000	−	1273.96i	−0.582199	−	1.33399i
$956$		0			0
$957$	187.830	+	481.170i	0.196269	+	0.502790i
$958$		0			0
$959$			717.287i			0.747953i
$960$		0			0
$961$	−952.000			−0.990635
$962$		0			0
$963$	1878.30			1.95046
$964$		0			0
$965$	156.525	+	358.643i	0.162202	+	0.371651i
$966$		0			0
$967$	−431.561			−0.446289			−0.223144	−	0.974785i	$-0.571632\pi$
$967$	−431.561			−0.446289			−0.223144	+	0.974785i	$0.571632\pi$
$968$		0			0
$969$	−735.000			−0.758514
$970$		0			0
$971$	1048.00			1.07930			0.539650	−	0.841890i	$-0.318557\pi$
$971$	1048.00			1.07930			0.539650	+	0.841890i	$0.318557\pi$
$972$		0			0
$973$		−	2520.42i		−	2.59036i
$974$		0			0
$975$	751.319	+	696.793i	0.770583	+	0.714659i
$976$		0			0
$977$		−	247.459i		−	0.253285i	−0.991948	−	0.126642i	$-0.959580\pi$
$977$		−	247.459i		−	0.253285i	0.991948	−	0.126642i	$-0.0404200\pi$
$978$		0			0
$979$	148.000	+	379.137i	0.151175	+	0.387270i
$980$		0			0
$981$		−	245.927i		−	0.250690i
$982$		0			0
$983$		−	27.4955i		−	0.0279710i	−0.999902	−	0.0139855i	$-0.995548\pi$
$983$		−	27.4955i		−	0.0279710i	0.999902	−	0.0139855i	$-0.00445186\pi$
$984$		0			0
$985$	393.548	+	901.732i	0.399541	+	0.915464i
$986$		0			0
$987$	−3287.02			−3.33031
$988$		0			0
$989$			614.817i			0.621655i
$990$		0			0
$991$	−1182.00			−1.19273			−0.596367	−	0.802712i	$-0.703390\pi$
$991$	−1182.00			−1.19273			−0.596367	+	0.802712i	$0.703390\pi$
$992$		0			0
$993$		−	632.395i		−	0.636853i
$994$		0			0
$995$	−106.000	−	242.877i	−0.106533	−	0.244097i
$996$		0			0
$997$	−1694.94			−1.70004			−0.850020	−	0.526751i	$-0.823410\pi$
$997$	−1694.94			−1.70004			−0.850020	+	0.526751i	$0.823410\pi$
$998$		0			0
$999$	−63.0000			−0.0630631

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.3.i.d.769.1		4
4.3	odd	2		55.3.d.d.54.4	yes	4
5.4	even	2	inner	880.3.i.d.769.4		4
11.10	odd	2	inner	880.3.i.d.769.2		4
12.11	even	2		495.3.h.d.109.2		4
20.3	even	4		275.3.c.e.76.1		4
20.7	even	4		275.3.c.e.76.4		4
20.19	odd	2		55.3.d.d.54.1	✓	4
44.43	even	2		55.3.d.d.54.2	yes	4
55.54	odd	2	inner	880.3.i.d.769.3		4
60.59	even	2		495.3.h.d.109.3		4
132.131	odd	2		495.3.h.d.109.4		4
220.43	odd	4		275.3.c.e.76.3		4
220.87	odd	4		275.3.c.e.76.2		4
220.219	even	2		55.3.d.d.54.3	yes	4
660.659	odd	2		495.3.h.d.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.d.d.54.1	✓	4	20.19	odd	2
55.3.d.d.54.2	yes	4	44.43	even	2
55.3.d.d.54.3	yes	4	220.219	even	2
55.3.d.d.54.4	yes	4	4.3	odd	2
275.3.c.e.76.1		4	20.3	even	4
275.3.c.e.76.2		4	220.87	odd	4
275.3.c.e.76.3		4	220.43	odd	4
275.3.c.e.76.4		4	20.7	even	4
495.3.h.d.109.1		4	660.659	odd	2
495.3.h.d.109.2		4	12.11	even	2
495.3.h.d.109.3		4	60.59	even	2
495.3.h.d.109.4		4	132.131	odd	2
880.3.i.d.769.1		4	1.1	even	1	trivial
880.3.i.d.769.2		4	11.10	odd	2	inner
880.3.i.d.769.3		4	55.54	odd	2	inner
880.3.i.d.769.4		4	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	880.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.58258i q^{3} +(-2.00000 - 4.58258i) q^{5} -11.1803 q^{7} -12.0000 q^{9} +(4.00000 + 10.2470i) q^{11} +8.94427 q^{13} +(-21.0000 + 9.16515i) q^{15} -15.6525 q^{17} +10.2470i q^{19} +51.2348i q^{21} -27.4955i q^{23} +(-17.0000 + 18.3303i) q^{25} +13.7477i q^{27} +10.2470i q^{29} +3.00000 q^{31} +(46.9574 - 18.3303i) q^{33} +(22.3607 + 51.2348i) q^{35} +4.58258i q^{37} -40.9878i q^{39} +20.4939i q^{41} -22.3607 q^{43} +(24.0000 + 54.9909i) q^{45} +64.1561i q^{47} +76.0000 q^{49} +71.7287i q^{51} +4.58258i q^{53} +(38.9574 - 38.8242i) q^{55} +46.9574 q^{57} +18.0000 q^{59} -71.7287i q^{61} +134.164 q^{63} +(-17.8885 - 40.9878i) q^{65} -27.4955i q^{67} -126.000 q^{69} -27.0000 q^{71} +58.1378 q^{73} +(84.0000 + 77.9038i) q^{75} +(-44.7214 - 114.564i) q^{77} -61.4817i q^{79} -45.0000 q^{81} +71.5542 q^{83} +(31.3050 + 71.7287i) q^{85} +46.9574 q^{87} +37.0000 q^{89} -100.000 q^{91} -13.7477i q^{93} +(46.9574 - 20.4939i) q^{95} +119.147i q^{97} +(-48.0000 - 122.963i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 48 q^{9} + 16 q^{11} - 84 q^{15} - 68 q^{25} + 12 q^{31} + 96 q^{45} + 304 q^{49} - 32 q^{55} + 72 q^{59} - 504 q^{69} - 108 q^{71} + 336 q^{75} - 180 q^{81} + 148 q^{89} - 400 q^{91} - 192 q^{99}+O(q^{100}) \) 4 * q - 8 * q^5 - 48 * q^9 + 16 * q^11 - 84 * q^15 - 68 * q^25 + 12 * q^31 + 96 * q^45 + 304 * q^49 - 32 * q^55 + 72 * q^59 - 504 * q^69 - 108 * q^71 + 336 * q^75 - 180 * q^81 + 148 * q^89 - 400 * q^91 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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