LMFDB - Embedded newform 720.3.c.a.449.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,3,Mod(449,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, 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("720.449"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	720.449
Dual form			720.3.c.a.449.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.64575 + 4.24264i) q^{5} -11.2250i q^{7} -4.24264i q^{11} -11.2250i q^{13} -10.5830 q^{17} -20.0000 q^{19} -5.29150 q^{23} +(-11.0000 + 22.4499i) q^{25} -8.48528i q^{29} -26.0000 q^{31} +(47.6235 - 29.6985i) q^{35} -33.6749i q^{37} -55.1543i q^{41} +22.4499i q^{43} -21.1660 q^{47} -77.0000 q^{49} +84.6640 q^{53} +(18.0000 - 11.2250i) q^{55} -46.6690i q^{59} -22.0000 q^{61} +(47.6235 - 29.6985i) q^{65} -89.7998i q^{67} +50.9117i q^{71} -67.3498i q^{73} -47.6235 q^{77} -14.0000 q^{79} +74.0810 q^{83} +(-28.0000 - 44.8999i) q^{85} +89.0955i q^{89} -126.000 q^{91} +(-52.9150 - 84.8528i) q^{95} -22.4499i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 80 q^{19} - 44 q^{25} - 104 q^{31} - 308 q^{49} + 72 q^{55} - 88 q^{61} - 56 q^{79} - 112 q^{85} - 504 q^{91}+O(q^{100}) $ 4 * q - 80 * q^19 - 44 * q^25 - 104 * q^31 - 308 * q^49 + 72 * q^55 - 88 * q^61 - 56 * q^79 - 112 * q^85 - 504 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.64575	+	4.24264i	0.529150	+	0.848528i
$5$	2.64575	+	4.24264i	0.529150	+	0.848528i
$6$		0			0
$7$		−	11.2250i		−	1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$		−	11.2250i		−	1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.24264i		−	0.385695i	−0.981229	−	0.192847i	$-0.938228\pi$
$11$		−	4.24264i		−	0.385695i	0.981229	−	0.192847i	$-0.0617722\pi$
$12$		0			0
$13$		−	11.2250i		−	0.863459i	−0.902003	−	0.431730i	$-0.857903\pi$
$13$		−	11.2250i		−	0.863459i	0.902003	−	0.431730i	$-0.142097\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−10.5830			−0.622530			−0.311265	−	0.950323i	$-0.600753\pi$
$17$	−10.5830			−0.622530			−0.311265	+	0.950323i	$0.600753\pi$
$18$		0			0
$19$	−20.0000			−1.05263			−0.526316	−	0.850289i	$-0.676427\pi$
$19$	−20.0000			−1.05263			−0.526316	+	0.850289i	$0.676427\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.29150			−0.230065			−0.115033	−	0.993362i	$-0.536697\pi$
$23$	−5.29150			−0.230065			−0.115033	+	0.993362i	$0.536697\pi$
$24$		0			0
$25$	−11.0000	+	22.4499i	−0.440000	+	0.897998i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.48528i		−	0.292596i	−0.989241	−	0.146298i	$-0.953264\pi$
$29$		−	8.48528i		−	0.292596i	0.989241	−	0.146298i	$-0.0467358\pi$
$30$		0			0
$31$	−26.0000			−0.838710			−0.419355	−	0.907822i	$-0.637744\pi$
$31$	−26.0000			−0.838710			−0.419355	+	0.907822i	$0.637744\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	47.6235	−	29.6985i	1.36067	−	0.848528i
$36$		0			0
$37$		−	33.6749i		−	0.910133i	−0.890457	−	0.455066i	$-0.849616\pi$
$37$		−	33.6749i		−	0.910133i	0.890457	−	0.455066i	$-0.150384\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	55.1543i		−	1.34523i	−0.739994	−	0.672614i	$-0.765172\pi$
$41$		−	55.1543i		−	1.34523i	0.739994	−	0.672614i	$-0.234828\pi$
$42$		0			0
$43$			22.4499i			0.522092i	0.965326	+	0.261046i	$0.0840674\pi$
$43$			22.4499i			0.522092i	−0.965326	+	0.261046i	$0.915933\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−21.1660			−0.450341			−0.225170	−	0.974319i	$-0.572294\pi$
$47$	−21.1660			−0.450341			−0.225170	+	0.974319i	$0.572294\pi$
$48$		0			0
$49$	−77.0000			−1.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	84.6640			1.59743			0.798717	−	0.601706i	$-0.205512\pi$
$53$	84.6640			1.59743			0.798717	+	0.601706i	$0.205512\pi$
$54$		0			0
$55$	18.0000	−	11.2250i	0.327273	−	0.204090i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	46.6690i		−	0.791001i	−0.918466	−	0.395500i	$-0.870571\pi$
$59$		−	46.6690i		−	0.791001i	0.918466	−	0.395500i	$-0.129429\pi$
$60$		0			0
$61$	−22.0000			−0.360656			−0.180328	−	0.983607i	$-0.557716\pi$
$61$	−22.0000			−0.360656			−0.180328	+	0.983607i	$0.557716\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	47.6235	−	29.6985i	0.732670	−	0.456900i
$66$		0			0
$67$		−	89.7998i		−	1.34030i	−0.742228	−	0.670148i	$-0.766231\pi$
$67$		−	89.7998i		−	1.34030i	0.742228	−	0.670148i	$-0.233769\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			50.9117i			0.717066i	0.933517	+	0.358533i	$0.116723\pi$
$71$			50.9117i			0.717066i	−0.933517	+	0.358533i	$0.883277\pi$
$72$		0			0
$73$		−	67.3498i		−	0.922600i	−0.887244	−	0.461300i	$-0.847383\pi$
$73$		−	67.3498i		−	0.922600i	0.887244	−	0.461300i	$-0.152617\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−47.6235			−0.618487
$78$		0			0
$79$	−14.0000			−0.177215			−0.0886076	−	0.996067i	$-0.528242\pi$
$79$	−14.0000			−0.177215			−0.0886076	+	0.996067i	$0.528242\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	74.0810			0.892543			0.446271	−	0.894898i	$-0.352752\pi$
$83$	74.0810			0.892543			0.446271	+	0.894898i	$0.352752\pi$
$84$		0			0
$85$	−28.0000	−	44.8999i	−0.329412	−	0.528234i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			89.0955i			1.00107i	0.865716	+	0.500536i	$0.166864\pi$
$89$			89.0955i			1.00107i	−0.865716	+	0.500536i	$0.833136\pi$
$90$		0			0
$91$	−126.000			−1.38462
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−52.9150	−	84.8528i	−0.557000	−	0.893188i
$96$		0			0
$97$		−	22.4499i		−	0.231443i	−0.993282	−	0.115721i	$-0.963082\pi$
$97$		−	22.4499i		−	0.231443i	0.993282	−	0.115721i	$-0.0369180\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	135.765i		−	1.34420i	−0.740459	−	0.672101i	$-0.765392\pi$
$101$		−	135.765i		−	1.34420i	0.740459	−	0.672101i	$-0.234608\pi$
$102$		0			0
$103$		−	56.1249i		−	0.544902i	−0.962170	−	0.272451i	$-0.912166\pi$
$103$		−	56.1249i		−	0.544902i	0.962170	−	0.272451i	$-0.0878342\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.5830			0.0989066			0.0494533	−	0.998776i	$-0.484252\pi$
$107$	10.5830			0.0989066			0.0494533	+	0.998776i	$0.484252\pi$
$108$		0			0
$109$	−70.0000			−0.642202			−0.321101	−	0.947045i	$-0.604053\pi$
$109$	−70.0000			−0.642202			−0.321101	+	0.947045i	$0.604053\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−137.579			−1.21751			−0.608757	−	0.793357i	$-0.708332\pi$
$113$	−137.579			−1.21751			−0.608757	+	0.793357i	$0.708332\pi$
$114$		0			0
$115$	−14.0000	−	22.4499i	−0.121739	−	0.195217i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			118.794i			0.998268i
$120$		0			0
$121$	103.000			0.851240
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−124.350	+	12.7279i	−0.994802	+	0.101823i
$126$		0			0
$127$			168.375i			1.32578i	0.748715	+	0.662892i	$0.230671\pi$
$127$			168.375i			1.32578i	−0.748715	+	0.662892i	$0.769329\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	148.492i		−	1.13353i	−0.823880	−	0.566765i	$-0.808195\pi$
$131$		−	148.492i		−	1.13353i	0.823880	−	0.566765i	$-0.191805\pi$
$132$		0			0
$133$			224.499i			1.68797i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	211.660			1.54496			0.772482	−	0.635036i	$-0.219015\pi$
$137$	211.660			1.54496			0.772482	+	0.635036i	$0.219015\pi$
$138$		0			0
$139$	−206.000			−1.48201			−0.741007	−	0.671497i	$-0.765652\pi$
$139$	−206.000			−1.48201			−0.741007	+	0.671497i	$0.765652\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−47.6235			−0.333032
$144$		0			0
$145$	36.0000	−	22.4499i	0.248276	−	0.154827i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			135.765i			0.911171i	0.890192	+	0.455586i	$0.150570\pi$
$149$			135.765i			0.911171i	−0.890192	+	0.455586i	$0.849430\pi$
$150$		0			0
$151$	202.000			1.33775			0.668874	−	0.743376i	$-0.266776\pi$
$151$	202.000			1.33775			0.668874	+	0.743376i	$0.266776\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−68.7895	−	110.309i	−0.443803	−	0.711669i
$156$		0			0
$157$			56.1249i			0.357483i	0.983896	+	0.178742i	$0.0572026\pi$
$157$			56.1249i			0.357483i	−0.983896	+	0.178742i	$0.942797\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			59.3970i			0.368925i
$162$		0			0
$163$		−	202.049i		−	1.23957i	−0.784773	−	0.619784i	$-0.787220\pi$
$163$		−	202.049i		−	1.23957i	0.784773	−	0.619784i	$-0.212780\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	185.203			1.10900			0.554499	−	0.832185i	$-0.312910\pi$
$167$	185.203			1.10900			0.554499	+	0.832185i	$0.312910\pi$
$168$		0			0
$169$	43.0000			0.254438
$170$		0			0
$171$		0			0
$172$		0			0
$173$	21.1660			0.122347			0.0611734	−	0.998127i	$-0.480516\pi$
$173$	21.1660			0.122347			0.0611734	+	0.998127i	$0.480516\pi$
$174$		0			0
$175$	252.000	+	123.475i	1.44000	+	0.705570i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			241.831i			1.35101i	0.737356	+	0.675504i	$0.236074\pi$
$179$			241.831i			1.35101i	−0.737356	+	0.675504i	$0.763926\pi$
$180$		0			0
$181$	74.0000			0.408840			0.204420	−	0.978883i	$-0.434469\pi$
$181$	74.0000			0.408840			0.204420	+	0.978883i	$0.434469\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	142.871	−	89.0955i	0.772273	−	0.481597i
$186$		0			0
$187$			44.8999i			0.240106i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			161.220i			0.844086i	0.906576	+	0.422043i	$0.138687\pi$
$191$			161.220i			0.844086i	−0.906576	+	0.422043i	$0.861313\pi$
$192$		0			0
$193$		−	179.600i		−	0.930568i	−0.885162	−	0.465284i	$-0.845952\pi$
$193$		−	179.600i		−	0.930568i	0.885162	−	0.465284i	$-0.154048\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	37.0405			0.188023			0.0940115	−	0.995571i	$-0.470031\pi$
$197$	37.0405			0.188023			0.0940115	+	0.995571i	$0.470031\pi$
$198$		0			0
$199$	250.000			1.25628			0.628141	−	0.778100i	$-0.283816\pi$
$199$	250.000			1.25628			0.628141	+	0.778100i	$0.283816\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−95.2470			−0.469197
$204$		0			0
$205$	234.000	−	145.925i	1.14146	−	0.711828i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			84.8528i			0.405994i
$210$		0			0
$211$	154.000			0.729858			0.364929	−	0.931035i	$-0.381093\pi$
$211$	154.000			0.729858			0.364929	+	0.931035i	$0.381093\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−95.2470	+	59.3970i	−0.443010	+	0.276265i
$216$		0			0
$217$			291.849i			1.34493i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			118.794i			0.537529i
$222$		0			0
$223$			392.874i			1.76177i	0.473333	+	0.880883i	$0.343051\pi$
$223$			392.874i			1.76177i	−0.473333	+	0.880883i	$0.656949\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.1660			−0.0932423			−0.0466212	−	0.998913i	$-0.514845\pi$
$227$	−21.1660			−0.0932423			−0.0466212	+	0.998913i	$0.514845\pi$
$228$		0			0
$229$	−118.000			−0.515284			−0.257642	−	0.966240i	$-0.582945\pi$
$229$	−118.000			−0.515284			−0.257642	+	0.966240i	$0.582945\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−391.571			−1.68056			−0.840282	−	0.542150i	$-0.817610\pi$
$233$	−391.571			−1.68056			−0.840282	+	0.542150i	$0.817610\pi$
$234$		0			0
$235$	−56.0000	−	89.7998i	−0.238298	−	0.382127i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			347.897i			1.45563i	0.685771	+	0.727817i	$0.259465\pi$
$239$			347.897i			1.45563i	−0.685771	+	0.727817i	$0.740535\pi$
$240$		0			0
$241$	−40.0000			−0.165975			−0.0829876	−	0.996551i	$-0.526446\pi$
$241$	−40.0000			−0.165975			−0.0829876	+	0.996551i	$0.526446\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−203.723	−	326.683i	−0.831522	−	1.33340i
$246$		0			0
$247$			224.499i			0.908905i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	241.831i		−	0.963468i	−0.876317	−	0.481734i	$-0.840007\pi$
$251$		−	241.831i		−	0.963468i	0.876317	−	0.481734i	$-0.159993\pi$
$252$		0			0
$253$			22.4499i			0.0887350i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−232.826			−0.905938			−0.452969	−	0.891526i	$-0.649635\pi$
$257$	−232.826			−0.905938			−0.452969	+	0.891526i	$0.649635\pi$
$258$		0			0
$259$	−378.000			−1.45946
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−164.037			−0.623713			−0.311857	−	0.950129i	$-0.600951\pi$
$263$	−164.037			−0.623713			−0.311857	+	0.950129i	$0.600951\pi$
$264$		0			0
$265$	224.000	+	359.199i	0.845283	+	1.35547i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			534.573i			1.98726i	0.112695	+	0.993630i	$0.464052\pi$
$269$			534.573i			1.98726i	−0.112695	+	0.993630i	$0.535948\pi$
$270$		0			0
$271$	286.000			1.05535			0.527675	−	0.849446i	$-0.323064\pi$
$271$	286.000			1.05535			0.527675	+	0.849446i	$0.323064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	95.2470	+	46.6690i	0.346353	+	0.169706i
$276$		0			0
$277$		−	190.825i		−	0.688897i	−0.938805	−	0.344449i	$-0.888066\pi$
$277$		−	190.825i		−	0.688897i	0.938805	−	0.344449i	$-0.111934\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			80.6102i			0.286869i	0.989660	+	0.143434i	$0.0458146\pi$
$281$			80.6102i			0.286869i	−0.989660	+	0.143434i	$0.954185\pi$
$282$		0			0
$283$		−	89.7998i		−	0.317314i	−0.987334	−	0.158657i	$-0.949284\pi$
$283$		−	89.7998i		−	0.317314i	0.987334	−	0.158657i	$-0.0507164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−619.106			−2.15716
$288$		0			0
$289$	−177.000			−0.612457
$290$		0			0
$291$		0			0
$292$		0			0
$293$	576.774			1.96851			0.984256	−	0.176751i	$-0.0565587\pi$
$293$	576.774			1.96851			0.984256	+	0.176751i	$0.0565587\pi$
$294$		0			0
$295$	198.000	−	123.475i	0.671186	−	0.418558i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			59.3970i			0.198652i
$300$		0			0
$301$	252.000			0.837209
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−58.2065	−	93.3381i	−0.190841	−	0.306027i
$306$		0			0
$307$		−	269.399i		−	0.877522i	−0.898604	−	0.438761i	$-0.855417\pi$
$307$		−	269.399i		−	0.877522i	0.898604	−	0.438761i	$-0.144583\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	59.3970i		−	0.190987i	−0.995430	−	0.0954935i	$-0.969557\pi$
$311$		−	59.3970i		−	0.190987i	0.995430	−	0.0954935i	$-0.0304429\pi$
$312$		0			0
$313$			179.600i			0.573800i	0.957960	+	0.286900i	$0.0926248\pi$
$313$			179.600i			0.573800i	−0.957960	+	0.286900i	$0.907375\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−312.199			−0.984854			−0.492427	−	0.870354i	$-0.663890\pi$
$317$	−312.199			−0.984854			−0.492427	+	0.870354i	$0.663890\pi$
$318$		0			0
$319$	−36.0000			−0.112853
$320$		0			0
$321$		0			0
$322$		0			0
$323$	211.660			0.655294
$324$		0			0
$325$	252.000	+	123.475i	0.775385	+	0.379922i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			237.588i			0.722152i
$330$		0			0
$331$	112.000			0.338369			0.169184	−	0.985584i	$-0.445887\pi$
$331$	112.000			0.338369			0.169184	+	0.985584i	$0.445887\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	380.988	−	237.588i	1.13728	−	0.709218i
$336$		0			0
$337$		−	112.250i		−	0.333085i	−0.986034	−	0.166543i	$-0.946740\pi$
$337$		−	112.250i		−	0.333085i	0.986034	−	0.166543i	$-0.0532603\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			110.309i			0.323486i
$342$		0			0
$343$			314.299i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	518.567			1.49443			0.747215	−	0.664582i	$-0.231390\pi$
$347$	518.567			1.49443			0.747215	+	0.664582i	$0.231390\pi$
$348$		0			0
$349$	122.000			0.349570			0.174785	−	0.984607i	$-0.444077\pi$
$349$	122.000			0.349570			0.174785	+	0.984607i	$0.444077\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	402.154			1.13925			0.569624	−	0.821906i	$-0.307089\pi$
$353$	402.154			1.13925			0.569624	+	0.821906i	$0.307089\pi$
$354$		0			0
$355$	−216.000	+	134.700i	−0.608451	+	0.379436i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			636.396i			1.77269i	0.463024	+	0.886346i	$0.346764\pi$
$359$			636.396i			1.77269i	−0.463024	+	0.886346i	$0.653236\pi$
$360$		0			0
$361$	39.0000			0.108033
$362$		0			0
$363$		0			0
$364$		0			0
$365$	285.741	−	178.191i	0.782852	−	0.488194i
$366$		0			0
$367$		−	684.723i		−	1.86573i	−0.360225	−	0.932866i	$-0.617300\pi$
$367$		−	684.723i		−	1.86573i	0.360225	−	0.932866i	$-0.382700\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	950.352i		−	2.56159i
$372$		0			0
$373$		−	145.925i		−	0.391219i	−0.980682	−	0.195609i	$-0.937332\pi$
$373$		−	145.925i		−	0.391219i	0.980682	−	0.195609i	$-0.0626685\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−95.2470			−0.252645
$378$		0			0
$379$	−362.000			−0.955145			−0.477573	−	0.878592i	$-0.658483\pi$
$379$	−362.000			−0.955145			−0.477573	+	0.878592i	$0.658483\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	42.3320			0.110527			0.0552637	−	0.998472i	$-0.482400\pi$
$383$	42.3320			0.110527			0.0552637	+	0.998472i	$0.482400\pi$
$384$		0			0
$385$	−126.000	−	202.049i	−0.327273	−	0.524804i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	263.044i		−	0.676205i	−0.941109	−	0.338102i	$-0.890215\pi$
$389$		−	263.044i		−	0.676205i	0.941109	−	0.338102i	$-0.109785\pi$
$390$		0			0
$391$	56.0000			0.143223
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−37.0405	−	59.3970i	−0.0937735	−	0.150372i
$396$		0			0
$397$			33.6749i			0.0848235i	0.999100	+	0.0424117i	$0.0135041\pi$
$397$			33.6749i			0.0848235i	−0.999100	+	0.0424117i	$0.986496\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	462.448i		−	1.15324i	−0.817014	−	0.576618i	$-0.804372\pi$
$401$		−	462.448i		−	1.15324i	0.817014	−	0.576618i	$-0.195628\pi$
$402$		0			0
$403$			291.849i			0.724192i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−142.871			−0.351033
$408$		0			0
$409$	−82.0000			−0.200489			−0.100244	−	0.994963i	$-0.531962\pi$
$409$	−82.0000			−0.200489			−0.100244	+	0.994963i	$0.531962\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−523.859			−1.26842
$414$		0			0
$415$	196.000	+	314.299i	0.472289	+	0.757348i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			207.889i			0.496156i	0.968740	+	0.248078i	$0.0797989\pi$
$419$			207.889i			0.496156i	−0.968740	+	0.248078i	$0.920201\pi$
$420$		0			0
$421$	−490.000			−1.16390			−0.581948	−	0.813226i	$-0.697709\pi$
$421$	−490.000			−1.16390			−0.581948	+	0.813226i	$0.697709\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	116.413	−	237.588i	0.273913	−	0.559030i
$426$		0			0
$427$			246.949i			0.578336i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			687.308i			1.59468i	0.603529	+	0.797341i	$0.293761\pi$
$431$			687.308i			1.59468i	−0.603529	+	0.797341i	$0.706239\pi$
$432$		0			0
$433$		−	202.049i		−	0.466627i	−0.972402	−	0.233314i	$-0.925043\pi$
$433$		−	202.049i		−	0.466627i	0.972402	−	0.233314i	$-0.0749568\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	105.830			0.242174
$438$		0			0
$439$	−302.000			−0.687927			−0.343964	−	0.938983i	$-0.611770\pi$
$439$	−302.000			−0.687927			−0.343964	+	0.938983i	$0.611770\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	264.575			0.597235			0.298618	−	0.954373i	$-0.403475\pi$
$443$	264.575			0.597235			0.298618	+	0.954373i	$0.403475\pi$
$444$		0			0
$445$	−378.000	+	235.724i	−0.849438	+	0.529718i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			216.375i			0.481904i	0.970537	+	0.240952i	$0.0774596\pi$
$449$			216.375i			0.481904i	−0.970537	+	0.240952i	$0.922540\pi$
$450$		0			0
$451$	−234.000			−0.518847
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−333.365	−	534.573i	−0.732670	−	1.17489i
$456$		0			0
$457$		−	561.249i		−	1.22812i	−0.789261	−	0.614058i	$-0.789536\pi$
$457$		−	561.249i		−	1.22812i	0.789261	−	0.614058i	$-0.210464\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	237.588i		−	0.515375i	−0.966228	−	0.257688i	$-0.917039\pi$
$461$		−	237.588i		−	0.515375i	0.966228	−	0.257688i	$-0.0829605\pi$
$462$		0			0
$463$		−	729.623i		−	1.57586i	−0.615765	−	0.787930i	$-0.711153\pi$
$463$		−	729.623i		−	1.57586i	0.615765	−	0.787930i	$-0.288847\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−465.652			−0.997114			−0.498557	−	0.866857i	$-0.666137\pi$
$467$	−465.652			−0.997114			−0.498557	+	0.866857i	$0.666137\pi$
$468$		0			0
$469$	−1008.00			−2.14925
$470$		0			0
$471$		0			0
$472$		0			0
$473$	95.2470			0.201368
$474$		0			0
$475$	220.000	−	448.999i	0.463158	−	0.945261i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	475.176i		−	0.992016i	−0.868318	−	0.496008i	$-0.834799\pi$
$479$		−	475.176i		−	0.992016i	0.868318	−	0.496008i	$-0.165201\pi$
$480$		0			0
$481$	−378.000			−0.785863
$482$		0			0
$483$		0			0
$484$		0			0
$485$	95.2470	−	59.3970i	0.196386	−	0.122468i
$486$		0			0
$487$			505.124i			1.03722i	0.855012	+	0.518608i	$0.173549\pi$
$487$			505.124i			1.03722i	−0.855012	+	0.518608i	$0.826451\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	275.772i		−	0.561653i	−0.959759	−	0.280827i	$-0.909391\pi$
$491$		−	275.772i		−	0.561653i	0.959759	−	0.280827i	$-0.0906086\pi$
$492$		0			0
$493$			89.7998i			0.182150i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	571.482			1.14986
$498$		0			0
$499$	−368.000			−0.737475			−0.368737	−	0.929534i	$-0.620210\pi$
$499$	−368.000			−0.737475			−0.368737	+	0.929534i	$0.620210\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−275.158			−0.547034			−0.273517	−	0.961867i	$-0.588187\pi$
$503$	−275.158			−0.547034			−0.273517	+	0.961867i	$0.588187\pi$
$504$		0			0
$505$	576.000	−	359.199i	1.14059	−	0.711285i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	118.794i		−	0.233387i	−0.993168	−	0.116693i	$-0.962770\pi$
$509$		−	118.794i		−	0.233387i	0.993168	−	0.116693i	$-0.0372295\pi$
$510$		0			0
$511$	−756.000			−1.47945
$512$		0			0
$513$		0			0
$514$		0			0
$515$	238.118	−	148.492i	0.462364	−	0.288335i
$516$		0			0
$517$			89.7998i			0.173694i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			89.0955i			0.171009i	0.996338	+	0.0855043i	$0.0272501\pi$
$521$			89.0955i			0.171009i	−0.996338	+	0.0855043i	$0.972750\pi$
$522$		0			0
$523$		−	875.548i		−	1.67409i	−0.547136	−	0.837044i	$-0.684282\pi$
$523$		−	875.548i		−	1.67409i	0.547136	−	0.837044i	$-0.315718\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	275.158			0.522122
$528$		0			0
$529$	−501.000			−0.947070
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−619.106			−1.16155
$534$		0			0
$535$	28.0000	+	44.8999i	0.0523364	+	0.0839250i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			326.683i			0.606092i
$540$		0			0
$541$	434.000			0.802218			0.401109	−	0.916030i	$-0.368625\pi$
$541$	434.000			0.802218			0.401109	+	0.916030i	$0.368625\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−185.203	−	296.985i	−0.339821	−	0.544926i
$546$		0			0
$547$		−	112.250i		−	0.205210i	−0.994722	−	0.102605i	$-0.967282\pi$
$547$		−	112.250i		−	0.205210i	0.994722	−	0.102605i	$-0.0327177\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			169.706i			0.307996i
$552$		0			0
$553$			157.150i			0.284177i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	465.652			0.836000			0.418000	−	0.908447i	$-0.362731\pi$
$557$	465.652			0.836000			0.418000	+	0.908447i	$0.362731\pi$
$558$		0			0
$559$	252.000			0.450805
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−52.9150			−0.0939876			−0.0469938	−	0.998895i	$-0.514964\pi$
$563$	−52.9150			−0.0939876			−0.0469938	+	0.998895i	$0.514964\pi$
$564$		0			0
$565$	−364.000	−	583.699i	−0.644248	−	1.03309i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			640.639i			1.12590i	0.826490	+	0.562951i	$0.190334\pi$
$569$			640.639i			1.12590i	−0.826490	+	0.562951i	$0.809666\pi$
$570$		0			0
$571$	568.000			0.994746			0.497373	−	0.867537i	$-0.334298\pi$
$571$	568.000			0.994746			0.497373	+	0.867537i	$0.334298\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	58.2065	−	118.794i	0.101229	−	0.206598i
$576$		0			0
$577$			67.3498i			0.116724i	0.998295	+	0.0583621i	$0.0185878\pi$
$577$			67.3498i			0.116724i	−0.998295	+	0.0583621i	$0.981412\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	831.558i		−	1.43125i
$582$		0			0
$583$		−	359.199i		−	0.616122i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1026.55			1.74881			0.874405	−	0.485197i	$-0.161252\pi$
$587$	1026.55			1.74881			0.874405	+	0.485197i	$0.161252\pi$
$588$		0			0
$589$	520.000			0.882852
$590$		0			0
$591$		0			0
$592$		0			0
$593$	656.146			1.10649			0.553243	−	0.833020i	$-0.313390\pi$
$593$	656.146			1.10649			0.553243	+	0.833020i	$0.313390\pi$
$594$		0			0
$595$	−504.000	+	314.299i	−0.847059	+	0.528234i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	924.896i		−	1.54407i	−0.635582	−	0.772033i	$-0.719240\pi$
$599$		−	924.896i		−	1.54407i	0.635582	−	0.772033i	$-0.280760\pi$
$600$		0			0
$601$	788.000			1.31115			0.655574	−	0.755131i	$-0.272427\pi$
$601$	788.000			1.31115			0.655574	+	0.755131i	$0.272427\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	272.512	+	436.992i	0.450434	+	0.722301i
$606$		0			0
$607$			11.2250i			0.0184925i	0.999957	+	0.00924627i	$0.00294322\pi$
$607$			11.2250i			0.0184925i	−0.999957	+	0.00924627i	$0.997057\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			237.588i			0.388851i
$612$		0			0
$613$		−	572.474i		−	0.933888i	−0.884287	−	0.466944i	$-0.845355\pi$
$613$		−	572.474i		−	0.933888i	0.884287	−	0.466944i	$-0.154645\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−423.320			−0.686094			−0.343047	−	0.939318i	$-0.611459\pi$
$617$	−423.320			−0.686094			−0.343047	+	0.939318i	$0.611459\pi$
$618$		0			0
$619$	−194.000			−0.313409			−0.156704	−	0.987646i	$-0.550087\pi$
$619$	−194.000			−0.313409			−0.156704	+	0.987646i	$0.550087\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1000.09			1.60529
$624$		0			0
$625$	−383.000	−	493.899i	−0.612800	−	0.790238i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			356.382i			0.566585i
$630$		0			0
$631$	−1190.00			−1.88590			−0.942948	−	0.332941i	$-0.891959\pi$
$631$	−1190.00			−1.88590			−0.942948	+	0.332941i	$0.891959\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−714.353	+	445.477i	−1.12497	+	0.701539i
$636$		0			0
$637$			864.323i			1.35686i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	708.521i		−	1.10534i	−0.833401	−	0.552668i	$-0.813610\pi$
$641$		−	708.521i		−	1.10534i	0.833401	−	0.552668i	$-0.186390\pi$
$642$		0			0
$643$			651.048i			1.01252i	0.862382	+	0.506258i	$0.168972\pi$
$643$			651.048i			1.01252i	−0.862382	+	0.506258i	$0.831028\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1058.30			1.63570			0.817852	−	0.575429i	$-0.195165\pi$
$647$	1058.30			1.63570			0.817852	+	0.575429i	$0.195165\pi$
$648$		0			0
$649$	−198.000			−0.305085
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−470.944			−0.721200			−0.360600	−	0.932721i	$-0.617428\pi$
$653$	−470.944			−0.721200			−0.360600	+	0.932721i	$0.617428\pi$
$654$		0			0
$655$	630.000	−	392.874i	0.961832	−	0.599808i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	80.6102i		−	0.122322i	−0.998128	−	0.0611610i	$-0.980520\pi$
$659$		−	80.6102i		−	0.122322i	0.998128	−	0.0611610i	$-0.0194803\pi$
$660$		0			0
$661$	170.000			0.257186			0.128593	−	0.991697i	$-0.458954\pi$
$661$	170.000			0.257186			0.128593	+	0.991697i	$0.458954\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−952.470	+	593.970i	−1.43229	+	0.893188i
$666$		0			0
$667$			44.8999i			0.0673162i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			93.3381i			0.139103i
$672$		0			0
$673$			1055.15i			1.56783i	0.620870	+	0.783913i	$0.286779\pi$
$673$			1055.15i			1.56783i	−0.620870	+	0.783913i	$0.713221\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	830.766			1.22713			0.613564	−	0.789645i	$-0.289735\pi$
$677$	830.766			1.22713			0.613564	+	0.789645i	$0.289735\pi$
$678$		0			0
$679$	−252.000			−0.371134
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1037.13			−1.51850			−0.759249	−	0.650800i	$-0.774434\pi$
$683$	−1037.13			−1.51850			−0.759249	+	0.650800i	$0.774434\pi$
$684$		0			0
$685$	560.000	+	897.998i	0.817518	+	1.31095i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	950.352i		−	1.37932i
$690$		0			0
$691$	652.000			0.943560			0.471780	−	0.881716i	$-0.343612\pi$
$691$	652.000			0.943560			0.471780	+	0.881716i	$0.343612\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−545.025	−	873.984i	−0.784208	−	1.25753i
$696$		0			0
$697$			583.699i			0.837444i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			763.675i			1.08941i	0.838628	+	0.544704i	$0.183358\pi$
$701$			763.675i			1.08941i	−0.838628	+	0.544704i	$0.816642\pi$
$702$		0			0
$703$			673.498i			0.958035i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1523.95			−2.15552
$708$		0			0
$709$	158.000			0.222849			0.111425	−	0.993773i	$-0.464459\pi$
$709$	158.000			0.222849			0.111425	+	0.993773i	$0.464459\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	137.579			0.192958
$714$		0			0
$715$	−126.000	−	202.049i	−0.176224	−	0.282587i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			127.279i			0.177023i	0.996075	+	0.0885113i	$0.0282109\pi$
$719$			127.279i			0.177023i	−0.996075	+	0.0885113i	$0.971789\pi$
$720$		0			0
$721$	−630.000			−0.873786
$722$		0			0
$723$		0			0
$724$		0			0
$725$	190.494	+	93.3381i	0.262750	+	0.128742i
$726$		0			0
$727$			662.273i			0.910967i	0.890244	+	0.455484i	$0.150534\pi$
$727$			662.273i			0.910967i	−0.890244	+	0.455484i	$0.849466\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	237.588i		−	0.325018i
$732$		0			0
$733$		−	684.723i		−	0.934138i	−0.884221	−	0.467069i	$-0.845310\pi$
$733$		−	684.723i		−	0.934138i	0.884221	−	0.467069i	$-0.154690\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−380.988			−0.516945
$738$		0			0
$739$	1240.00			1.67794			0.838972	−	0.544175i	$-0.183157\pi$
$739$	1240.00			1.67794			0.838972	+	0.544175i	$0.183157\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	42.3320			0.0569745			0.0284872	−	0.999594i	$-0.490931\pi$
$743$	42.3320			0.0569745			0.0284872	+	0.999594i	$0.490931\pi$
$744$		0			0
$745$	−576.000	+	359.199i	−0.773154	+	0.482146i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	118.794i		−	0.158603i
$750$		0			0
$751$	154.000			0.205060			0.102530	−	0.994730i	$-0.467306\pi$
$751$	154.000			0.205060			0.102530	+	0.994730i	$0.467306\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	534.442	+	857.013i	0.707870	+	1.13512i
$756$		0			0
$757$		−	1313.32i		−	1.73490i	−0.497522	−	0.867452i	$-0.665756\pi$
$757$		−	1313.32i		−	1.73490i	0.497522	−	0.867452i	$-0.334244\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			504.874i			0.663435i	0.943379	+	0.331718i	$0.107628\pi$
$761$			504.874i			0.663435i	−0.943379	+	0.331718i	$0.892372\pi$
$762$		0			0
$763$			785.748i			1.02981i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−523.859			−0.682997
$768$		0			0
$769$	368.000			0.478544			0.239272	−	0.970953i	$-0.423091\pi$
$769$	368.000			0.478544			0.239272	+	0.970953i	$0.423091\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−153.454			−0.198517			−0.0992585	−	0.995062i	$-0.531647\pi$
$773$	−153.454			−0.198517			−0.0992585	+	0.995062i	$0.531647\pi$
$774$		0			0
$775$	286.000	−	583.699i	0.369032	−	0.753159i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1103.09i			1.41603i
$780$		0			0
$781$	216.000			0.276569
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−238.118	+	148.492i	−0.303335	+	0.189162i
$786$		0			0
$787$		−	426.549i		−	0.541994i	−0.962580	−	0.270997i	$-0.912647\pi$
$787$		−	426.549i		−	0.541994i	0.962580	−	0.270997i	$-0.0873533\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1544.32i			1.95237i
$792$		0			0
$793$			246.949i			0.311412i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−232.826			−0.292128			−0.146064	−	0.989275i	$-0.546661\pi$
$797$	−232.826			−0.292128			−0.146064	+	0.989275i	$0.546661\pi$
$798$		0			0
$799$	224.000			0.280350
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−285.741			−0.355842
$804$		0			0
$805$	−252.000	+	157.150i	−0.313043	+	0.195217i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			420.021i			0.519186i	0.965718	+	0.259593i	$0.0835884\pi$
$809$			420.021i			0.519186i	−0.965718	+	0.259593i	$0.916412\pi$
$810$		0			0
$811$	970.000			1.19605			0.598027	−	0.801476i	$-0.295951\pi$
$811$	970.000			1.19605			0.598027	+	0.801476i	$0.295951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	857.223	−	534.573i	1.05181	−	0.655917i
$816$		0			0
$817$		−	448.999i		−	0.549570i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			42.4264i			0.0516765i	0.999666	+	0.0258383i	$0.00822549\pi$
$821$			42.4264i			0.0516765i	−0.999666	+	0.0258383i	$0.991775\pi$
$822$		0			0
$823$		−	392.874i		−	0.477368i	−0.971097	−	0.238684i	$-0.923284\pi$
$823$		−	392.874i		−	0.477368i	0.971097	−	0.238684i	$-0.0767160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−560.899			−0.678234			−0.339117	−	0.940744i	$-0.610128\pi$
$827$	−560.899			−0.678234			−0.339117	+	0.940744i	$0.610128\pi$
$828$		0			0
$829$	1010.00			1.21834			0.609168	−	0.793041i	$-0.291504\pi$
$829$	1010.00			1.21834			0.609168	+	0.793041i	$0.291504\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	814.891			0.978261
$834$		0			0
$835$	490.000	+	785.748i	0.586826	+	0.941016i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	449.720i		−	0.536019i	−0.963416	−	0.268009i	$-0.913634\pi$
$839$		−	449.720i		−	0.536019i	0.963416	−	0.268009i	$-0.0863659\pi$
$840$		0			0
$841$	769.000			0.914388
$842$		0			0
$843$		0			0
$844$		0			0
$845$	113.767	+	182.434i	0.134636	+	0.215898i
$846$		0			0
$847$		−	1156.17i		−	1.36502i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			178.191i			0.209390i
$852$		0			0
$853$		−	931.673i		−	1.09223i	−0.837710	−	0.546115i	$-0.816106\pi$
$853$		−	931.673i		−	1.09223i	0.837710	−	0.546115i	$-0.183894\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1312.29			−1.53126			−0.765632	−	0.643279i	$-0.777573\pi$
$857$	−1312.29			−1.53126			−0.765632	+	0.643279i	$0.777573\pi$
$858$		0			0
$859$	694.000			0.807916			0.403958	−	0.914777i	$-0.367634\pi$
$859$	694.000			0.807916			0.403958	+	0.914777i	$0.367634\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−799.017			−0.925860			−0.462930	−	0.886395i	$-0.653202\pi$
$863$	−799.017			−0.925860			−0.462930	+	0.886395i	$0.653202\pi$
$864$		0			0
$865$	56.0000	+	89.7998i	0.0647399	+	0.103815i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			59.3970i			0.0683509i
$870$		0			0
$871$	−1008.00			−1.15729
$872$		0			0
$873$		0			0
$874$		0			0
$875$	142.871	+	1395.83i	0.163281	+	1.59523i
$876$		0			0
$877$			796.973i			0.908749i	0.890811	+	0.454375i	$0.150137\pi$
$877$			796.973i			0.908749i	−0.890811	+	0.454375i	$0.849863\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			827.315i			0.939063i	0.882916	+	0.469532i	$0.155577\pi$
$881$			827.315i			0.939063i	−0.882916	+	0.469532i	$0.844423\pi$
$882$		0			0
$883$			471.449i			0.533917i	0.963708	+	0.266959i	$0.0860187\pi$
$883$			471.449i			0.533917i	−0.963708	+	0.266959i	$0.913981\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1137.67			1.28261			0.641304	−	0.767287i	$-0.278394\pi$
$887$	1137.67			1.28261			0.641304	+	0.767287i	$0.278394\pi$
$888$		0			0
$889$	1890.00			2.12598
$890$		0			0
$891$		0			0
$892$		0			0
$893$	423.320			0.474043
$894$		0			0
$895$	−1026.00	+	639.823i	−1.14637	+	0.714886i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			220.617i			0.245403i
$900$		0			0
$901$	−896.000			−0.994451
$902$		0			0
$903$		0			0
$904$		0			0
$905$	195.786	+	313.955i	0.216338	+	0.346912i
$906$		0			0
$907$			1234.75i			1.36135i	0.732584	+	0.680676i	$0.238314\pi$
$907$			1234.75i			1.36135i	−0.732584	+	0.680676i	$0.761686\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			288.500i			0.316684i	0.987384	+	0.158342i	$0.0506149\pi$
$911$			288.500i			0.316684i	−0.987384	+	0.158342i	$0.949385\pi$
$912$		0			0
$913$		−	314.299i		−	0.344249i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1666.82			−1.81769
$918$		0			0
$919$	1078.00			1.17301			0.586507	−	0.809944i	$-0.300503\pi$
$919$	1078.00			1.17301			0.586507	+	0.809944i	$0.300503\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	571.482			0.619157
$924$		0			0
$925$	756.000	+	370.424i	0.817297	+	0.400458i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1752.21i		−	1.88613i	−0.332615	−	0.943063i	$-0.607931\pi$
$929$		−	1752.21i		−	1.88613i	0.332615	−	0.943063i	$-0.392069\pi$
$930$		0			0
$931$	1540.00			1.65414
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−190.494	+	118.794i	−0.203737	+	0.127052i
$936$		0			0
$937$			942.898i			1.00629i	0.864201	+	0.503147i	$0.167825\pi$
$937$			942.898i			1.00629i	−0.864201	+	0.503147i	$0.832175\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1026.72i			1.09109i	0.838080	+	0.545547i	$0.183678\pi$
$941$			1026.72i			1.09109i	−0.838080	+	0.545547i	$0.816322\pi$
$942$		0			0
$943$			291.849i			0.309490i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−497.401			−0.525239			−0.262619	−	0.964899i	$-0.584586\pi$
$947$	−497.401			−0.525239			−0.262619	+	0.964899i	$0.584586\pi$
$948$		0			0
$949$	−756.000			−0.796628
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−486.818			−0.510827			−0.255414	−	0.966832i	$-0.582212\pi$
$953$	−486.818			−0.510827			−0.255414	+	0.966832i	$0.582212\pi$
$954$		0			0
$955$	−684.000	+	426.549i	−0.716230	+	0.446648i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2375.88i		−	2.47745i
$960$		0			0
$961$	−285.000			−0.296566
$962$		0			0
$963$		0			0
$964$		0			0
$965$	761.976	−	475.176i	0.789613	−	0.492410i
$966$		0			0
$967$		−	662.273i		−	0.684874i	−0.939541	−	0.342437i	$-0.888748\pi$
$967$		−	662.273i		−	0.684874i	0.939541	−	0.342437i	$-0.111252\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1641.90i		−	1.69094i	−0.534024	−	0.845470i	$-0.679321\pi$
$971$		−	1641.90i		−	1.69094i	0.534024	−	0.845470i	$-0.320679\pi$
$972$		0			0
$973$			2312.34i			2.37651i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1291.13			1.32152			0.660761	−	0.750597i	$-0.270234\pi$
$977$	1291.13			1.32152			0.660761	+	0.750597i	$0.270234\pi$
$978$		0			0
$979$	378.000			0.386108
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−592.648			−0.602898			−0.301449	−	0.953482i	$-0.597470\pi$
$983$	−592.648			−0.602898			−0.301449	+	0.953482i	$0.597470\pi$
$984$		0			0
$985$	98.0000	+	157.150i	0.0994924	+	0.159543i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	118.794i		−	0.120115i
$990$		0			0
$991$	−1694.00			−1.70938			−0.854692	−	0.519135i	$-0.826254\pi$
$991$	−1694.00			−1.70938			−0.854692	+	0.519135i	$0.826254\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	661.438	+	1060.66i	0.664762	+	1.06599i
$996$		0			0
$997$			954.123i			0.956994i	0.878089	+	0.478497i	$0.158818\pi$
$997$			954.123i			0.956994i	−0.878089	+	0.478497i	$0.841182\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.c.a.449.4		4
3.2	odd	2	inner	720.3.c.a.449.1		4
4.3	odd	2		45.3.d.a.44.2	yes	4
5.2	odd	4		3600.3.l.s.1601.3		4
5.3	odd	4		3600.3.l.s.1601.1		4
5.4	even	2	inner	720.3.c.a.449.2		4
8.3	odd	2		2880.3.c.b.449.1		4
8.5	even	2		2880.3.c.g.449.1		4
12.11	even	2		45.3.d.a.44.3	yes	4
15.2	even	4		3600.3.l.s.1601.4		4
15.8	even	4		3600.3.l.s.1601.2		4
15.14	odd	2	inner	720.3.c.a.449.3		4
20.3	even	4		225.3.c.d.26.4		4
20.7	even	4		225.3.c.d.26.1		4
20.19	odd	2		45.3.d.a.44.4	yes	4
24.5	odd	2		2880.3.c.g.449.4		4
24.11	even	2		2880.3.c.b.449.4		4
36.7	odd	6		405.3.h.j.134.3		8
36.11	even	6		405.3.h.j.134.2		8
36.23	even	6		405.3.h.j.269.1		8
36.31	odd	6		405.3.h.j.269.4		8
40.19	odd	2		2880.3.c.b.449.3		4
40.29	even	2		2880.3.c.g.449.3		4
60.23	odd	4		225.3.c.d.26.2		4
60.47	odd	4		225.3.c.d.26.3		4
60.59	even	2		45.3.d.a.44.1	✓	4
120.29	odd	2		2880.3.c.g.449.2		4
120.59	even	2		2880.3.c.b.449.2		4
180.59	even	6		405.3.h.j.269.3		8
180.79	odd	6		405.3.h.j.134.1		8
180.119	even	6		405.3.h.j.134.4		8
180.139	odd	6		405.3.h.j.269.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.d.a.44.1	✓	4	60.59	even	2
45.3.d.a.44.2	yes	4	4.3	odd	2
45.3.d.a.44.3	yes	4	12.11	even	2
45.3.d.a.44.4	yes	4	20.19	odd	2
225.3.c.d.26.1		4	20.7	even	4
225.3.c.d.26.2		4	60.23	odd	4
225.3.c.d.26.3		4	60.47	odd	4
225.3.c.d.26.4		4	20.3	even	4
405.3.h.j.134.1		8	180.79	odd	6
405.3.h.j.134.2		8	36.11	even	6
405.3.h.j.134.3		8	36.7	odd	6
405.3.h.j.134.4		8	180.119	even	6
405.3.h.j.269.1		8	36.23	even	6
405.3.h.j.269.2		8	180.139	odd	6
405.3.h.j.269.3		8	180.59	even	6
405.3.h.j.269.4		8	36.31	odd	6
720.3.c.a.449.1		4	3.2	odd	2	inner
720.3.c.a.449.2		4	5.4	even	2	inner
720.3.c.a.449.3		4	15.14	odd	2	inner
720.3.c.a.449.4		4	1.1	even	1	trivial
2880.3.c.b.449.1		4	8.3	odd	2
2880.3.c.b.449.2		4	120.59	even	2
2880.3.c.b.449.3		4	40.19	odd	2
2880.3.c.b.449.4		4	24.11	even	2
2880.3.c.g.449.1		4	8.5	even	2
2880.3.c.g.449.2		4	120.29	odd	2
2880.3.c.g.449.3		4	40.29	even	2
2880.3.c.g.449.4		4	24.5	odd	2
3600.3.l.s.1601.1		4	5.3	odd	4
3600.3.l.s.1601.2		4	15.8	even	4
3600.3.l.s.1601.3		4	5.2	odd	4
3600.3.l.s.1601.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.64575 + 4.24264i) q^{5} -11.2250i q^{7} -4.24264i q^{11} -11.2250i q^{13} -10.5830 q^{17} -20.0000 q^{19} -5.29150 q^{23} +(-11.0000 + 22.4499i) q^{25} -8.48528i q^{29} -26.0000 q^{31} +(47.6235 - 29.6985i) q^{35} -33.6749i q^{37} -55.1543i q^{41} +22.4499i q^{43} -21.1660 q^{47} -77.0000 q^{49} +84.6640 q^{53} +(18.0000 - 11.2250i) q^{55} -46.6690i q^{59} -22.0000 q^{61} +(47.6235 - 29.6985i) q^{65} -89.7998i q^{67} +50.9117i q^{71} -67.3498i q^{73} -47.6235 q^{77} -14.0000 q^{79} +74.0810 q^{83} +(-28.0000 - 44.8999i) q^{85} +89.0955i q^{89} -126.000 q^{91} +(-52.9150 - 84.8528i) q^{95} -22.4499i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 80 q^{19} - 44 q^{25} - 104 q^{31} - 308 q^{49} + 72 q^{55} - 88 q^{61} - 56 q^{79} - 112 q^{85} - 504 q^{91}+O(q^{100}) \) 4 * q - 80 * q^19 - 44 * q^25 - 104 * q^31 - 308 * q^49 + 72 * q^55 - 88 * q^61 - 56 * q^79 - 112 * q^85 - 504 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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