LMFDB - Embedded newform 1152.3.m.a.415.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			415.3
Root			$-0.671462 + 1.24464i$ of defining polynomial
Character	$\chi$	$=$	1152.415
Dual form			1152.3.m.a.991.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.68585 + 3.68585i) q^{5} -9.66442 q^{7} +O(q^{10})$	$q+(3.68585 + 3.68585i) q^{5} -9.66442 q^{7} +(5.51806 - 5.51806i) q^{11} +(6.27131 - 6.27131i) q^{13} +6.78623 q^{17} +(-13.5181 - 13.5181i) q^{19} -17.0790 q^{23} +2.17092i q^{25} +(4.85677 - 4.85677i) q^{29} +5.25662i q^{31} +(-35.6216 - 35.6216i) q^{35} +(18.1856 + 18.1856i) q^{37} -48.2302i q^{41} +(54.5113 - 54.5113i) q^{43} +40.4015i q^{47} +44.4011 q^{49} +(10.8996 + 10.8996i) q^{53} +40.6774 q^{55} +(50.8898 - 50.8898i) q^{59} +(17.0147 - 17.0147i) q^{61} +46.2302 q^{65} +(-22.9191 - 22.9191i) q^{67} +51.6047 q^{71} -78.5032i q^{73} +(-53.3288 + 53.3288i) q^{77} -108.512i q^{79} +(57.3173 + 57.3173i) q^{83} +(25.0130 + 25.0130i) q^{85} +44.1276i q^{89} +(-60.6086 + 60.6086i) q^{91} -99.6510i q^{95} +112.700 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 6 * q - 2 * q^5 - 4 * q^7	$ 6 q - 2 q^{5} - 4 q^{7} - 18 q^{11} + 2 q^{13} + 4 q^{17} - 30 q^{19} - 60 q^{23} - 18 q^{29} - 100 q^{35} - 46 q^{37} + 114 q^{43} - 46 q^{49} + 78 q^{53} + 252 q^{55} + 206 q^{59} - 30 q^{61} - 12 q^{65} + 226 q^{67} + 260 q^{71} - 212 q^{77} + 318 q^{83} + 212 q^{85} - 188 q^{91} - 4 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 - 4 * q^7 - 18 * q^11 + 2 * q^13 + 4 * q^17 - 30 * q^19 - 60 * q^23 - 18 * q^29 - 100 * q^35 - 46 * q^37 + 114 * q^43 - 46 * q^49 + 78 * q^53 + 252 * q^55 + 206 * q^59 - 30 * q^61 - 12 * q^65 + 226 * q^67 + 260 * q^71 - 212 * q^77 + 318 * q^83 + 212 * q^85 - 188 * q^91 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	3.68585	+	3.68585i	0.737169	+	0.737169i	0.972029	−	0.234860i	$-0.0754632\pi$
$5$	3.68585	+	3.68585i	0.737169	+	0.737169i	−0.234860	+	0.972029i	$0.575463\pi$
$6$		0			0
$7$	−9.66442			−1.38063			−0.690316	−	0.723508i	$-0.742528\pi$
$7$	−9.66442			−1.38063			−0.690316	+	0.723508i	$0.742528\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.51806	−	5.51806i	0.501642	−	0.501642i	−0.410306	−	0.911948i	$-0.634578\pi$
$11$	5.51806	−	5.51806i	0.501642	−	0.501642i	0.911948	+	0.410306i	$0.134578\pi$
$12$		0			0
$13$	6.27131	−	6.27131i	0.482408	−	0.482408i	−0.423492	−	0.905900i	$-0.639196\pi$
$13$	6.27131	−	6.27131i	0.482408	−	0.482408i	0.905900	+	0.423492i	$0.139196\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.78623			0.399190			0.199595	−	0.979878i	$-0.436037\pi$
$17$	6.78623			0.399190			0.199595	+	0.979878i	$0.436037\pi$
$18$		0			0
$19$	−13.5181	−	13.5181i	−0.711477	−	0.711477i	0.255367	−	0.966844i	$-0.417804\pi$
$19$	−13.5181	−	13.5181i	−0.711477	−	0.711477i	−0.966844	+	0.255367i	$0.917804\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.0790			−0.742564			−0.371282	−	0.928520i	$-0.621082\pi$
$23$	−17.0790			−0.742564			−0.371282	+	0.928520i	$0.621082\pi$
$24$		0			0
$25$			2.17092i			0.0868370i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.85677	−	4.85677i	0.167475	−	0.167475i	−0.618394	−	0.785868i	$-0.712216\pi$
$29$	4.85677	−	4.85677i	0.167475	−	0.167475i	0.785868	+	0.618394i	$0.212216\pi$
$30$		0			0
$31$			5.25662i			0.169568i	0.996399	+	0.0847841i	$0.0270201\pi$
$31$			5.25662i			0.169568i	−0.996399	+	0.0847841i	$0.972980\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−35.6216	−	35.6216i	−1.01776	−	1.01776i
$36$		0			0
$37$	18.1856	+	18.1856i	0.491503	+	0.491503i	0.908780	−	0.417276i	$-0.137015\pi$
$37$	18.1856	+	18.1856i	0.491503	+	0.491503i	−0.417276	+	0.908780i	$0.637015\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	48.2302i		−	1.17635i	−0.808735	−	0.588173i	$-0.799848\pi$
$41$		−	48.2302i		−	1.17635i	0.808735	−	0.588173i	$-0.200152\pi$
$42$		0			0
$43$	54.5113	−	54.5113i	1.26771	−	1.26771i	0.320435	−	0.947271i	$-0.396171\pi$
$43$	54.5113	−	54.5113i	1.26771	−	1.26771i	0.947271	−	0.320435i	$-0.103829\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			40.4015i			0.859607i	0.902922	+	0.429804i	$0.141417\pi$
$47$			40.4015i			0.859607i	−0.902922	+	0.429804i	$0.858583\pi$
$48$		0			0
$49$	44.4011			0.906144
$50$		0			0
$51$		0			0
$52$		0			0
$53$	10.8996	+	10.8996i	0.205653	+	0.205653i	0.802417	−	0.596764i	$-0.203547\pi$
$53$	10.8996	+	10.8996i	0.205653	+	0.205653i	−0.596764	+	0.802417i	$0.703547\pi$
$54$		0			0
$55$	40.6774			0.739590
$56$		0			0
$57$		0			0
$58$		0			0
$59$	50.8898	−	50.8898i	0.862538	−	0.862538i	−0.129094	−	0.991632i	$-0.541207\pi$
$59$	50.8898	−	50.8898i	0.862538	−	0.862538i	0.991632	+	0.129094i	$0.0412070\pi$
$60$		0			0
$61$	17.0147	−	17.0147i	0.278929	−	0.278929i	−0.553752	−	0.832682i	$-0.686804\pi$
$61$	17.0147	−	17.0147i	0.278929	−	0.278929i	0.832682	+	0.553752i	$0.186804\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	46.2302			0.711233
$66$		0			0
$67$	−22.9191	−	22.9191i	−0.342077	−	0.342077i	0.515071	−	0.857148i	$-0.327766\pi$
$67$	−22.9191	−	22.9191i	−0.342077	−	0.342077i	−0.857148	+	0.515071i	$0.827766\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	51.6047			0.726827			0.363414	−	0.931628i	$-0.381611\pi$
$71$	51.6047			0.726827			0.363414	+	0.931628i	$0.381611\pi$
$72$		0			0
$73$		−	78.5032i		−	1.07539i	−0.843141	−	0.537693i	$-0.819296\pi$
$73$		−	78.5032i		−	1.07539i	0.843141	−	0.537693i	$-0.180704\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−53.3288	+	53.3288i	−0.692582	+	0.692582i
$78$		0			0
$79$		−	108.512i		−	1.37357i	−0.726859	−	0.686787i	$-0.759021\pi$
$79$		−	108.512i		−	1.37357i	0.726859	−	0.686787i	$-0.240979\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	57.3173	+	57.3173i	0.690570	+	0.690570i	0.962357	−	0.271788i	$-0.0876148\pi$
$83$	57.3173	+	57.3173i	0.690570	+	0.690570i	−0.271788	+	0.962357i	$0.587615\pi$
$84$		0			0
$85$	25.0130	+	25.0130i	0.294271	+	0.294271i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			44.1276i			0.495816i	0.968784	+	0.247908i	$0.0797431\pi$
$89$			44.1276i			0.495816i	−0.968784	+	0.247908i	$0.920257\pi$
$90$		0			0
$91$	−60.6086	+	60.6086i	−0.666028	+	0.666028i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	99.6510i		−	1.04896i
$96$		0			0
$97$	112.700			1.16185			0.580926	−	0.813956i	$-0.302691\pi$
$97$	112.700			1.16185			0.580926	+	0.813956i	$0.302691\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	97.3859	+	97.3859i	0.964217	+	0.964217i	0.999382	−	0.0351644i	$-0.0111955\pi$
$101$	97.3859	+	97.3859i	0.964217	+	0.964217i	−0.0351644	+	0.999382i	$0.511195\pi$
$102$		0			0
$103$	−138.698			−1.34658			−0.673290	−	0.739379i	$-0.735119\pi$
$103$	−138.698			−1.34658			−0.673290	+	0.739379i	$0.735119\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−31.7386	+	31.7386i	−0.296622	+	0.296622i	−0.839689	−	0.543067i	$-0.817263\pi$
$107$	−31.7386	+	31.7386i	−0.296622	+	0.296622i	0.543067	+	0.839689i	$0.317263\pi$
$108$		0			0
$109$	−0.712308	+	0.712308i	−0.00653493	+	0.00653493i	−0.710367	−	0.703832i	$-0.751471\pi$
$109$	−0.712308	+	0.712308i	−0.00653493	+	0.00653493i	0.703832	+	0.710367i	$0.251471\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.8888			−0.131759			−0.0658795	−	0.997828i	$-0.520985\pi$
$113$	−14.8888			−0.131759			−0.0658795	+	0.997828i	$0.520985\pi$
$114$		0			0
$115$	−62.9504	−	62.9504i	−0.547395	−	0.547395i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−65.5850			−0.551134
$120$		0			0
$121$			60.1021i			0.496711i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	84.1445	−	84.1445i	0.673156	−	0.673156i
$126$		0			0
$127$		−	106.861i		−	0.841425i	−0.907194	−	0.420712i	$-0.861780\pi$
$127$		−	106.861i		−	0.841425i	0.907194	−	0.420712i	$-0.138220\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−153.198	−	153.198i	−1.16945	−	1.16945i	−0.982338	−	0.187116i	$-0.940086\pi$
$131$	−153.198	−	153.198i	−1.16945	−	1.16945i	−0.187116	−	0.982338i	$-0.559914\pi$
$132$		0			0
$133$	130.644	+	130.644i	0.982287	+	0.982287i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	75.1700i		−	0.548686i	−0.961632	−	0.274343i	$-0.911540\pi$
$137$		−	75.1700i		−	0.548686i	0.961632	−	0.274343i	$-0.0884604\pi$
$138$		0			0
$139$	107.425	−	107.425i	0.772843	−	0.772843i	−0.205760	−	0.978603i	$-0.565966\pi$
$139$	107.425	−	107.425i	0.772843	−	0.772843i	0.978603	+	0.205760i	$0.0659665\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	69.2109i		−	0.483992i
$144$		0			0
$145$	35.8026			0.246915
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−146.031	−	146.031i	−0.980074	−	0.980074i	0.0197310	−	0.999805i	$-0.493719\pi$
$149$	−146.031	−	146.031i	−0.980074	−	0.980074i	−0.999805	+	0.0197310i	$0.993719\pi$
$150$		0			0
$151$	220.513			1.46035			0.730175	−	0.683260i	$-0.239439\pi$
$151$	220.513			1.46035			0.730175	+	0.683260i	$0.239439\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−19.3751	+	19.3751i	−0.125000	+	0.125000i
$156$		0			0
$157$	109.561	−	109.561i	0.697839	−	0.697839i	−0.266105	−	0.963944i	$-0.585737\pi$
$157$	109.561	−	109.561i	0.697839	−	0.697839i	0.963944	+	0.266105i	$0.0857369\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	165.058			1.02521
$162$		0			0
$163$	−56.7781	−	56.7781i	−0.348332	−	0.348332i	0.511156	−	0.859488i	$-0.329217\pi$
$163$	−56.7781	−	56.7781i	−0.348332	−	0.348332i	−0.859488	+	0.511156i	$0.829217\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−106.677			−0.638781			−0.319391	−	0.947623i	$-0.603478\pi$
$167$	−106.677			−0.638781			−0.319391	+	0.947623i	$0.603478\pi$
$168$		0			0
$169$			90.3414i			0.534564i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	178.360	−	178.360i	1.03098	−	1.03098i	0.0314805	−	0.999504i	$-0.489978\pi$
$173$	178.360	−	178.360i	1.03098	−	1.03098i	0.999504	−	0.0314805i	$-0.0100222\pi$
$174$		0			0
$175$		−	20.9807i		−	0.119890i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	60.4622	+	60.4622i	0.337778	+	0.337778i	0.855530	−	0.517753i	$-0.173231\pi$
$179$	60.4622	+	60.4622i	0.337778	+	0.337778i	−0.517753	+	0.855530i	$0.673231\pi$
$180$		0			0
$181$	−147.113	−	147.113i	−0.812779	−	0.812779i	0.172271	−	0.985050i	$-0.444890\pi$
$181$	−147.113	−	147.113i	−0.812779	−	0.812779i	−0.985050	+	0.172271i	$0.944890\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			134.059i			0.724642i
$186$		0			0
$187$	37.4468	−	37.4468i	0.200250	−	0.200250i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	106.861i		−	0.559481i	−0.960076	−	0.279741i	$-0.909752\pi$
$191$		−	106.861i		−	0.559481i	0.960076	−	0.279741i	$-0.0902485\pi$
$192$		0			0
$193$	68.1873			0.353302			0.176651	−	0.984274i	$-0.443474\pi$
$193$	68.1873			0.353302			0.176651	+	0.984274i	$0.443474\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−61.8529	−	61.8529i	−0.313974	−	0.313974i	0.532473	−	0.846447i	$-0.321263\pi$
$197$	−61.8529	−	61.8529i	−0.313974	−	0.313974i	−0.846447	+	0.532473i	$0.821263\pi$
$198$		0			0
$199$	−158.466			−0.796310			−0.398155	−	0.917318i	$-0.630349\pi$
$199$	−158.466			−0.796310			−0.398155	+	0.917318i	$0.630349\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−46.9379	+	46.9379i	−0.231221	+	0.231221i
$204$		0			0
$205$	177.769	−	177.769i	0.867166	−	0.867166i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−149.187			−0.713813
$210$		0			0
$211$	197.031	+	197.031i	0.933798	+	0.933798i	0.997941	−	0.0641430i	$-0.0204314\pi$
$211$	197.031	+	197.031i	0.933798	+	0.933798i	−0.0641430	+	0.997941i	$0.520431\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	401.841			1.86903
$216$		0			0
$217$		−	50.8022i		−	0.234111i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	42.5585	−	42.5585i	0.192573	−	0.192573i
$222$		0			0
$223$		−	15.7698i		−	0.0707168i	−0.999375	−	0.0353584i	$-0.988743\pi$
$223$		−	15.7698i		−	0.0707168i	0.999375	−	0.0353584i	$-0.0112573\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	199.289	+	199.289i	0.877927	+	0.877927i	0.993320	−	0.115393i	$-0.0368127\pi$
$227$	199.289	+	199.289i	0.877927	+	0.877927i	−0.115393	+	0.993320i	$0.536813\pi$
$228$		0			0
$229$	−230.522	−	230.522i	−1.00664	−	1.00664i	−0.999978	−	0.00666715i	$-0.997878\pi$
$229$	−230.522	−	230.522i	−1.00664	−	1.00664i	−0.00666715	−	0.999978i	$-0.502122\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			344.791i			1.47979i	0.672722	+	0.739895i	$0.265125\pi$
$233$			344.791i			1.47979i	−0.672722	+	0.739895i	$0.734875\pi$
$234$		0			0
$235$	−148.914	+	148.914i	−0.633676	+	0.633676i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	77.1978i		−	0.323004i	−0.986872	−	0.161502i	$-0.948366\pi$
$239$		−	77.1978i		−	0.323004i	0.986872	−	0.161502i	$-0.0516337\pi$
$240$		0			0
$241$	−293.483			−1.21777			−0.608885	−	0.793259i	$-0.708383\pi$
$241$	−293.483			−1.21777			−0.608885	+	0.793259i	$0.708383\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	163.656	+	163.656i	0.667982	+	0.667982i
$246$		0			0
$247$	−169.552			−0.686445
$248$		0			0
$249$		0			0
$250$		0			0
$251$	79.6322	−	79.6322i	0.317260	−	0.317260i	−0.530454	−	0.847714i	$-0.677979\pi$
$251$	79.6322	−	79.6322i	0.317260	−	0.317260i	0.847714	+	0.530454i	$0.177979\pi$
$252$		0			0
$253$	−94.2427	+	94.2427i	−0.372501	+	0.372501i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−221.860			−0.863270			−0.431635	−	0.902048i	$-0.642063\pi$
$257$	−221.860			−0.863270			−0.431635	+	0.902048i	$0.642063\pi$
$258$		0			0
$259$	−175.753	−	175.753i	−0.678585	−	0.678585i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−374.223			−1.42290			−0.711451	−	0.702736i	$-0.751961\pi$
$263$	−374.223			−1.42290			−0.711451	+	0.702736i	$0.751961\pi$
$264$		0			0
$265$			80.3486i			0.303202i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−357.970	+	357.970i	−1.33075	+	1.33075i	−0.426042	+	0.904704i	$0.640092\pi$
$269$	−357.970	+	357.970i	−1.33075	+	1.33075i	−0.904704	+	0.426042i	$0.859908\pi$
$270$		0			0
$271$			359.030i			1.32484i	0.749135	+	0.662418i	$0.230470\pi$
$271$			359.030i			1.32484i	−0.749135	+	0.662418i	$0.769530\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.9793	+	11.9793i	0.0435610	+	0.0435610i
$276$		0			0
$277$	351.765	+	351.765i	1.26991	+	1.26991i	0.946134	+	0.323775i	$0.104952\pi$
$277$	351.765	+	351.765i	1.26991	+	1.26991i	0.323775	+	0.946134i	$0.395048\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	191.390i		−	0.681103i	−0.940226	−	0.340552i	$-0.889386\pi$
$281$		−	191.390i		−	0.681103i	0.940226	−	0.340552i	$-0.110614\pi$
$282$		0			0
$283$	31.3119	−	31.3119i	0.110643	−	0.110643i	−0.649618	−	0.760261i	$-0.725071\pi$
$283$	31.3119	−	31.3119i	0.110643	−	0.110643i	0.760261	+	0.649618i	$0.225071\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			466.117i			1.62410i
$288$		0			0
$289$	−242.947			−0.840647
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−92.0889	−	92.0889i	−0.314297	−	0.314297i	0.532275	−	0.846572i	$-0.321337\pi$
$293$	−92.0889	−	92.0889i	−0.314297	−	0.314297i	−0.846572	+	0.532275i	$0.821337\pi$
$294$		0			0
$295$	375.144			1.27167
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−107.107	+	107.107i	−0.358219	+	0.358219i
$300$		0			0
$301$	−526.821	+	526.821i	−1.75023	+	1.75023i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	125.427			0.411236
$306$		0			0
$307$	257.566	+	257.566i	0.838978	+	0.838978i	0.988724	−	0.149746i	$-0.0478457\pi$
$307$	257.566	+	257.566i	0.838978	+	0.838978i	−0.149746	+	0.988724i	$0.547846\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	130.914			0.420946			0.210473	−	0.977600i	$-0.432500\pi$
$311$	130.914			0.420946			0.210473	+	0.977600i	$0.432500\pi$
$312$		0			0
$313$			51.8354i			0.165608i	0.996566	+	0.0828041i	$0.0263876\pi$
$313$			51.8354i			0.165608i	−0.996566	+	0.0828041i	$0.973612\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−109.636	+	109.636i	−0.345856	+	0.345856i	−0.858563	−	0.512707i	$-0.828643\pi$
$317$	−109.636	+	109.636i	−0.345856	+	0.345856i	0.512707	+	0.858563i	$0.328643\pi$
$318$		0			0
$319$		−	53.5999i		−	0.168025i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−91.7367	−	91.7367i	−0.284014	−	0.284014i
$324$		0			0
$325$	13.6145	+	13.6145i	0.0418909	+	0.0418909i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	390.458i		−	1.18680i
$330$		0			0
$331$	−323.226	+	323.226i	−0.976515	+	0.976515i	−0.999730	−	0.0232157i	$-0.992610\pi$
$331$	−323.226	+	323.226i	−0.976515	+	0.976515i	0.0232157	+	0.999730i	$0.492610\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	168.953i		−	0.504337i
$336$		0			0
$337$	315.159			0.935191			0.467596	−	0.883943i	$-0.345120\pi$
$337$	315.159			0.935191			0.467596	+	0.883943i	$0.345120\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	29.0063	+	29.0063i	0.0850625	+	0.0850625i
$342$		0			0
$343$	44.4459			0.129580
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−307.568	+	307.568i	−0.886363	+	0.886363i	−0.994172	−	0.107809i	$-0.965616\pi$
$347$	−307.568	+	307.568i	−0.886363	+	0.886363i	0.107809	+	0.994172i	$0.465616\pi$
$348$		0			0
$349$	−170.461	+	170.461i	−0.488427	+	0.488427i	−0.907810	−	0.419382i	$-0.862247\pi$
$349$	−170.461	+	170.461i	−0.488427	+	0.488427i	0.419382	+	0.907810i	$0.362247\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−238.136			−0.674606			−0.337303	−	0.941396i	$-0.609515\pi$
$353$	−238.136			−0.674606			−0.337303	+	0.941396i	$0.609515\pi$
$354$		0			0
$355$	190.207	+	190.207i	0.535795	+	0.535795i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−33.6470			−0.0937241			−0.0468620	−	0.998901i	$-0.514922\pi$
$359$	−33.6470			−0.0937241			−0.0468620	+	0.998901i	$0.514922\pi$
$360$		0			0
$361$			4.47577i			0.0123983i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	289.351	−	289.351i	0.792741	−	0.792741i
$366$		0			0
$367$			240.758i			0.656016i	0.944675	+	0.328008i	$0.106377\pi$
$367$			240.758i			0.656016i	−0.944675	+	0.328008i	$0.893623\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−105.339	−	105.339i	−0.283931	−	0.283931i
$372$		0			0
$373$	−432.504	−	432.504i	−1.15953	−	1.15953i	−0.984577	−	0.174951i	$-0.944023\pi$
$373$	−432.504	−	432.504i	−1.15953	−	1.15953i	−0.174951	−	0.984577i	$-0.555977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	60.9166i		−	0.161583i
$378$		0			0
$379$	174.716	−	174.716i	0.460993	−	0.460993i	−0.437988	−	0.898981i	$-0.644309\pi$
$379$	174.716	−	174.716i	0.460993	−	0.460993i	0.898981	+	0.437988i	$0.144309\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			673.381i			1.75817i	0.476661	+	0.879087i	$0.341847\pi$
$383$			673.381i			1.75817i	−0.476661	+	0.879087i	$0.658153\pi$
$384$		0			0
$385$	−393.124			−1.02110
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−274.646	−	274.646i	−0.706031	−	0.706031i	0.259667	−	0.965698i	$-0.416387\pi$
$389$	−274.646	−	274.646i	−0.706031	−	0.706031i	−0.965698	+	0.259667i	$0.916387\pi$
$390$		0			0
$391$	−115.902			−0.296424
$392$		0			0
$393$		0			0
$394$		0			0
$395$	399.960	−	399.960i	1.01256	−	1.01256i
$396$		0			0
$397$	271.254	−	271.254i	0.683259	−	0.683259i	−0.277474	−	0.960733i	$-0.589497\pi$
$397$	271.254	−	271.254i	0.683259	−	0.683259i	0.960733	+	0.277474i	$0.0894972\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	415.193			1.03539			0.517697	−	0.855564i	$-0.326790\pi$
$401$	415.193			1.03539			0.517697	+	0.855564i	$0.326790\pi$
$402$		0			0
$403$	32.9659	+	32.9659i	0.0818011	+	0.0818011i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	200.699			0.493117
$408$		0			0
$409$		−	634.686i		−	1.55180i	−0.630856	−	0.775900i	$-0.717296\pi$
$409$		−	634.686i		−	1.55180i	0.630856	−	0.775900i	$-0.282704\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−491.820	+	491.820i	−1.19085	+	1.19085i
$414$		0			0
$415$			422.525i			1.01813i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.2687	+	19.2687i	0.0459873	+	0.0459873i	0.729726	−	0.683739i	$-0.239647\pi$
$419$	19.2687	+	19.2687i	0.0459873	+	0.0459873i	−0.683739	+	0.729726i	$0.739647\pi$
$420$		0			0
$421$	−244.505	−	244.505i	−0.580773	−	0.580773i	0.354343	−	0.935116i	$-0.384705\pi$
$421$	−244.505	−	244.505i	−0.580773	−	0.580773i	−0.935116	+	0.354343i	$0.884705\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			14.7324i			0.0346644i
$426$		0			0
$427$	−164.437	+	164.437i	−0.385099	+	0.385099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	337.331i		−	0.782670i	−0.920248	−	0.391335i	$-0.872013\pi$
$431$		−	337.331i		−	0.782670i	0.920248	−	0.391335i	$-0.127987\pi$
$432$		0			0
$433$	−424.560			−0.980508			−0.490254	−	0.871580i	$-0.663096\pi$
$433$	−424.560			−0.980508			−0.490254	+	0.871580i	$0.663096\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	230.874	+	230.874i	0.528317	+	0.528317i
$438$		0			0
$439$	−162.004			−0.369029			−0.184514	−	0.982830i	$-0.559071\pi$
$439$	−162.004			−0.369029			−0.184514	+	0.982830i	$0.559071\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	492.189	−	492.189i	1.11104	−	1.11104i	0.118026	−	0.993010i	$-0.462343\pi$
$443$	492.189	−	492.189i	1.11104	−	1.11104i	0.993010	−	0.118026i	$-0.0376567\pi$
$444$		0			0
$445$	−162.648	+	162.648i	−0.365500	+	0.365500i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−195.434			−0.435266			−0.217633	−	0.976031i	$-0.569834\pi$
$449$	−195.434			−0.435266			−0.217633	+	0.976031i	$0.569834\pi$
$450$		0			0
$451$	−266.137	−	266.137i	−0.590104	−	0.590104i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−446.788			−0.981951
$456$		0			0
$457$		−	386.874i		−	0.846552i	−0.906001	−	0.423276i	$-0.860880\pi$
$457$		−	386.874i		−	0.846552i	0.906001	−	0.423276i	$-0.139120\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−174.401	+	174.401i	−0.378310	+	0.378310i	−0.870492	−	0.492182i	$-0.836199\pi$
$461$	−174.401	+	174.401i	−0.378310	+	0.378310i	0.492182	+	0.870492i	$0.336199\pi$
$462$		0			0
$463$		−	60.5295i		−	0.130733i	−0.997861	−	0.0653666i	$-0.979178\pi$
$463$		−	60.5295i		−	0.130733i	0.997861	−	0.0653666i	$-0.0208217\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	306.482	+	306.482i	0.656279	+	0.656279i	0.954497	−	0.298219i	$-0.0963926\pi$
$467$	306.482	+	306.482i	0.656279	+	0.656279i	−0.298219	+	0.954497i	$0.596393\pi$
$468$		0			0
$469$	221.500	+	221.500i	0.472282	+	0.472282i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	601.593i		−	1.27187i
$474$		0			0
$475$	29.3467	−	29.3467i	0.0617825	−	0.0617825i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			376.452i			0.785912i	0.919557	+	0.392956i	$0.128547\pi$
$479$			376.452i			0.785912i	−0.919557	+	0.392956i	$0.871453\pi$
$480$		0			0
$481$	228.095			0.474210
$482$		0			0
$483$		0			0
$484$		0			0
$485$	415.393	+	415.393i	0.856481	+	0.856481i
$486$		0			0
$487$	77.2033			0.158528			0.0792641	−	0.996854i	$-0.474743\pi$
$487$	77.2033			0.158528			0.0792641	+	0.996854i	$0.474743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−581.438	+	581.438i	−1.18419	+	1.18419i	−0.205543	+	0.978648i	$0.565896\pi$
$491$	−581.438	+	581.438i	−1.18419	+	1.18419i	−0.978648	+	0.205543i	$0.934104\pi$
$492$		0			0
$493$	32.9592	−	32.9592i	0.0668543	−	0.0668543i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−498.730			−1.00348
$498$		0			0
$499$	174.006	+	174.006i	0.348709	+	0.348709i	0.859629	−	0.510920i	$-0.170695\pi$
$499$	174.006	+	174.006i	0.348709	+	0.348709i	−0.510920	+	0.859629i	$0.670695\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	355.262			0.706286			0.353143	−	0.935569i	$-0.385113\pi$
$503$	355.262			0.706286			0.353143	+	0.935569i	$0.385113\pi$
$504$		0			0
$505$			717.899i			1.42158i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	279.667	−	279.667i	0.549444	−	0.549444i	−0.376836	−	0.926280i	$-0.622988\pi$
$509$	279.667	−	279.667i	0.549444	−	0.549444i	0.926280	+	0.376836i	$0.122988\pi$
$510$		0			0
$511$			758.688i			1.48471i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−511.218	−	511.218i	−0.992657	−	0.992657i
$516$		0			0
$517$	222.938	+	222.938i	0.431215	+	0.431215i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			705.745i			1.35460i	0.735708	+	0.677299i	$0.236849\pi$
$521$			705.745i			1.35460i	−0.735708	+	0.677299i	$0.763151\pi$
$522$		0			0
$523$	186.762	−	186.762i	0.357098	−	0.357098i	−0.505644	−	0.862742i	$-0.668745\pi$
$523$	186.762	−	186.762i	0.357098	−	0.357098i	0.862742	+	0.505644i	$0.168745\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			35.6726i			0.0676899i
$528$		0			0
$529$	−237.309			−0.448599
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−302.466	−	302.466i	−0.567479	−	0.567479i
$534$		0			0
$535$	−233.967			−0.437321
$536$		0			0
$537$		0			0
$538$		0			0
$539$	245.008	−	245.008i	0.454560	−	0.454560i
$540$		0			0
$541$	−119.274	+	119.274i	−0.220470	+	0.220470i	−0.808696	−	0.588226i	$-0.799826\pi$
$541$	−119.274	+	119.274i	−0.220470	+	0.220470i	0.588226	+	0.808696i	$0.299826\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.25091			−0.00963470
$546$		0			0
$547$	141.472	+	141.472i	0.258632	+	0.258632i	0.824498	−	0.565865i	$-0.191458\pi$
$547$	141.472	+	141.472i	0.258632	+	0.258632i	−0.565865	+	0.824498i	$0.691458\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−131.308			−0.238309
$552$		0			0
$553$			1048.71i			1.89640i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	375.881	−	375.881i	0.674831	−	0.674831i	−0.283995	−	0.958826i	$-0.591660\pi$
$557$	375.881	−	375.881i	0.674831	−	0.674831i	0.958826	+	0.283995i	$0.0916599\pi$
$558$		0			0
$559$		−	683.715i		−	1.22310i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	305.349	+	305.349i	0.542360	+	0.542360i	0.924220	−	0.381860i	$-0.124716\pi$
$563$	305.349	+	305.349i	0.542360	+	0.542360i	−0.381860	+	0.924220i	$0.624716\pi$
$564$		0			0
$565$	−54.8777	−	54.8777i	−0.0971287	−	0.0971287i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			296.778i			0.521578i	0.965396	+	0.260789i	$0.0839827\pi$
$569$			296.778i			0.521578i	−0.965396	+	0.260789i	$0.916017\pi$
$570$		0			0
$571$	347.717	−	347.717i	0.608961	−	0.608961i	−0.333714	−	0.942674i	$-0.608302\pi$
$571$	347.717	−	347.717i	0.608961	−	0.608961i	0.942674	+	0.333714i	$0.108302\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	37.0771i		−	0.0644820i
$576$		0			0
$577$	−189.382			−0.328218			−0.164109	−	0.986442i	$-0.552475\pi$
$577$	−189.382			−0.328218			−0.164109	+	0.986442i	$0.552475\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−553.939	−	553.939i	−0.953423	−	0.953423i
$582$		0			0
$583$	120.289			0.206328
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−641.187	+	641.187i	−1.09231	+	1.09231i	−0.0970301	+	0.995281i	$0.530934\pi$
$587$	−641.187	+	641.187i	−1.09231	+	1.09231i	−0.995281	+	0.0970301i	$0.969066\pi$
$588$		0			0
$589$	71.0592	−	71.0592i	0.120644	−	0.120644i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	127.909			0.215697			0.107849	−	0.994167i	$-0.465604\pi$
$593$	127.909			0.215697			0.107849	+	0.994167i	$0.465604\pi$
$594$		0			0
$595$	−241.736	−	241.736i	−0.406279	−	0.406279i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−794.804			−1.32688			−0.663442	−	0.748227i	$-0.730905\pi$
$599$	−794.804			−1.32688			−0.663442	+	0.748227i	$0.730905\pi$
$600$		0			0
$601$			89.2746i			0.148543i	0.997238	+	0.0742717i	$0.0236632\pi$
$601$			89.2746i			0.148543i	−0.997238	+	0.0742717i	$0.976337\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−221.527	+	221.527i	−0.366160	+	0.366160i
$606$		0			0
$607$			316.002i			0.520596i	0.965528	+	0.260298i	$0.0838208\pi$
$607$			316.002i			0.520596i	−0.965528	+	0.260298i	$0.916179\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	253.370	+	253.370i	0.414682	+	0.414682i
$612$		0			0
$613$	192.003	+	192.003i	0.313219	+	0.313219i	0.846155	−	0.532936i	$-0.178911\pi$
$613$	192.003	+	192.003i	0.313219	+	0.313219i	−0.532936	+	0.846155i	$0.678911\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	105.762i		−	0.171413i	−0.996320	−	0.0857066i	$-0.972685\pi$
$617$		−	105.762i		−	0.171413i	0.996320	−	0.0857066i	$-0.0273148\pi$
$618$		0			0
$619$	−553.819	+	553.819i	−0.894699	+	0.894699i	−0.994961	−	0.100262i	$-0.968032\pi$
$619$	−553.819	+	553.819i	−0.894699	+	0.894699i	0.100262	+	0.994961i	$0.468032\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	426.468i		−	0.684539i
$624$		0			0
$625$	674.560			1.07930
$626$		0			0
$627$		0			0
$628$		0			0
$629$	123.412	+	123.412i	0.196203	+	0.196203i
$630$		0			0
$631$	762.907			1.20904			0.604522	−	0.796589i	$-0.293364\pi$
$631$	762.907			1.20904			0.604522	+	0.796589i	$0.293364\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	393.873	−	393.873i	0.620272	−	0.620272i
$636$		0			0
$637$	278.453	−	278.453i	0.437132	−	0.437132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	412.834			0.644046			0.322023	−	0.946732i	$-0.395637\pi$
$641$	412.834			0.644046			0.322023	+	0.946732i	$0.395637\pi$
$642$		0			0
$643$	372.515	+	372.515i	0.579339	+	0.579339i	0.934721	−	0.355382i	$-0.115649\pi$
$643$	372.515	+	372.515i	0.579339	+	0.579339i	−0.355382	+	0.934721i	$0.615649\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1170.94			1.80980			0.904899	−	0.425627i	$-0.139946\pi$
$647$	1170.94			1.80980			0.904899	+	0.425627i	$0.139946\pi$
$648$		0			0
$649$		−	561.625i		−	0.865370i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.7523	−	13.7523i	0.0210602	−	0.0210602i	−0.696498	−	0.717558i	$-0.745260\pi$
$653$	13.7523	−	13.7523i	0.0210602	−	0.0210602i	0.717558	+	0.696498i	$0.245260\pi$
$654$		0			0
$655$		−	1129.33i		−	1.72417i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	283.149	+	283.149i	0.429664	+	0.429664i	0.888514	−	0.458850i	$-0.151738\pi$
$659$	283.149	+	283.149i	0.429664	+	0.429664i	−0.458850	+	0.888514i	$0.651738\pi$
$660$		0			0
$661$	−287.535	−	287.535i	−0.435000	−	0.435000i	0.455325	−	0.890325i	$-0.349523\pi$
$661$	−287.535	−	287.535i	−0.435000	−	0.435000i	−0.890325	+	0.455325i	$0.849523\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			963.069i			1.44822i
$666$		0			0
$667$	−82.9486	+	82.9486i	−0.124361	+	0.124361i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	187.776i		−	0.279845i
$672$		0			0
$673$	−45.5265			−0.0676471			−0.0338236	−	0.999428i	$-0.510768\pi$
$673$	−45.5265			−0.0676471			−0.0338236	+	0.999428i	$0.510768\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	208.341	+	208.341i	0.307742	+	0.307742i	0.844033	−	0.536291i	$-0.180175\pi$
$677$	208.341	+	208.341i	0.307742	+	0.307742i	−0.536291	+	0.844033i	$0.680175\pi$
$678$		0			0
$679$	−1089.18			−1.60409
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−219.645	+	219.645i	−0.321589	+	0.321589i	−0.849377	−	0.527787i	$-0.823022\pi$
$683$	−219.645	+	219.645i	−0.321589	+	0.321589i	0.527787	+	0.849377i	$0.323022\pi$
$684$		0			0
$685$	277.065	−	277.065i	0.404475	−	0.404475i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	136.710			0.198418
$690$		0			0
$691$	−692.991	−	692.991i	−1.00288	−	1.00288i	−0.999996	−	0.00288571i	$-0.999081\pi$
$691$	−692.991	−	692.991i	−1.00288	−	1.00288i	−0.00288571	−	0.999996i	$-0.500919\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	791.905			1.13943
$696$		0			0
$697$		−	327.301i		−	0.469585i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	195.377	−	195.377i	0.278712	−	0.278712i	−0.553883	−	0.832595i	$-0.686854\pi$
$701$	195.377	−	195.377i	0.278712	−	0.278712i	0.832595	+	0.553883i	$0.186854\pi$
$702$		0			0
$703$		−	491.668i		−	0.699386i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−941.179	−	941.179i	−1.33123	−	1.33123i
$708$		0			0
$709$	−318.083	−	318.083i	−0.448636	−	0.448636i	0.446265	−	0.894901i	$-0.352754\pi$
$709$	−318.083	−	318.083i	−0.448636	−	0.448636i	−0.894901	+	0.446265i	$0.852754\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	89.7775i		−	0.125915i
$714$		0			0
$715$	255.101	−	255.101i	0.356784	−	0.356784i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1122.38i		−	1.56103i	−0.625139	−	0.780514i	$-0.714958\pi$
$719$		−	1122.38i		−	1.56103i	0.625139	−	0.780514i	$-0.285042\pi$
$720$		0			0
$721$	1340.43			1.85913
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.5437	+	10.5437i	0.0145430	+	0.0145430i
$726$		0			0
$727$	529.192			0.727911			0.363956	−	0.931416i	$-0.381426\pi$
$727$	529.192			0.727911			0.363956	+	0.931416i	$0.381426\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	369.926	−	369.926i	0.506055	−	0.506055i
$732$		0			0
$733$	263.121	−	263.121i	0.358965	−	0.358965i	−0.504466	−	0.863431i	$-0.668311\pi$
$733$	263.121	−	263.121i	0.358965	−	0.358965i	0.863431	+	0.504466i	$0.168311\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−252.938			−0.343200
$738$		0			0
$739$	44.5459	+	44.5459i	0.0602787	+	0.0602787i	0.736603	−	0.676325i	$-0.236428\pi$
$739$	44.5459	+	44.5459i	0.0602787	+	0.0602787i	−0.676325	+	0.736603i	$0.736428\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−762.894			−1.02678			−0.513388	−	0.858157i	$-0.671610\pi$
$743$	−762.894			−1.02678			−0.513388	+	0.858157i	$0.671610\pi$
$744$		0			0
$745$		−	1076.50i		−	1.44496i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	306.735	−	306.735i	0.409526	−	0.409526i
$750$		0			0
$751$		−	1342.93i		−	1.78819i	−0.447876	−	0.894095i	$-0.647820\pi$
$751$		−	1342.93i		−	1.78819i	0.447876	−	0.894095i	$-0.352180\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	812.776	+	812.776i	1.07652	+	1.07652i
$756$		0			0
$757$	394.830	+	394.830i	0.521573	+	0.521573i	0.918046	−	0.396474i	$-0.129766\pi$
$757$	394.830	+	394.830i	0.521573	+	0.521573i	−0.396474	+	0.918046i	$0.629766\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	480.213i		−	0.631029i	−0.948921	−	0.315514i	$-0.897823\pi$
$761$		−	480.213i		−	0.631029i	0.948921	−	0.315514i	$-0.102177\pi$
$762$		0			0
$763$	6.88404	−	6.88404i	0.00902233	−	0.00902233i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	638.291i		−	0.832191i
$768$		0			0
$769$	472.763			0.614777			0.307388	−	0.951584i	$-0.400545\pi$
$769$	472.763			0.614777			0.307388	+	0.951584i	$0.400545\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	857.735	+	857.735i	1.10962	+	1.10962i	0.993200	+	0.116418i	$0.0371412\pi$
$773$	857.735	+	857.735i	1.10962	+	1.10962i	0.116418	+	0.993200i	$0.462859\pi$
$774$		0			0
$775$	−11.4117			−0.0147248
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−651.978	+	651.978i	−0.836942	+	0.836942i
$780$		0			0
$781$	284.758	−	284.758i	0.364607	−	0.364607i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	807.648			1.02885
$786$		0			0
$787$	−170.355	−	170.355i	−0.216462	−	0.216462i	0.590544	−	0.807006i	$-0.298913\pi$
$787$	−170.355	−	170.355i	−0.216462	−	0.216462i	−0.807006	+	0.590544i	$0.798913\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	143.891			0.181911
$792$		0			0
$793$		−	213.409i		−	0.269116i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	835.571	−	835.571i	1.04840	−	1.04840i	0.0496277	−	0.998768i	$-0.484197\pi$
$797$	835.571	−	835.571i	1.04840	−	1.04840i	0.998768	−	0.0496277i	$-0.0158035\pi$
$798$		0			0
$799$			274.174i			0.343147i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−433.185	−	433.185i	−0.539458	−	0.539458i
$804$		0			0
$805$	608.380	+	608.380i	0.755751	+	0.755751i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	371.926i		−	0.459735i	−0.973222	−	0.229868i	$-0.926171\pi$
$809$		−	371.926i		−	0.459735i	0.973222	−	0.229868i	$-0.0738293\pi$
$810$		0			0
$811$	275.629	−	275.629i	0.339863	−	0.339863i	−0.516453	−	0.856316i	$-0.672748\pi$
$811$	275.629	−	275.629i	0.339863	−	0.339863i	0.856316	+	0.516453i	$0.172748\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	418.550i		−	0.513559i
$816$		0			0
$817$	−1473.77			−1.80389
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−904.923	−	904.923i	−1.10222	−	1.10222i	−0.994142	−	0.108079i	$-0.965530\pi$
$821$	−904.923	−	904.923i	−1.10222	−	1.10222i	−0.108079	−	0.994142i	$-0.534470\pi$
$822$		0			0
$823$	523.237			0.635768			0.317884	−	0.948130i	$-0.397028\pi$
$823$	523.237			0.635768			0.317884	+	0.948130i	$0.397028\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	722.805	−	722.805i	0.874008	−	0.874008i	−0.118898	−	0.992906i	$-0.537936\pi$
$827$	722.805	−	722.805i	0.874008	−	0.874008i	0.992906	+	0.118898i	$0.0379363\pi$
$828$		0			0
$829$	286.380	−	286.380i	0.345453	−	0.345453i	−0.512960	−	0.858413i	$-0.671451\pi$
$829$	286.380	−	286.380i	0.345453	−	0.345453i	0.858413	+	0.512960i	$0.171451\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	301.316			0.361724
$834$		0			0
$835$	−393.193	−	393.193i	−0.470890	−	0.470890i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1353.58			−1.61333			−0.806666	−	0.591008i	$-0.798730\pi$
$839$	−1353.58			−1.61333			−0.806666	+	0.591008i	$0.798730\pi$
$840$		0			0
$841$			793.824i			0.943904i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−332.984	+	332.984i	−0.394064	+	0.394064i
$846$		0			0
$847$		−	580.852i		−	0.685776i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−310.591	−	310.591i	−0.364972	−	0.364972i
$852$		0			0
$853$	668.253	+	668.253i	0.783415	+	0.783415i	0.980405	−	0.196990i	$-0.0631167\pi$
$853$	668.253	+	668.253i	0.783415	+	0.783415i	−0.196990	+	0.980405i	$0.563117\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	488.688i		−	0.570230i	−0.958493	−	0.285115i	$-0.907968\pi$
$857$		−	488.688i		−	0.570230i	0.958493	−	0.285115i	$-0.0920319\pi$
$858$		0			0
$859$	268.818	−	268.818i	0.312943	−	0.312943i	−0.533106	−	0.846048i	$-0.678975\pi$
$859$	268.818	−	268.818i	0.312943	−	0.312943i	0.846048	+	0.533106i	$0.178975\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			152.667i			0.176903i	0.996080	+	0.0884514i	$0.0281918\pi$
$863$			152.667i			0.176903i	−0.996080	+	0.0884514i	$0.971808\pi$
$864$		0			0
$865$	1314.82			1.52002
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−598.777	−	598.777i	−0.689042	−	0.689042i
$870$		0			0
$871$	−287.466			−0.330041
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−813.208	+	813.208i	−0.929380	+	0.929380i
$876$		0			0
$877$	−162.637	+	162.637i	−0.185447	+	0.185447i	−0.793725	−	0.608277i	$-0.791861\pi$
$877$	−162.637	+	162.637i	−0.185447	+	0.185447i	0.608277	+	0.793725i	$0.291861\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−873.243			−0.991196			−0.495598	−	0.868552i	$-0.665051\pi$
$881$	−873.243			−0.991196			−0.495598	+	0.868552i	$0.665051\pi$
$882$		0			0
$883$	230.025	+	230.025i	0.260504	+	0.260504i	0.825259	−	0.564755i	$-0.191029\pi$
$883$	230.025	+	230.025i	0.260504	+	0.260504i	−0.564755	+	0.825259i	$0.691029\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−430.685			−0.485552			−0.242776	−	0.970082i	$-0.578058\pi$
$887$	−430.685			−0.485552			−0.242776	+	0.970082i	$0.578058\pi$
$888$		0			0
$889$			1032.75i			1.16170i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	546.150	−	546.150i	0.611590	−	0.611590i
$894$		0			0
$895$			445.709i			0.497999i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	25.5302	+	25.5302i	0.0283984	+	0.0283984i
$900$		0			0
$901$	73.9673	+	73.9673i	0.0820947	+	0.0820947i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1084.47i		−	1.19831i
$906$		0			0
$907$	−22.2262	+	22.2262i	−0.0245052	+	0.0245052i	−0.719253	−	0.694748i	$-0.755516\pi$
$907$	−22.2262	+	22.2262i	−0.0245052	+	0.0245052i	0.694748	+	0.719253i	$0.255516\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1399.85i			1.53661i	0.640083	+	0.768306i	$0.278900\pi$
$911$			1399.85i			1.53661i	−0.640083	+	0.768306i	$0.721100\pi$
$912$		0			0
$913$	632.560			0.692837
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1480.57	+	1480.57i	1.61458	+	1.61458i
$918$		0			0
$919$	−806.944			−0.878068			−0.439034	−	0.898470i	$-0.644679\pi$
$919$	−806.944			−0.878068			−0.439034	+	0.898470i	$0.644679\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	323.629	−	323.629i	0.350628	−	0.350628i
$924$		0			0
$925$	−39.4796	+	39.4796i	−0.0426806	+	0.0426806i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1620.69			1.74455			0.872276	−	0.489013i	$-0.162643\pi$
$929$	1620.69			1.74455			0.872276	+	0.489013i	$0.162643\pi$
$930$		0			0
$931$	−600.216	−	600.216i	−0.644701	−	0.644701i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	276.046			0.295237
$936$		0			0
$937$			598.181i			0.638400i	0.947687	+	0.319200i	$0.103414\pi$
$937$			598.181i			0.638400i	−0.947687	+	0.319200i	$0.896586\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−977.842	+	977.842i	−1.03915	+	1.03915i	−0.0399498	+	0.999202i	$0.512720\pi$
$941$	−977.842	+	977.842i	−1.03915	+	1.03915i	−0.999202	+	0.0399498i	$0.987280\pi$
$942$		0			0
$943$			823.721i			0.873511i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−827.881	−	827.881i	−0.874215	−	0.874215i	0.118714	−	0.992929i	$-0.462123\pi$
$947$	−827.881	−	827.881i	−0.874215	−	0.874215i	−0.992929	+	0.118714i	$0.962123\pi$
$948$		0			0
$949$	−492.317	−	492.317i	−0.518775	−	0.518775i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1846.78i			1.93786i	0.247333	+	0.968930i	$0.420446\pi$
$953$			1846.78i			1.93786i	−0.247333	+	0.968930i	$0.579554\pi$
$954$		0			0
$955$	393.873	−	393.873i	0.412432	−	0.412432i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			726.475i			0.757534i
$960$		0			0
$961$	933.368			0.971247
$962$		0			0
$963$		0			0
$964$		0			0
$965$	251.328	+	251.328i	0.260443	+	0.260443i
$966$		0			0
$967$	−363.922			−0.376341			−0.188170	−	0.982136i	$-0.560256\pi$
$967$	−363.922			−0.376341			−0.188170	+	0.982136i	$0.560256\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1161.30	−	1161.30i	1.19598	−	1.19598i	0.220619	−	0.975360i	$-0.429192\pi$
$971$	1161.30	−	1161.30i	1.19598	−	1.19598i	0.975360	−	0.220619i	$-0.0708076\pi$
$972$		0			0
$973$	−1038.20	+	1038.20i	−1.06701	+	1.06701i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1159.63			−1.18693			−0.593467	−	0.804858i	$-0.702241\pi$
$977$	−1159.63			−1.18693			−0.593467	+	0.804858i	$0.702241\pi$
$978$		0			0
$979$	243.499	+	243.499i	0.248722	+	0.248722i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1780.51			1.81131			0.905653	−	0.424020i	$-0.139382\pi$
$983$	1780.51			1.81131			0.905653	+	0.424020i	$0.139382\pi$
$984$		0			0
$985$		−	455.961i		−	0.462904i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−930.997	+	930.997i	−0.941352	+	0.941352i
$990$		0			0
$991$		−	675.783i		−	0.681920i	−0.940078	−	0.340960i	$-0.889248\pi$
$991$		−	675.783i		−	0.681920i	0.940078	−	0.340960i	$-0.110752\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−584.080	−	584.080i	−0.587015	−	0.587015i
$996$		0			0
$997$	−9.44963	−	9.44963i	−0.00947806	−	0.00947806i	0.702352	−	0.711830i	$-0.252133\pi$
$997$	−9.44963	−	9.44963i	−0.00947806	−	0.00947806i	−0.711830	+	0.702352i	$0.752133\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.a.415.3		6
3.2	odd	2		128.3.f.b.31.2		6
4.3	odd	2		1152.3.m.b.415.3		6
8.3	odd	2		576.3.m.a.271.1		6
8.5	even	2		144.3.m.a.91.2		6
12.11	even	2		128.3.f.a.31.2		6
16.3	odd	4	inner	1152.3.m.a.991.3		6
16.5	even	4		576.3.m.a.559.1		6
16.11	odd	4		144.3.m.a.19.2		6
16.13	even	4		1152.3.m.b.991.3		6
24.5	odd	2		16.3.f.a.11.2	yes	6
24.11	even	2		64.3.f.a.15.2		6
48.5	odd	4		64.3.f.a.47.2		6
48.11	even	4		16.3.f.a.3.2	✓	6
48.29	odd	4		128.3.f.a.95.2		6
48.35	even	4		128.3.f.b.95.2		6
96.5	odd	8		1024.3.d.k.511.7		12
96.11	even	8		1024.3.d.k.511.8		12
96.29	odd	8		1024.3.c.j.1023.6		12
96.35	even	8		1024.3.c.j.1023.8		12
96.53	odd	8		1024.3.d.k.511.6		12
96.59	even	8		1024.3.d.k.511.5		12
96.77	odd	8		1024.3.c.j.1023.7		12
96.83	even	8		1024.3.c.j.1023.5		12
120.29	odd	2		400.3.r.c.251.2		6
120.53	even	4		400.3.k.d.299.1		6
120.77	even	4		400.3.k.c.299.3		6
240.59	even	4		400.3.r.c.51.2		6
240.107	odd	4		400.3.k.d.99.1		6
240.203	odd	4		400.3.k.c.99.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.2	✓	6	48.11	even	4
16.3.f.a.11.2	yes	6	24.5	odd	2
64.3.f.a.15.2		6	24.11	even	2
64.3.f.a.47.2		6	48.5	odd	4
128.3.f.a.31.2		6	12.11	even	2
128.3.f.a.95.2		6	48.29	odd	4
128.3.f.b.31.2		6	3.2	odd	2
128.3.f.b.95.2		6	48.35	even	4
144.3.m.a.19.2		6	16.11	odd	4
144.3.m.a.91.2		6	8.5	even	2
400.3.k.c.99.3		6	240.203	odd	4
400.3.k.c.299.3		6	120.77	even	4
400.3.k.d.99.1		6	240.107	odd	4
400.3.k.d.299.1		6	120.53	even	4
400.3.r.c.51.2		6	240.59	even	4
400.3.r.c.251.2		6	120.29	odd	2
576.3.m.a.271.1		6	8.3	odd	2
576.3.m.a.559.1		6	16.5	even	4
1024.3.c.j.1023.5		12	96.83	even	8
1024.3.c.j.1023.6		12	96.29	odd	8
1024.3.c.j.1023.7		12	96.77	odd	8
1024.3.c.j.1023.8		12	96.35	even	8
1024.3.d.k.511.5		12	96.59	even	8
1024.3.d.k.511.6		12	96.53	odd	8
1024.3.d.k.511.7		12	96.5	odd	8
1024.3.d.k.511.8		12	96.11	even	8
1152.3.m.a.415.3		6	1.1	even	1	trivial
1152.3.m.a.991.3		6	16.3	odd	4	inner
1152.3.m.b.415.3		6	4.3	odd	2
1152.3.m.b.991.3		6	16.13	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.68585 + 3.68585i) q^{5} -9.66442 q^{7} +O(q^{10})\)	\(q+(3.68585 + 3.68585i) q^{5} -9.66442 q^{7} +(5.51806 - 5.51806i) q^{11} +(6.27131 - 6.27131i) q^{13} +6.78623 q^{17} +(-13.5181 - 13.5181i) q^{19} -17.0790 q^{23} +2.17092i q^{25} +(4.85677 - 4.85677i) q^{29} +5.25662i q^{31} +(-35.6216 - 35.6216i) q^{35} +(18.1856 + 18.1856i) q^{37} -48.2302i q^{41} +(54.5113 - 54.5113i) q^{43} +40.4015i q^{47} +44.4011 q^{49} +(10.8996 + 10.8996i) q^{53} +40.6774 q^{55} +(50.8898 - 50.8898i) q^{59} +(17.0147 - 17.0147i) q^{61} +46.2302 q^{65} +(-22.9191 - 22.9191i) q^{67} +51.6047 q^{71} -78.5032i q^{73} +(-53.3288 + 53.3288i) q^{77} -108.512i q^{79} +(57.3173 + 57.3173i) q^{83} +(25.0130 + 25.0130i) q^{85} +44.1276i q^{89} +(-60.6086 + 60.6086i) q^{91} -99.6510i q^{95} +112.700 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 6 * q - 2 * q^5 - 4 * q^7	\( 6 q - 2 q^{5} - 4 q^{7} - 18 q^{11} + 2 q^{13} + 4 q^{17} - 30 q^{19} - 60 q^{23} - 18 q^{29} - 100 q^{35} - 46 q^{37} + 114 q^{43} - 46 q^{49} + 78 q^{53} + 252 q^{55} + 206 q^{59} - 30 q^{61} - 12 q^{65} + 226 q^{67} + 260 q^{71} - 212 q^{77} + 318 q^{83} + 212 q^{85} - 188 q^{91} - 4 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 - 4 * q^7 - 18 * q^11 + 2 * q^13 + 4 * q^17 - 30 * q^19 - 60 * q^23 - 18 * q^29 - 100 * q^35 - 46 * q^37 + 114 * q^43 - 46 * q^49 + 78 * q^53 + 252 * q^55 + 206 * q^59 - 30 * q^61 - 12 * q^65 + 226 * q^67 + 260 * q^71 - 212 * q^77 + 318 * q^83 + 212 * q^85 - 188 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more