LMFDB - Embedded newform 1008.3.f.g.433.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(433,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1008.433");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1008.433
Dual form			1008.3.f.g.433.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.91359i q^{5} +(-6.24264 - 3.16693i) q^{7} +O(q^{10})$	$q-5.91359i q^{5} +(-6.24264 - 3.16693i) q^{7} -1.75736 q^{11} -18.7554i q^{13} +23.4803i q^{17} -23.0600i q^{19} +18.7279 q^{23} -9.97056 q^{25} -30.0000 q^{29} +8.60927i q^{31} +(-18.7279 + 36.9164i) q^{35} -70.9117 q^{37} +41.3951i q^{41} -10.4264 q^{43} -38.6995i q^{47} +(28.9411 + 39.5400i) q^{49} +37.0294 q^{53} +10.3923i q^{55} +97.4872i q^{59} -16.7262i q^{61} -110.912 q^{65} -60.9706 q^{67} -110.610 q^{71} +56.7585i q^{73} +(10.9706 + 5.56543i) q^{77} +69.8234 q^{79} -6.43583i q^{83} +138.853 q^{85} -42.0915i q^{89} +(-59.3970 + 117.083i) q^{91} -136.368 q^{95} -51.7153i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 24 q^{11} + 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} - 80 q^{37} + 128 q^{43} - 20 q^{49} + 216 q^{53} - 240 q^{65} - 176 q^{67} - 120 q^{71} - 24 q^{77} - 128 q^{79} + 216 q^{85} - 240 q^{95}+O(q^{100}) $ 4 * q - 8 * q^7 - 24 * q^11 + 24 * q^23 + 28 * q^25 - 120 * q^29 - 24 * q^35 - 80 * q^37 + 128 * q^43 - 20 * q^49 + 216 * q^53 - 240 * q^65 - 176 * q^67 - 120 * q^71 - 24 * q^77 - 128 * q^79 + 216 * q^85 - 240 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$		−	5.91359i		−	1.18272i	−0.806408	−	0.591359i	$-0.798592\pi$
$5$		−	5.91359i		−	1.18272i	0.806408	−	0.591359i	$-0.201408\pi$
$6$		0			0
$7$	−6.24264	−	3.16693i	−0.891806	−	0.452418i
$7$	−6.24264	−	3.16693i	−0.891806	−	0.452418i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.75736			−0.159760			−0.0798800	−	0.996804i	$-0.525454\pi$
$11$	−1.75736			−0.159760			−0.0798800	+	0.996804i	$0.525454\pi$
$12$		0			0
$13$		−	18.7554i		−	1.44272i	−0.692559	−	0.721361i	$-0.743517\pi$
$13$		−	18.7554i		−	1.44272i	0.692559	−	0.721361i	$-0.256483\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			23.4803i			1.38119i	0.723240	+	0.690597i	$0.242652\pi$
$17$			23.4803i			1.38119i	−0.723240	+	0.690597i	$0.757348\pi$
$18$		0			0
$19$		−	23.0600i		−	1.21369i	−0.794822	−	0.606843i	$-0.792436\pi$
$19$		−	23.0600i		−	1.21369i	0.794822	−	0.606843i	$-0.207564\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	18.7279			0.814257			0.407129	−	0.913371i	$-0.366530\pi$
$23$	18.7279			0.814257			0.407129	+	0.913371i	$0.366530\pi$
$24$		0			0
$25$	−9.97056			−0.398823
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−30.0000			−1.03448			−0.517241	−	0.855840i	$-0.673041\pi$
$29$	−30.0000			−1.03448			−0.517241	+	0.855840i	$0.673041\pi$
$30$		0			0
$31$			8.60927i			0.277718i	0.990312	+	0.138859i	$0.0443435\pi$
$31$			8.60927i			0.277718i	−0.990312	+	0.138859i	$0.955656\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−18.7279	+	36.9164i	−0.535083	+	1.05476i
$36$		0			0
$37$	−70.9117			−1.91653			−0.958266	−	0.285878i	$-0.907715\pi$
$37$	−70.9117			−1.91653			−0.958266	+	0.285878i	$0.907715\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			41.3951i			1.00964i	0.863225	+	0.504819i	$0.168441\pi$
$41$			41.3951i			1.00964i	−0.863225	+	0.504819i	$0.831559\pi$
$42$		0			0
$43$	−10.4264			−0.242475			−0.121237	−	0.992624i	$-0.538686\pi$
$43$	−10.4264			−0.242475			−0.121237	+	0.992624i	$0.538686\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	38.6995i		−	0.823393i	−0.911321	−	0.411696i	$-0.864936\pi$
$47$		−	38.6995i		−	0.823393i	0.911321	−	0.411696i	$-0.135064\pi$
$48$		0			0
$49$	28.9411	+	39.5400i	0.590635	+	0.806939i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	37.0294			0.698669			0.349334	−	0.936998i	$-0.386408\pi$
$53$	37.0294			0.698669			0.349334	+	0.936998i	$0.386408\pi$
$54$		0			0
$55$			10.3923i			0.188951i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			97.4872i			1.65233i	0.563431	+	0.826163i	$0.309481\pi$
$59$			97.4872i			1.65233i	−0.563431	+	0.826163i	$0.690519\pi$
$60$		0			0
$61$		−	16.7262i		−	0.274199i	−0.990557	−	0.137100i	$-0.956222\pi$
$61$		−	16.7262i		−	0.274199i	0.990557	−	0.137100i	$-0.0437781\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−110.912			−1.70633
$66$		0			0
$67$	−60.9706			−0.910008			−0.455004	−	0.890489i	$-0.650362\pi$
$67$	−60.9706			−0.910008			−0.455004	+	0.890489i	$0.650362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−110.610			−1.55789			−0.778945	−	0.627092i	$-0.784245\pi$
$71$	−110.610			−1.55789			−0.778945	+	0.627092i	$0.784245\pi$
$72$		0			0
$73$			56.7585i			0.777514i	0.921340	+	0.388757i	$0.127095\pi$
$73$			56.7585i			0.777514i	−0.921340	+	0.388757i	$0.872905\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.9706	+	5.56543i	0.142475	+	0.0722783i
$78$		0			0
$79$	69.8234			0.883840			0.441920	−	0.897054i	$-0.354297\pi$
$79$	69.8234			0.883840			0.441920	+	0.897054i	$0.354297\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.43583i		−	0.0775401i	−0.999248	−	0.0387701i	$-0.987656\pi$
$83$		−	6.43583i		−	0.0775401i	0.999248	−	0.0387701i	$-0.0123440\pi$
$84$		0			0
$85$	138.853			1.63356
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	42.0915i		−	0.472938i	−0.971639	−	0.236469i	$-0.924010\pi$
$89$		−	42.0915i		−	0.472938i	0.971639	−	0.236469i	$-0.0759901\pi$
$90$		0			0
$91$	−59.3970	+	117.083i	−0.652714	+	1.28663i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−136.368			−1.43545
$96$		0			0
$97$		−	51.7153i		−	0.533148i	−0.963814	−	0.266574i	$-0.914108\pi$
$97$		−	51.7153i		−	0.533148i	0.963814	−	0.266574i	$-0.0858916\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			6.60991i			0.0654447i	0.999464	+	0.0327223i	$0.0104177\pi$
$101$			6.60991i			0.0654447i	−0.999464	+	0.0327223i	$0.989582\pi$
$102$		0			0
$103$			175.871i			1.70748i	0.520696	+	0.853742i	$0.325673\pi$
$103$			175.871i			1.70748i	−0.520696	+	0.853742i	$0.674327\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−46.2426			−0.432174			−0.216087	−	0.976374i	$-0.569330\pi$
$107$	−46.2426			−0.432174			−0.216087	+	0.976374i	$0.569330\pi$
$108$		0			0
$109$	−35.9411			−0.329735			−0.164868	−	0.986316i	$-0.552720\pi$
$109$	−35.9411			−0.329735			−0.164868	+	0.986316i	$0.552720\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	73.0294			0.646278			0.323139	−	0.946351i	$-0.395262\pi$
$113$	73.0294			0.646278			0.323139	+	0.946351i	$0.395262\pi$
$114$		0			0
$115$		−	110.749i		−	0.963037i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	74.3604	−	146.579i	0.624877	−	1.23176i
$120$		0			0
$121$	−117.912			−0.974477
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	88.8780i		−	0.711024i
$126$		0			0
$127$	−89.9411			−0.708198			−0.354099	−	0.935208i	$-0.615212\pi$
$127$	−89.9411			−0.708198			−0.354099	+	0.935208i	$0.615212\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.1753i		−	0.0929415i	−0.998920	−	0.0464708i	$-0.985203\pi$
$131$		−	12.1753i		−	0.0929415i	0.998920	−	0.0464708i	$-0.0147974\pi$
$132$		0			0
$133$	−73.0294	+	143.955i	−0.549094	+	1.08237i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−165.765			−1.20996			−0.604980	−	0.796241i	$-0.706819\pi$
$137$	−165.765			−1.20996			−0.604980	+	0.796241i	$0.706819\pi$
$138$		0			0
$139$		−	220.514i		−	1.58643i	−0.608941	−	0.793215i	$-0.708406\pi$
$139$		−	220.514i		−	1.58643i	0.608941	−	0.793215i	$-0.291594\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			32.9600i			0.230489i
$144$		0			0
$145$			177.408i			1.22350i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−210.853			−1.41512			−0.707560	−	0.706653i	$-0.750204\pi$
$149$	−210.853			−1.41512			−0.707560	+	0.706653i	$0.750204\pi$
$150$		0			0
$151$	72.3675			0.479255			0.239628	−	0.970865i	$-0.422975\pi$
$151$	72.3675			0.479255			0.239628	+	0.970865i	$0.422975\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	50.9117			0.328463
$156$		0			0
$157$		−	233.674i		−	1.48837i	−0.667974	−	0.744184i	$-0.732838\pi$
$157$		−	233.674i		−	1.48837i	0.667974	−	0.744184i	$-0.267162\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−116.912	−	59.3100i	−0.726160	−	0.368385i
$162$		0			0
$163$	73.0883			0.448395			0.224197	−	0.974544i	$-0.428024\pi$
$163$	73.0883			0.448395			0.224197	+	0.974544i	$0.428024\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			39.3958i			0.235903i	0.993019	+	0.117951i	$0.0376327\pi$
$167$			39.3958i			0.235903i	−0.993019	+	0.117951i	$0.962367\pi$
$168$		0			0
$169$	−182.765			−1.08145
$170$		0			0
$171$		0			0
$172$		0			0
$173$			23.8284i			0.137737i	0.997626	+	0.0688683i	$0.0219388\pi$
$173$			23.8284i			0.137737i	−0.997626	+	0.0688683i	$0.978061\pi$
$174$		0			0
$175$	62.2426	+	31.5761i	0.355672	+	0.180435i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.9045			−0.0720924			−0.0360462	−	0.999350i	$-0.511476\pi$
$179$	−12.9045			−0.0720924			−0.0360462	+	0.999350i	$0.511476\pi$
$180$		0			0
$181$		−	65.3678i		−	0.361148i	−0.983561	−	0.180574i	$-0.942204\pi$
$181$		−	65.3678i		−	0.361148i	0.983561	−	0.180574i	$-0.0577955\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			419.343i			2.26672i
$186$		0			0
$187$		−	41.2633i		−	0.220659i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−100.066			−0.523906			−0.261953	−	0.965081i	$-0.584367\pi$
$191$	−100.066			−0.523906			−0.261953	+	0.965081i	$0.584367\pi$
$192$		0			0
$193$	78.9117			0.408869			0.204434	−	0.978880i	$-0.434464\pi$
$193$	78.9117			0.408869			0.204434	+	0.978880i	$0.434464\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	183.941			0.933711			0.466856	−	0.884334i	$-0.345387\pi$
$197$	183.941			0.933711			0.466856	+	0.884334i	$0.345387\pi$
$198$		0			0
$199$			170.029i			0.854419i	0.904153	+	0.427210i	$0.140503\pi$
$199$			170.029i			0.854419i	−0.904153	+	0.427210i	$0.859497\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	187.279	+	95.0079i	0.922558	+	0.468019i
$204$		0			0
$205$	244.794			1.19412
$206$		0			0
$207$		0			0
$208$		0			0
$209$			40.5247i			0.193898i
$210$		0			0
$211$	−21.5736			−0.102245			−0.0511223	−	0.998692i	$-0.516280\pi$
$211$	−21.5736			−0.102245			−0.0511223	+	0.998692i	$0.516280\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			61.6575i			0.286779i
$216$		0			0
$217$	27.2649	−	53.7446i	0.125645	−	0.247671i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	440.382			1.99268
$222$		0			0
$223$		−	119.359i		−	0.535240i	−0.963525	−	0.267620i	$-0.913763\pi$
$223$		−	119.359i		−	0.535240i	0.963525	−	0.267620i	$-0.0862372\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			169.843i			0.748207i	0.927387	+	0.374103i	$0.122049\pi$
$227$			169.843i			0.748207i	−0.927387	+	0.374103i	$0.877951\pi$
$228$		0			0
$229$			110.011i			0.480396i	0.970724	+	0.240198i	$0.0772124\pi$
$229$			110.011i			0.480396i	−0.970724	+	0.240198i	$0.922788\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−57.2649			−0.245772			−0.122886	−	0.992421i	$-0.539215\pi$
$233$	−57.2649			−0.245772			−0.122886	+	0.992421i	$0.539215\pi$
$234$		0			0
$235$	−228.853			−0.973842
$236$		0			0
$237$		0			0
$238$		0			0
$239$	281.522			1.17792			0.588958	−	0.808164i	$-0.299538\pi$
$239$	281.522			1.17792			0.588958	+	0.808164i	$0.299538\pi$
$240$		0			0
$241$		−	168.306i		−	0.698366i	−0.937055	−	0.349183i	$-0.886459\pi$
$241$		−	168.306i		−	0.698366i	0.937055	−	0.349183i	$-0.113541\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	233.823	−	171.146i	0.954381	−	0.698555i
$246$		0			0
$247$	−432.500			−1.75101
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	106.096i		−	0.422695i	−0.977411	−	0.211348i	$-0.932215\pi$
$251$		−	106.096i		−	0.422695i	0.977411	−	0.211348i	$-0.0677852\pi$
$252$		0			0
$253$	−32.9117			−0.130086
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	290.462i		−	1.13020i	−0.825021	−	0.565102i	$-0.808837\pi$
$257$		−	290.462i		−	1.13020i	0.825021	−	0.565102i	$-0.191163\pi$
$258$		0			0
$259$	442.676	+	224.572i	1.70917	+	0.867074i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−89.0223			−0.338488			−0.169244	−	0.985574i	$-0.554133\pi$
$263$	−89.0223			−0.338488			−0.169244	+	0.985574i	$0.554133\pi$
$264$		0			0
$265$		−	218.977i		−	0.826328i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			191.498i			0.711888i	0.934507	+	0.355944i	$0.115841\pi$
$269$			191.498i			0.711888i	−0.934507	+	0.355944i	$0.884159\pi$
$270$		0			0
$271$		−	217.440i		−	0.802362i	−0.915999	−	0.401181i	$-0.868600\pi$
$271$		−	217.440i		−	0.802362i	0.915999	−	0.401181i	$-0.131400\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	17.5219			0.0637159
$276$		0			0
$277$	290.676			1.04937			0.524686	−	0.851296i	$-0.324183\pi$
$277$	290.676			1.04937			0.524686	+	0.851296i	$0.324183\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.8528			−0.0670919			−0.0335459	−	0.999437i	$-0.510680\pi$
$281$	−18.8528			−0.0670919			−0.0335459	+	0.999437i	$0.510680\pi$
$282$		0			0
$283$		−	401.734i		−	1.41955i	−0.704426	−	0.709777i	$-0.748796\pi$
$283$		−	401.734i		−	1.41955i	0.704426	−	0.709777i	$-0.251204\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	131.095	−	258.415i	0.456779	−	0.900401i
$288$		0			0
$289$	−262.324			−0.907695
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	280.893i		−	0.958679i	−0.877629	−	0.479340i	$-0.840876\pi$
$293$		−	280.893i		−	0.958679i	0.877629	−	0.479340i	$-0.159124\pi$
$294$		0			0
$295$	576.500			1.95424
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	351.249i		−	1.17475i
$300$		0			0
$301$	65.0883	+	33.0197i	0.216240	+	0.109700i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−98.9117			−0.324301
$306$		0			0
$307$			152.318i			0.496151i	0.968741	+	0.248076i	$0.0797982\pi$
$307$			152.318i			0.496151i	−0.968741	+	0.248076i	$0.920202\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			283.156i			0.910470i	0.890371	+	0.455235i	$0.150445\pi$
$311$			283.156i			0.910470i	−0.890371	+	0.455235i	$0.849555\pi$
$312$		0			0
$313$		−	48.5819i		−	0.155214i	−0.996984	−	0.0776069i	$-0.975272\pi$
$313$		−	48.5819i		−	0.155214i	0.996984	−	0.0776069i	$-0.0247279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	578.029			1.82343			0.911717	−	0.410819i	$-0.134757\pi$
$317$	578.029			1.82343			0.911717	+	0.410819i	$0.134757\pi$
$318$		0			0
$319$	52.7208			0.165269
$320$		0			0
$321$		0			0
$322$		0			0
$323$	541.456			1.67633
$324$		0			0
$325$			187.002i			0.575390i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−122.558	+	241.587i	−0.372518	+	0.734307i
$330$		0			0
$331$	−332.368			−1.00413			−0.502066	−	0.864829i	$-0.667426\pi$
$331$	−332.368			−1.00413			−0.502066	+	0.864829i	$0.667426\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			360.555i			1.07628i
$336$		0			0
$337$	88.1766			0.261652			0.130826	−	0.991405i	$-0.458237\pi$
$337$	88.1766			0.261652			0.130826	+	0.991405i	$0.458237\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	15.1296i		−	0.0443683i
$342$		0			0
$343$	−55.4487	−	338.488i	−0.161658	−	0.986847i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−320.080			−0.922422			−0.461211	−	0.887291i	$-0.652585\pi$
$347$	−320.080			−0.922422			−0.461211	+	0.887291i	$0.652585\pi$
$348$		0			0
$349$			333.046i			0.954287i	0.878825	+	0.477143i	$0.158328\pi$
$349$			333.046i			0.954287i	−0.878825	+	0.477143i	$0.841672\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	655.712i		−	1.85754i	−0.370654	−	0.928771i	$-0.620866\pi$
$353$		−	655.712i		−	1.85754i	0.370654	−	0.928771i	$-0.379134\pi$
$354$		0			0
$355$			654.103i			1.84254i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−97.7574			−0.272305			−0.136152	−	0.990688i	$-0.543474\pi$
$359$	−97.7574			−0.272305			−0.136152	+	0.990688i	$0.543474\pi$
$360$		0			0
$361$	−170.765			−0.473032
$362$		0			0
$363$		0			0
$364$		0			0
$365$	335.647			0.919580
$366$		0			0
$367$		−	321.057i		−	0.874815i	−0.899263	−	0.437408i	$-0.855897\pi$
$367$		−	321.057i		−	0.874815i	0.899263	−	0.437408i	$-0.144103\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−231.161	−	117.270i	−0.623077	−	0.316091i
$372$		0			0
$373$	187.470			0.502601			0.251300	−	0.967909i	$-0.419142\pi$
$373$	187.470			0.502601			0.251300	+	0.967909i	$0.419142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			562.662i			1.49247i
$378$		0			0
$379$	357.103			0.942223			0.471112	−	0.882074i	$-0.343853\pi$
$379$	357.103			0.942223			0.471112	+	0.882074i	$0.343853\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	622.230i		−	1.62462i	−0.583225	−	0.812311i	$-0.698209\pi$
$383$		−	622.230i		−	1.62462i	0.583225	−	0.812311i	$-0.301791\pi$
$384$		0			0
$385$	32.9117	−	64.8754i	0.0854849	−	0.168508i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−227.470			−0.584756			−0.292378	−	0.956303i	$-0.594447\pi$
$389$	−227.470			−0.584756			−0.292378	+	0.956303i	$0.594447\pi$
$390$		0			0
$391$			439.737i			1.12465i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	412.907i		−	1.04533i
$396$		0			0
$397$		−	720.329i		−	1.81443i	−0.420666	−	0.907216i	$-0.638204\pi$
$397$		−	720.329i		−	1.81443i	0.420666	−	0.907216i	$-0.361796\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−697.176			−1.73859			−0.869296	−	0.494291i	$-0.835428\pi$
$401$	−697.176			−1.73859			−0.869296	+	0.494291i	$0.835428\pi$
$402$		0			0
$403$	161.470			0.400670
$404$		0			0
$405$		0			0
$406$		0			0
$407$	124.617			0.306185
$408$		0			0
$409$		−	102.386i		−	0.250333i	−0.992136	−	0.125166i	$-0.960053\pi$
$409$		−	102.386i		−	0.250333i	0.992136	−	0.125166i	$-0.0399465\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	308.735	−	608.578i	0.747543	−	1.47355i
$414$		0			0
$415$	−38.0589			−0.0917081
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	391.426i		−	0.934191i	−0.884207	−	0.467095i	$-0.845300\pi$
$419$		−	391.426i		−	0.934191i	0.884207	−	0.467095i	$-0.154700\pi$
$420$		0			0
$421$	354.441			0.841902			0.420951	−	0.907083i	$-0.361696\pi$
$421$	354.441			0.841902			0.420951	+	0.907083i	$0.361696\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	234.112i		−	0.550851i
$426$		0			0
$427$	−52.9706	+	104.415i	−0.124053	+	0.244533i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	585.286			1.35797			0.678987	−	0.734151i	$-0.262419\pi$
$431$	585.286			1.35797			0.678987	+	0.734151i	$0.262419\pi$
$432$		0			0
$433$			392.207i			0.905789i	0.891564	+	0.452895i	$0.149609\pi$
$433$			392.207i			0.905789i	−0.891564	+	0.452895i	$0.850391\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	431.866i		−	0.988252i
$438$		0			0
$439$		−	392.513i		−	0.894106i	−0.894507	−	0.447053i	$-0.852473\pi$
$439$		−	392.513i		−	0.894106i	0.894507	−	0.447053i	$-0.147527\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−814.742			−1.83915			−0.919574	−	0.392918i	$-0.871466\pi$
$443$	−814.742			−1.83915			−0.919574	+	0.392918i	$0.871466\pi$
$444$		0			0
$445$	−248.912			−0.559352
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−180.323			−0.401610			−0.200805	−	0.979631i	$-0.564356\pi$
$449$	−180.323			−0.401610			−0.200805	+	0.979631i	$0.564356\pi$
$450$		0			0
$451$		−	72.7461i		−	0.161300i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	692.382	+	351.249i	1.52172	+	0.771977i
$456$		0			0
$457$	280.177			0.613078			0.306539	−	0.951858i	$-0.400829\pi$
$457$	280.177			0.613078			0.306539	+	0.951858i	$0.400829\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			406.297i			0.881338i	0.897670	+	0.440669i	$0.145259\pi$
$461$			406.297i			0.881338i	−0.897670	+	0.440669i	$0.854741\pi$
$462$		0			0
$463$	−457.470			−0.988056			−0.494028	−	0.869446i	$-0.664476\pi$
$463$	−457.470			−0.988056			−0.494028	+	0.869446i	$0.664476\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			643.711i			1.37840i	0.724573	+	0.689198i	$0.242037\pi$
$467$			643.711i			1.37840i	−0.724573	+	0.689198i	$0.757963\pi$
$468$		0			0
$469$	380.617	+	193.089i	0.811551	+	0.411705i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18.3229			0.0387377
$474$		0			0
$475$			229.921i			0.484045i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	168.535i		−	0.351847i	−0.984404	−	0.175924i	$-0.943709\pi$
$479$		−	168.535i		−	0.351847i	0.984404	−	0.175924i	$-0.0562912\pi$
$480$		0			0
$481$			1329.98i			2.76502i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−305.823			−0.630564
$486$		0			0
$487$	2.77965			0.00570771			0.00285385	−	0.999996i	$-0.499092\pi$
$487$	2.77965			0.00570771			0.00285385	+	0.999996i	$0.499092\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−247.477			−0.504027			−0.252014	−	0.967724i	$-0.581093\pi$
$491$	−247.477			−0.504027			−0.252014	+	0.967724i	$0.581093\pi$
$492$		0			0
$493$		−	704.409i		−	1.42882i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	690.500	+	350.295i	1.38934	+	0.704818i
$498$		0			0
$499$	483.426			0.968789			0.484394	−	0.874850i	$-0.339040\pi$
$499$	483.426			0.968789			0.484394	+	0.874850i	$0.339040\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	58.7033i		−	0.116706i	−0.998296	−	0.0583532i	$-0.981415\pi$
$503$		−	58.7033i		−	0.116706i	0.998296	−	0.0583532i	$-0.0185849\pi$
$504$		0			0
$505$	39.0883			0.0774026
$506$		0			0
$507$		0			0
$508$		0			0
$509$			68.8793i			0.135323i	0.997708	+	0.0676614i	$0.0215537\pi$
$509$			68.8793i			0.135323i	−0.997708	+	0.0676614i	$0.978446\pi$
$510$		0			0
$511$	179.750	−	354.323i	0.351762	−	0.693392i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1040.03			2.01947
$516$		0			0
$517$			68.0089i			0.131545i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			292.720i			0.561843i	0.959731	+	0.280922i	$0.0906401\pi$
$521$			292.720i			0.561843i	−0.959731	+	0.280922i	$0.909360\pi$
$522$		0			0
$523$			493.056i			0.942746i	0.881934	+	0.471373i	$0.156241\pi$
$523$			493.056i			0.942746i	−0.881934	+	0.471373i	$0.843759\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−202.148			−0.383583
$528$		0			0
$529$	−178.265			−0.336985
$530$		0			0
$531$		0			0
$532$		0			0
$533$	776.382			1.45663
$534$		0			0
$535$			273.460i			0.511140i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−50.8600	−	69.4860i	−0.0943598	−	0.128916i
$540$		0			0
$541$	−1037.85			−1.91840			−0.959198	−	0.282736i	$-0.908758\pi$
$541$	−1037.85			−1.91840			−0.959198	+	0.282736i	$0.908758\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			212.541i			0.389984i
$546$		0			0
$547$	130.530			0.238629			0.119314	−	0.992857i	$-0.461930\pi$
$547$	130.530			0.238629			0.119314	+	0.992857i	$0.461930\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			691.801i			1.25554i
$552$		0			0
$553$	−435.882	−	221.126i	−0.788214	−	0.399866i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−665.147			−1.19416			−0.597080	−	0.802182i	$-0.703673\pi$
$557$	−665.147			−1.19416			−0.597080	+	0.802182i	$0.703673\pi$
$558$		0			0
$559$			195.551i			0.349823i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	829.295i		−	1.47299i	−0.676441	−	0.736497i	$-0.736479\pi$
$563$		−	829.295i		−	1.47299i	0.676441	−	0.736497i	$-0.263521\pi$
$564$		0			0
$565$		−	431.866i		−	0.764365i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−706.971			−1.24248			−0.621240	−	0.783621i	$-0.713371\pi$
$569$	−706.971			−1.24248			−0.621240	+	0.783621i	$0.713371\pi$
$570$		0			0
$571$	366.912			0.642577			0.321289	−	0.946981i	$-0.395884\pi$
$571$	366.912			0.642577			0.321289	+	0.946981i	$0.395884\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−186.728			−0.324744
$576$		0			0
$577$			390.357i			0.676528i	0.941051	+	0.338264i	$0.109840\pi$
$577$			390.357i			0.676528i	−0.941051	+	0.338264i	$0.890160\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20.3818	+	40.1766i	−0.0350806	+	0.0691507i
$582$		0			0
$583$	−65.0740			−0.111619
$584$		0			0
$585$		0			0
$586$		0			0
$587$			702.499i			1.19676i	0.801212	+	0.598381i	$0.204189\pi$
$587$			702.499i			1.19676i	−0.801212	+	0.598381i	$0.795811\pi$
$588$		0			0
$589$	198.530			0.337063
$590$		0			0
$591$		0			0
$592$		0			0
$593$			942.519i			1.58941i	0.606997	+	0.794704i	$0.292374\pi$
$593$			942.519i			1.58941i	−0.606997	+	0.794704i	$0.707626\pi$
$594$		0			0
$595$	−866.808	−	439.737i	−1.45682	−	0.739054i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−952.109			−1.58950			−0.794749	−	0.606939i	$-0.792397\pi$
$599$	−952.109			−1.58950			−0.794749	+	0.606939i	$0.792397\pi$
$600$		0			0
$601$		−	729.804i		−	1.21432i	−0.794581	−	0.607158i	$-0.792310\pi$
$601$		−	729.804i		−	1.21432i	0.794581	−	0.607158i	$-0.207690\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			697.282i			1.15253i
$606$		0			0
$607$		−	1006.34i		−	1.65789i	−0.559332	−	0.828944i	$-0.688942\pi$
$607$		−	1006.34i		−	1.65789i	0.559332	−	0.828944i	$-0.311058\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−725.823			−1.18793
$612$		0			0
$613$	−199.588			−0.325592			−0.162796	−	0.986660i	$-0.552051\pi$
$613$	−199.588			−0.325592			−0.162796	+	0.986660i	$0.552051\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	353.294			0.572599			0.286299	−	0.958140i	$-0.407575\pi$
$617$	353.294			0.572599			0.286299	+	0.958140i	$0.407575\pi$
$618$		0			0
$619$		−	56.2064i		−	0.0908020i	−0.998969	−	0.0454010i	$-0.985543\pi$
$619$		−	56.2064i		−	0.0908020i	0.998969	−	0.0454010i	$-0.0144566\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−133.301	+	262.762i	−0.213966	+	0.421769i
$624$		0			0
$625$	−774.852			−1.23976
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1665.03i		−	2.64710i
$630$		0			0
$631$	−807.322			−1.27943			−0.639716	−	0.768611i	$-0.720948\pi$
$631$	−807.322			−1.27943			−0.639716	+	0.768611i	$0.720948\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			531.875i			0.837599i
$636$		0			0
$637$	741.588	−	542.802i	1.16419	−	0.852122i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1016.35			−1.58557			−0.792786	−	0.609500i	$-0.791370\pi$
$641$	−1016.35			−1.58557			−0.792786	+	0.609500i	$0.791370\pi$
$642$		0			0
$643$		−	404.688i		−	0.629375i	−0.949195	−	0.314687i	$-0.898100\pi$
$643$		−	404.688i		−	0.629375i	0.949195	−	0.314687i	$-0.101900\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			940.604i			1.45379i	0.686747	+	0.726896i	$0.259038\pi$
$647$			940.604i			1.45379i	−0.686747	+	0.726896i	$0.740962\pi$
$648$		0			0
$649$		−	171.320i		−	0.263975i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−731.970			−1.12093			−0.560467	−	0.828177i	$-0.689378\pi$
$653$	−731.970			−1.12093			−0.560467	+	0.828177i	$0.689378\pi$
$654$		0			0
$655$	−72.0000			−0.109924
$656$		0			0
$657$		0			0
$658$		0			0
$659$	904.316			1.37225			0.686127	−	0.727481i	$-0.259309\pi$
$659$	904.316			1.37225			0.686127	+	0.727481i	$0.259309\pi$
$660$		0			0
$661$			335.881i			0.508141i	0.967186	+	0.254070i	$0.0817695\pi$
$661$			335.881i			0.508141i	−0.967186	+	0.254070i	$0.918231\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	851.294	+	431.866i	1.28014	+	0.649423i
$666$		0			0
$667$	−561.838			−0.842335
$668$		0			0
$669$		0			0
$670$		0			0
$671$			29.3939i			0.0438061i
$672$		0			0
$673$	1191.44			1.77034			0.885171	−	0.465266i	$-0.154041\pi$
$673$	1191.44			1.77034			0.885171	+	0.465266i	$0.154041\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1211.07i			1.78888i	0.447186	+	0.894441i	$0.352426\pi$
$677$			1211.07i			1.78888i	−0.447186	+	0.894441i	$0.647574\pi$
$678$		0			0
$679$	−163.779	+	322.840i	−0.241206	+	0.475464i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1233.89			−1.80657			−0.903287	−	0.429038i	$-0.858853\pi$
$683$	−1233.89			−1.80657			−0.903287	+	0.429038i	$0.858853\pi$
$684$		0			0
$685$			980.264i			1.43104i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	694.501i		−	1.00798i
$690$		0			0
$691$		−	86.7045i		−	0.125477i	−0.998030	−	0.0627384i	$-0.980017\pi$
$691$		−	86.7045i		−	0.125477i	0.998030	−	0.0627384i	$-0.0199834\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1304.03			−1.87630
$696$		0			0
$697$	−971.970			−1.39450
$698$		0			0
$699$		0			0
$700$		0			0
$701$	149.147			0.212763			0.106382	−	0.994325i	$-0.466073\pi$
$701$	149.147			0.212763			0.106382	+	0.994325i	$0.466073\pi$
$702$		0			0
$703$			1635.22i			2.32607i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	20.9331	−	41.2633i	0.0296084	−	0.0583639i
$708$		0			0
$709$	189.647			0.267485			0.133742	−	0.991016i	$-0.457301\pi$
$709$	189.647			0.267485			0.133742	+	0.991016i	$0.457301\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			161.234i			0.226134i
$714$		0			0
$715$	194.912			0.272604
$716$		0			0
$717$		0			0
$718$		0			0
$719$			12.0064i			0.0166987i	0.999965	+	0.00834937i	$0.00265772\pi$
$719$			12.0064i			0.0166987i	−0.999965	+	0.00834937i	$0.997342\pi$
$720$		0			0
$721$	556.971	−	1097.90i	0.772497	−	1.52274i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	299.117			0.412575
$726$		0			0
$727$		−	417.169i		−	0.573823i	−0.957957	−	0.286911i	$-0.907371\pi$
$727$		−	417.169i		−	0.573823i	0.957957	−	0.286911i	$-0.0926286\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	244.815i		−	0.334904i
$732$		0			0
$733$		−	1286.99i		−	1.75578i	−0.478859	−	0.877892i	$-0.658949\pi$
$733$		−	1286.99i		−	1.75578i	0.478859	−	0.877892i	$-0.341051\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	107.147			0.145383
$738$		0			0
$739$	−1333.47			−1.80443			−0.902213	−	0.431292i	$-0.858058\pi$
$739$	−1333.47			−1.80443			−0.902213	+	0.431292i	$0.858058\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−776.476			−1.04506			−0.522528	−	0.852622i	$-0.675011\pi$
$743$	−776.476			−1.04506			−0.522528	+	0.852622i	$0.675011\pi$
$744$		0			0
$745$			1246.90i			1.67369i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	288.676	+	146.447i	0.385415	+	0.195524i
$750$		0			0
$751$	−48.8385			−0.0650313			−0.0325157	−	0.999471i	$-0.510352\pi$
$751$	−48.8385			−0.0650313			−0.0325157	+	0.999471i	$0.510352\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	427.952i		−	0.566824i
$756$		0			0
$757$	−1279.47			−1.69019			−0.845093	−	0.534620i	$-0.820455\pi$
$757$	−1279.47			−1.69019			−0.845093	+	0.534620i	$0.820455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1316.12i		−	1.72947i	−0.502231	−	0.864734i	$-0.667487\pi$
$761$		−	1316.12i		−	1.72947i	0.502231	−	0.864734i	$-0.332513\pi$
$762$		0			0
$763$	224.368	+	113.823i	0.294060	+	0.149178i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1828.41			2.38385
$768$		0			0
$769$			110.324i			0.143464i	0.997424	+	0.0717320i	$0.0228526\pi$
$769$			110.324i			0.143464i	−0.997424	+	0.0717320i	$0.977147\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	717.634i		−	0.928375i	−0.885737	−	0.464187i	$-0.846346\pi$
$773$		−	717.634i		−	0.928375i	0.885737	−	0.464187i	$-0.153654\pi$
$774$		0			0
$775$		−	85.8392i		−	0.110760i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	954.573			1.22538
$780$		0			0
$781$	194.382			0.248888
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1381.85			−1.76032
$786$		0			0
$787$		−	347.191i		−	0.441158i	−0.975369	−	0.220579i	$-0.929205\pi$
$787$		−	347.191i		−	0.441158i	0.975369	−	0.220579i	$-0.0707946\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−455.897	−	231.279i	−0.576355	−	0.292388i
$792$		0			0
$793$	−313.706			−0.395593
$794$		0			0
$795$		0			0
$796$		0			0
$797$			361.246i			0.453257i	0.973981	+	0.226629i	$0.0727704\pi$
$797$			361.246i			0.453257i	−0.973981	+	0.226629i	$0.927230\pi$
$798$		0			0
$799$	908.674			1.13726
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	99.7451i		−	0.124216i
$804$		0			0
$805$	−350.735	+	691.368i	−0.435696	+	0.858842i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	113.147			0.139861			0.0699303	−	0.997552i	$-0.477722\pi$
$809$	113.147			0.139861			0.0699303	+	0.997552i	$0.477722\pi$
$810$		0			0
$811$			134.182i			0.165453i	0.996572	+	0.0827264i	$0.0263628\pi$
$811$			134.182i			0.165453i	−0.996572	+	0.0827264i	$0.973637\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	432.214i		−	0.530324i
$816$		0			0
$817$			240.433i			0.294288i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−491.677			−0.598876			−0.299438	−	0.954116i	$-0.596799\pi$
$821$	−491.677			−0.598876			−0.299438	+	0.954116i	$0.596799\pi$
$822$		0			0
$823$	941.852			1.14441			0.572207	−	0.820110i	$-0.306088\pi$
$823$	941.852			1.14441			0.572207	+	0.820110i	$0.306088\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	966.978			1.16926			0.584630	−	0.811300i	$-0.301240\pi$
$827$	966.978			1.16926			0.584630	+	0.811300i	$0.301240\pi$
$828$		0			0
$829$			1185.53i			1.43007i	0.699088	+	0.715035i	$0.253589\pi$
$829$			1185.53i			1.43007i	−0.699088	+	0.715035i	$0.746411\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−928.410	+	679.546i	−1.11454	+	0.815781i
$834$		0			0
$835$	232.971			0.279007
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1376.91i		−	1.64113i	−0.571550	−	0.820567i	$-0.693658\pi$
$839$		−	1376.91i		−	1.64113i	0.571550	−	0.820567i	$-0.306342\pi$
$840$		0			0
$841$	59.0000			0.0701546
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1080.79i			1.27905i
$846$		0			0
$847$	736.080	+	373.418i	0.869044	+	0.440871i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1328.03			−1.56055
$852$		0			0
$853$			175.006i			0.205165i	0.994725	+	0.102582i	$0.0327105\pi$
$853$			175.006i			0.205165i	−0.994725	+	0.102582i	$0.967289\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	61.7471i		−	0.0720503i	−0.999351	−	0.0360252i	$-0.988530\pi$
$857$		−	61.7471i		−	0.0720503i	0.999351	−	0.0360252i	$-0.0114696\pi$
$858$		0			0
$859$		−	191.306i		−	0.222708i	−0.993781	−	0.111354i	$-0.964481\pi$
$859$		−	191.306i		−	0.222708i	0.993781	−	0.111354i	$-0.0355188\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−494.507			−0.573009			−0.286504	−	0.958079i	$-0.592493\pi$
$863$	−494.507			−0.573009			−0.286504	+	0.958079i	$0.592493\pi$
$864$		0			0
$865$	140.912			0.162904
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−122.705			−0.141202
$870$		0			0
$871$			1143.53i			1.31289i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−281.470	+	554.833i	−0.321680	+	0.634095i
$876$		0			0
$877$	−454.353			−0.518077			−0.259038	−	0.965867i	$-0.583406\pi$
$877$	−454.353			−0.518077			−0.259038	+	0.965867i	$0.583406\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			143.493i			0.162875i	0.996678	+	0.0814375i	$0.0259511\pi$
$881$			143.493i			0.162875i	−0.996678	+	0.0814375i	$0.974049\pi$
$882$		0			0
$883$	927.986			1.05095			0.525473	−	0.850810i	$-0.323888\pi$
$883$	927.986			1.05095			0.525473	+	0.850810i	$0.323888\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	958.265i		−	1.08034i	−0.841555	−	0.540172i	$-0.818359\pi$
$887$		−	958.265i		−	1.08034i	0.841555	−	0.540172i	$-0.181641\pi$
$888$		0			0
$889$	561.470	+	284.837i	0.631575	+	0.320402i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−892.410			−0.999340
$894$		0			0
$895$			76.3122i			0.0852650i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	258.278i		−	0.287295i
$900$		0			0
$901$			869.462i			0.964996i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−386.558			−0.427136
$906$		0			0
$907$	552.721			0.609394			0.304697	−	0.952449i	$-0.401445\pi$
$907$	552.721			0.609394			0.304697	+	0.952449i	$0.401445\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	142.742			0.156687			0.0783437	−	0.996926i	$-0.475037\pi$
$911$	142.742			0.156687			0.0783437	+	0.996926i	$0.475037\pi$
$912$		0			0
$913$			11.3101i			0.0123878i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−38.5584	+	76.0063i	−0.0420485	+	0.0828858i
$918$		0			0
$919$	−1086.45			−1.18221			−0.591107	−	0.806593i	$-0.701309\pi$
$919$	−1086.45			−1.18221			−0.591107	+	0.806593i	$0.701309\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2074.54i			2.24760i
$924$		0			0
$925$	707.029			0.764356
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1550.75i		−	1.66927i	−0.550806	−	0.834633i	$-0.685680\pi$
$929$		−	1550.75i		−	1.66927i	0.550806	−	0.834633i	$-0.314320\pi$
$930$		0			0
$931$	911.793	−	667.383i	0.979370	−	0.716845i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−244.014			−0.260978
$936$		0			0
$937$		−	759.317i		−	0.810370i	−0.914235	−	0.405185i	$-0.867207\pi$
$937$		−	759.317i		−	0.810370i	0.914235	−	0.405185i	$-0.132793\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			36.3519i			0.0386312i	0.999813	+	0.0193156i	$0.00614873\pi$
$941$			36.3519i			0.0386312i	−0.999813	+	0.0193156i	$0.993851\pi$
$942$		0			0
$943$			775.245i			0.822105i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	92.5370			0.0977160			0.0488580	−	0.998806i	$-0.484442\pi$
$947$	92.5370			0.0977160			0.0488580	+	0.998806i	$0.484442\pi$
$948$		0			0
$949$	1064.53			1.12174
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1361.29			1.42843			0.714215	−	0.699927i	$-0.246784\pi$
$953$	1361.29			1.42843			0.714215	+	0.699927i	$0.246784\pi$
$954$		0			0
$955$			591.750i			0.619633i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1034.81	+	524.964i	1.07905	+	0.547408i
$960$		0			0
$961$	886.881			0.922873
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	466.651i		−	0.483577i
$966$		0			0
$967$	−481.677			−0.498115			−0.249057	−	0.968489i	$-0.580121\pi$
$967$	−481.677			−0.498115			−0.249057	+	0.968489i	$0.580121\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1427.34i			1.46997i	0.678082	+	0.734987i	$0.262812\pi$
$971$			1427.34i			1.46997i	−0.678082	+	0.734987i	$0.737188\pi$
$972$		0			0
$973$	−698.352	+	1376.59i	−0.717730	+	1.41479i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1356.26			1.38819			0.694095	−	0.719883i	$-0.255805\pi$
$977$	1356.26			1.38819			0.694095	+	0.719883i	$0.255805\pi$
$978$		0			0
$979$			73.9698i			0.0755565i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1315.51i		−	1.33826i	−0.743146	−	0.669129i	$-0.766667\pi$
$983$		−	1315.51i		−	1.33826i	0.743146	−	0.669129i	$-0.233333\pi$
$984$		0			0
$985$		−	1087.75i		−	1.10432i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−195.265			−0.197437
$990$		0			0
$991$	−1043.28			−1.05275			−0.526377	−	0.850251i	$-0.676450\pi$
$991$	−1043.28			−1.05275			−0.526377	+	0.850251i	$0.676450\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1005.48			1.01054
$996$		0			0
$997$		−	820.328i		−	0.822796i	−0.911456	−	0.411398i	$-0.865041\pi$
$997$		−	820.328i		−	0.822796i	0.911456	−	0.411398i	$-0.134959\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.f.g.433.1		4
3.2	odd	2		336.3.f.c.97.2		4
4.3	odd	2		126.3.c.b.55.3		4
7.6	odd	2	inner	1008.3.f.g.433.4		4
12.11	even	2		42.3.c.a.13.2	yes	4
21.20	even	2		336.3.f.c.97.3		4
24.5	odd	2		1344.3.f.e.769.3		4
24.11	even	2		1344.3.f.f.769.1		4
28.3	even	6		882.3.n.a.19.1		4
28.11	odd	6		882.3.n.d.19.1		4
28.19	even	6		882.3.n.d.325.1		4
28.23	odd	6		882.3.n.a.325.1		4
28.27	even	2		126.3.c.b.55.4		4
60.23	odd	4		1050.3.h.a.349.6		8
60.47	odd	4		1050.3.h.a.349.3		8
60.59	even	2		1050.3.f.a.601.3		4
84.11	even	6		294.3.g.c.19.2		4
84.23	even	6		294.3.g.b.31.2		4
84.47	odd	6		294.3.g.c.31.2		4
84.59	odd	6		294.3.g.b.19.2		4
84.83	odd	2		42.3.c.a.13.1	✓	4
168.83	odd	2		1344.3.f.f.769.4		4
168.125	even	2		1344.3.f.e.769.2		4
420.83	even	4		1050.3.h.a.349.7		8
420.167	even	4		1050.3.h.a.349.2		8
420.419	odd	2		1050.3.f.a.601.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.1	✓	4	84.83	odd	2
42.3.c.a.13.2	yes	4	12.11	even	2
126.3.c.b.55.3		4	4.3	odd	2
126.3.c.b.55.4		4	28.27	even	2
294.3.g.b.19.2		4	84.59	odd	6
294.3.g.b.31.2		4	84.23	even	6
294.3.g.c.19.2		4	84.11	even	6
294.3.g.c.31.2		4	84.47	odd	6
336.3.f.c.97.2		4	3.2	odd	2
336.3.f.c.97.3		4	21.20	even	2
882.3.n.a.19.1		4	28.3	even	6
882.3.n.a.325.1		4	28.23	odd	6
882.3.n.d.19.1		4	28.11	odd	6
882.3.n.d.325.1		4	28.19	even	6
1008.3.f.g.433.1		4	1.1	even	1	trivial
1008.3.f.g.433.4		4	7.6	odd	2	inner
1050.3.f.a.601.3		4	60.59	even	2
1050.3.f.a.601.4		4	420.419	odd	2
1050.3.h.a.349.2		8	420.167	even	4
1050.3.h.a.349.3		8	60.47	odd	4
1050.3.h.a.349.6		8	60.23	odd	4
1050.3.h.a.349.7		8	420.83	even	4
1344.3.f.e.769.2		4	168.125	even	2
1344.3.f.e.769.3		4	24.5	odd	2
1344.3.f.f.769.1		4	24.11	even	2
1344.3.f.f.769.4		4	168.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.91359i q^{5} +(-6.24264 - 3.16693i) q^{7} +O(q^{10})\)	\(q-5.91359i q^{5} +(-6.24264 - 3.16693i) q^{7} -1.75736 q^{11} -18.7554i q^{13} +23.4803i q^{17} -23.0600i q^{19} +18.7279 q^{23} -9.97056 q^{25} -30.0000 q^{29} +8.60927i q^{31} +(-18.7279 + 36.9164i) q^{35} -70.9117 q^{37} +41.3951i q^{41} -10.4264 q^{43} -38.6995i q^{47} +(28.9411 + 39.5400i) q^{49} +37.0294 q^{53} +10.3923i q^{55} +97.4872i q^{59} -16.7262i q^{61} -110.912 q^{65} -60.9706 q^{67} -110.610 q^{71} +56.7585i q^{73} +(10.9706 + 5.56543i) q^{77} +69.8234 q^{79} -6.43583i q^{83} +138.853 q^{85} -42.0915i q^{89} +(-59.3970 + 117.083i) q^{91} -136.368 q^{95} -51.7153i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 24 q^{11} + 24 q^{23} + 28 q^{25} - 120 q^{29} - 24 q^{35} - 80 q^{37} + 128 q^{43} - 20 q^{49} + 216 q^{53} - 240 q^{65} - 176 q^{67} - 120 q^{71} - 24 q^{77} - 128 q^{79} + 216 q^{85} - 240 q^{95}+O(q^{100}) \) 4 * q - 8 * q^7 - 24 * q^11 + 24 * q^23 + 28 * q^25 - 120 * q^29 - 24 * q^35 - 80 * q^37 + 128 * q^43 - 20 * q^49 + 216 * q^53 - 240 * q^65 - 176 * q^67 - 120 * q^71 - 24 * q^77 - 128 * q^79 + 216 * q^85 - 240 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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