LMFDB - Embedded newform 1344.3.f.f.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,3,Mod(769,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, 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("1344.769");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.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:	$36.6213475300$
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:	$\Z[a_1, \ldots, a_{5}]$
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			769.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.f.769.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -5.91359i q^{5} +(6.24264 + 3.16693i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -5.91359i q^{5} +(6.24264 + 3.16693i) q^{7} -3.00000 q^{9} +1.75736 q^{11} +18.7554i q^{13} -10.2426 q^{15} -23.4803i q^{17} -23.0600i q^{19} +(5.48528 - 10.8126i) q^{21} +18.7279 q^{23} -9.97056 q^{25} +5.19615i q^{27} -30.0000 q^{29} -8.60927i q^{31} -3.04384i q^{33} +(18.7279 - 36.9164i) q^{35} +70.9117 q^{37} +32.4853 q^{39} -41.3951i q^{41} -10.4264 q^{43} +17.7408i q^{45} -38.6995i q^{47} +(28.9411 + 39.5400i) q^{49} -40.6690 q^{51} +37.0294 q^{53} -10.3923i q^{55} -39.9411 q^{57} -97.4872i q^{59} +16.7262i q^{61} +(-18.7279 - 9.50079i) q^{63} +110.912 q^{65} -60.9706 q^{67} -32.4377i q^{69} -110.610 q^{71} +56.7585i q^{73} +17.2695i q^{75} +(10.9706 + 5.56543i) q^{77} -69.8234 q^{79} +9.00000 q^{81} +6.43583i q^{83} -138.853 q^{85} +51.9615i q^{87} +42.0915i q^{89} +(-59.3970 + 117.083i) q^{91} -14.9117 q^{93} -136.368 q^{95} -51.7153i q^{97} -5.27208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^7 - 12 * q^9	$ 4 q + 8 q^{7} - 12 q^{9} + 24 q^{11} - 24 q^{15} - 12 q^{21} + 24 q^{23} + 28 q^{25} - 120 q^{29} + 24 q^{35} + 80 q^{37} + 96 q^{39} + 128 q^{43} - 20 q^{49} + 24 q^{51} + 216 q^{53} - 24 q^{57} - 24 q^{63} + 240 q^{65} - 176 q^{67} - 120 q^{71} - 24 q^{77} + 128 q^{79} + 36 q^{81} - 216 q^{85} + 144 q^{93} - 240 q^{95} - 72 q^{99}+O(q^{100}) $ 4 * q + 8 * q^7 - 12 * q^9 + 24 * q^11 - 24 * q^15 - 12 * q^21 + 24 * q^23 + 28 * q^25 - 120 * q^29 + 24 * q^35 + 80 * q^37 + 96 * q^39 + 128 * q^43 - 20 * q^49 + 24 * q^51 + 216 * q^53 - 24 * q^57 - 24 * q^63 + 240 * q^65 - 176 * q^67 - 120 * q^71 - 24 * q^77 + 128 * q^79 + 36 * q^81 - 216 * q^85 + 144 * q^93 - 240 * q^95 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		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
$9$	−3.00000			−0.333333
$10$		0			0
$11$	1.75736			0.159760			0.0798800	−	0.996804i	$-0.474546\pi$
$11$	1.75736			0.159760			0.0798800	+	0.996804i	$0.474546\pi$
$12$		0			0
$13$			18.7554i			1.44272i	0.692559	+	0.721361i	$0.256483\pi$
$13$			18.7554i			1.44272i	−0.692559	+	0.721361i	$0.743517\pi$
$14$		0			0
$15$	−10.2426			−0.682843
$16$		0			0
$17$		−	23.4803i		−	1.38119i	−0.723240	−	0.690597i	$-0.757348\pi$
$17$		−	23.4803i		−	1.38119i	0.723240	−	0.690597i	$-0.242652\pi$
$18$		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
$21$	5.48528	−	10.8126i	0.261204	−	0.514884i
$22$		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
$27$			5.19615i			0.192450i
$28$		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	$-0.955656\pi$
$31$		−	8.60927i		−	0.277718i	0.990312	−	0.138859i	$-0.0443435\pi$
$32$		0			0
$33$		−	3.04384i		−	0.0922374i
$34$		0			0
$35$	18.7279	−	36.9164i	0.535083	−	1.05476i
$36$		0			0
$37$	70.9117			1.91653			0.958266	−	0.285878i	$-0.0922852\pi$
$37$	70.9117			1.91653			0.958266	+	0.285878i	$0.0922852\pi$
$38$		0			0
$39$	32.4853			0.832956
$40$		0			0
$41$		−	41.3951i		−	1.00964i	−0.863225	−	0.504819i	$-0.831559\pi$
$41$		−	41.3951i		−	1.00964i	0.863225	−	0.504819i	$-0.168441\pi$
$42$		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
$45$			17.7408i			0.394239i
$46$		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
$51$	−40.6690			−0.797432
$52$		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
$57$	−39.9411			−0.700721
$58$		0			0
$59$		−	97.4872i		−	1.65233i	−0.563431	−	0.826163i	$-0.690519\pi$
$59$		−	97.4872i		−	1.65233i	0.563431	−	0.826163i	$-0.309481\pi$
$60$		0			0
$61$			16.7262i			0.274199i	0.990557	+	0.137100i	$0.0437781\pi$
$61$			16.7262i			0.274199i	−0.990557	+	0.137100i	$0.956222\pi$
$62$		0			0
$63$	−18.7279	−	9.50079i	−0.297269	−	0.150806i
$64$		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
$69$		−	32.4377i		−	0.470112i
$70$		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
$75$			17.2695i			0.230260i
$76$		0			0
$77$	10.9706	+	5.56543i	0.142475	+	0.0722783i
$78$		0			0
$79$	−69.8234			−0.883840			−0.441920	−	0.897054i	$-0.645703\pi$
$79$	−69.8234			−0.883840			−0.441920	+	0.897054i	$0.645703\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			6.43583i			0.0775401i	0.999248	+	0.0387701i	$0.0123440\pi$
$83$			6.43583i			0.0775401i	−0.999248	+	0.0387701i	$0.987656\pi$
$84$		0			0
$85$	−138.853			−1.63356
$86$		0			0
$87$			51.9615i			0.597259i
$88$		0			0
$89$			42.0915i			0.472938i	0.971639	+	0.236469i	$0.0759901\pi$
$89$			42.0915i			0.472938i	−0.971639	+	0.236469i	$0.924010\pi$
$90$		0			0
$91$	−59.3970	+	117.083i	−0.652714	+	1.28663i
$92$		0			0
$93$	−14.9117			−0.160341
$94$		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
$99$	−5.27208			−0.0532533
$100$		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	$-0.674327\pi$
$103$		−	175.871i		−	1.70748i	0.520696	−	0.853742i	$-0.325673\pi$
$104$		0			0
$105$	−63.9411	−	32.4377i	−0.608963	−	0.308931i
$106$		0			0
$107$	46.2426			0.432174			0.216087	−	0.976374i	$-0.430670\pi$
$107$	46.2426			0.432174			0.216087	+	0.976374i	$0.430670\pi$
$108$		0			0
$109$	35.9411			0.329735			0.164868	−	0.986316i	$-0.447280\pi$
$109$	35.9411			0.329735			0.164868	+	0.986316i	$0.447280\pi$
$110$		0			0
$111$		−	122.823i		−	1.10651i
$112$		0			0
$113$	−73.0294			−0.646278			−0.323139	−	0.946351i	$-0.604738\pi$
$113$	−73.0294			−0.646278			−0.323139	+	0.946351i	$0.604738\pi$
$114$		0			0
$115$		−	110.749i		−	0.963037i
$116$		0			0
$117$		−	56.2662i		−	0.480907i
$118$		0			0
$119$	74.3604	−	146.579i	0.624877	−	1.23176i
$120$		0			0
$121$	−117.912			−0.974477
$122$		0			0
$123$	−71.6985			−0.582915
$124$		0			0
$125$		−	88.8780i		−	0.711024i
$126$		0			0
$127$	89.9411			0.708198			0.354099	−	0.935208i	$-0.384788\pi$
$127$	89.9411			0.708198			0.354099	+	0.935208i	$0.384788\pi$
$128$		0			0
$129$			18.0591i			0.139993i
$130$		0			0
$131$			12.1753i			0.0929415i	0.998920	+	0.0464708i	$0.0147974\pi$
$131$			12.1753i			0.0929415i	−0.998920	+	0.0464708i	$0.985203\pi$
$132$		0			0
$133$	73.0294	−	143.955i	0.549094	−	1.08237i
$134$		0			0
$135$	30.7279			0.227614
$136$		0			0
$137$	165.765			1.20996			0.604980	−	0.796241i	$-0.293181\pi$
$137$	165.765			1.20996			0.604980	+	0.796241i	$0.293181\pi$
$138$		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
$141$	−67.0294			−0.475386
$142$		0			0
$143$			32.9600i			0.230489i
$144$		0			0
$145$			177.408i			1.22350i
$146$		0			0
$147$	68.4853	−	50.1275i	0.465886	−	0.341003i
$148$		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	$-0.577025\pi$
$151$	−72.3675			−0.479255			−0.239628	+	0.970865i	$0.577025\pi$
$152$		0			0
$153$			70.4409i			0.460398i
$154$		0			0
$155$	−50.9117			−0.328463
$156$		0			0
$157$			233.674i			1.48837i	0.667974	+	0.744184i	$0.267162\pi$
$157$			233.674i			1.48837i	−0.667974	+	0.744184i	$0.732838\pi$
$158$		0			0
$159$		−	64.1369i		−	0.403377i
$160$		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
$165$	−18.0000			−0.109091
$166$		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
$171$			69.1801i			0.404562i
$172$		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
$177$	−168.853			−0.953971
$178$		0			0
$179$	12.9045			0.0720924			0.0360462	−	0.999350i	$-0.488524\pi$
$179$	12.9045			0.0720924			0.0360462	+	0.999350i	$0.488524\pi$
$180$		0			0
$181$			65.3678i			0.361148i	0.983561	+	0.180574i	$0.0577955\pi$
$181$			65.3678i			0.361148i	−0.983561	+	0.180574i	$0.942204\pi$
$182$		0			0
$183$	28.9706			0.158309
$184$		0			0
$185$		−	419.343i		−	2.26672i
$186$		0			0
$187$		−	41.2633i		−	0.220659i
$188$		0			0
$189$	−16.4558	+	32.4377i	−0.0870680	+	0.171628i
$190$		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
$195$		−	192.105i		−	0.985152i
$196$		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	$-0.859497\pi$
$199$		−	170.029i		−	0.854419i	0.904153	−	0.427210i	$-0.140503\pi$
$200$		0			0
$201$			105.604i			0.525394i
$202$		0			0
$203$	−187.279	−	95.0079i	−0.922558	−	0.468019i
$204$		0			0
$205$	−244.794			−1.19412
$206$		0			0
$207$	−56.1838			−0.271419
$208$		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
$213$			191.582i			0.899448i
$214$		0			0
$215$			61.6575i			0.286779i
$216$		0			0
$217$	27.2649	−	53.7446i	0.125645	−	0.247671i
$218$		0			0
$219$	98.3087			0.448898
$220$		0			0
$221$	440.382			1.99268
$222$		0			0
$223$			119.359i			0.535240i	0.963525	+	0.267620i	$0.0862372\pi$
$223$			119.359i			0.535240i	−0.963525	+	0.267620i	$0.913763\pi$
$224$		0			0
$225$	29.9117			0.132941
$226$		0			0
$227$		−	169.843i		−	0.748207i	−0.927387	−	0.374103i	$-0.877951\pi$
$227$		−	169.843i		−	0.748207i	0.927387	−	0.374103i	$-0.122049\pi$
$228$		0			0
$229$		−	110.011i		−	0.480396i	−0.970724	−	0.240198i	$-0.922788\pi$
$229$		−	110.011i		−	0.480396i	0.970724	−	0.240198i	$-0.0772124\pi$
$230$		0			0
$231$	9.63961	−	19.0016i	0.0417299	−	0.0822579i
$232$		0			0
$233$	57.2649			0.245772			0.122886	−	0.992421i	$-0.460785\pi$
$233$	57.2649			0.245772			0.122886	+	0.992421i	$0.460785\pi$
$234$		0			0
$235$	−228.853			−0.973842
$236$		0			0
$237$			120.938i			0.510285i
$238$		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
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	233.823	−	171.146i	0.954381	−	0.698555i
$246$		0			0
$247$	432.500			1.75101
$248$		0			0
$249$	11.1472			0.0447678
$250$		0			0
$251$			106.096i			0.422695i	0.977411	+	0.211348i	$0.0677852\pi$
$251$			106.096i			0.422695i	−0.977411	+	0.211348i	$0.932215\pi$
$252$		0			0
$253$	32.9117			0.130086
$254$		0			0
$255$			240.500i			0.943138i
$256$		0			0
$257$			290.462i			1.13020i	0.825021	+	0.565102i	$0.191163\pi$
$257$			290.462i			1.13020i	−0.825021	+	0.565102i	$0.808837\pi$
$258$		0			0
$259$	442.676	+	224.572i	1.70917	+	0.867074i
$260$		0			0
$261$	90.0000			0.344828
$262$		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
$267$	72.9045			0.273051
$268$		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	$0.131400\pi$
$271$			217.440i			0.802362i	−0.915999	+	0.401181i	$0.868600\pi$
$272$		0			0
$273$	202.794	+	102.879i	0.742835	+	0.376845i
$274$		0			0
$275$	−17.5219			−0.0637159
$276$		0			0
$277$	−290.676			−1.04937			−0.524686	−	0.851296i	$-0.675817\pi$
$277$	−290.676			−1.04937			−0.524686	+	0.851296i	$0.675817\pi$
$278$		0			0
$279$			25.8278i			0.0925728i
$280$		0			0
$281$	18.8528			0.0670919			0.0335459	−	0.999437i	$-0.489320\pi$
$281$	18.8528			0.0670919			0.0335459	+	0.999437i	$0.489320\pi$
$282$		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
$285$			236.195i			0.828756i
$286$		0			0
$287$	131.095	−	258.415i	0.456779	−	0.900401i
$288$		0			0
$289$	−262.324			−0.907695
$290$		0			0
$291$	−89.5736			−0.307813
$292$		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
$297$			9.13151i			0.0307458i
$298$		0			0
$299$			351.249i			1.17475i
$300$		0			0
$301$	−65.0883	−	33.0197i	−0.216240	−	0.109700i
$302$		0			0
$303$	11.4487			0.0377845
$304$		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
$309$	−304.617			−0.985817
$310$		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
$315$	−56.1838	+	110.749i	−0.178361	+	0.351585i
$316$		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
$321$		−	80.0946i		−	0.249516i
$322$		0			0
$323$	−541.456			−1.67633
$324$		0			0
$325$		−	187.002i		−	0.575390i
$326$		0			0
$327$		−	62.2519i		−	0.190373i
$328$		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
$333$	−212.735			−0.638844
$334$		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
$339$			126.491i			0.373129i
$340$		0			0
$341$		−	15.1296i		−	0.0443683i
$342$		0			0
$343$	55.4487	+	338.488i	0.161658	+	0.986847i
$344$		0			0
$345$	−191.823			−0.556010
$346$		0			0
$347$	320.080			0.922422			0.461211	−	0.887291i	$-0.347415\pi$
$347$	320.080			0.922422			0.461211	+	0.887291i	$0.347415\pi$
$348$		0			0
$349$		−	333.046i		−	0.954287i	−0.878825	−	0.477143i	$-0.841672\pi$
$349$		−	333.046i		−	0.954287i	0.878825	−	0.477143i	$-0.158328\pi$
$350$		0			0
$351$	−97.4558			−0.277652
$352$		0			0
$353$			655.712i			1.85754i	0.370654	+	0.928771i	$0.379134\pi$
$353$			655.712i			1.85754i	−0.370654	+	0.928771i	$0.620866\pi$
$354$		0			0
$355$			654.103i			1.84254i
$356$		0			0
$357$	−253.882	−	128.796i	−0.711155	−	0.360773i
$358$		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
$363$			204.229i			0.562614i
$364$		0			0
$365$	335.647			0.919580
$366$		0			0
$367$			321.057i			0.874815i	0.899263	+	0.437408i	$0.144103\pi$
$367$			321.057i			0.874815i	−0.899263	+	0.437408i	$0.855897\pi$
$368$		0			0
$369$			124.185i			0.336546i
$370$		0			0
$371$	231.161	+	117.270i	0.623077	+	0.316091i
$372$		0			0
$373$	−187.470			−0.502601			−0.251300	−	0.967909i	$-0.580858\pi$
$373$	−187.470			−0.502601			−0.251300	+	0.967909i	$0.580858\pi$
$374$		0			0
$375$	−153.941			−0.410510
$376$		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
$381$		−	155.783i		−	0.408878i
$382$		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
$387$	31.2792			0.0808249
$388$		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
$393$	21.0883			0.0536598
$394$		0			0
$395$			412.907i			1.04533i
$396$		0			0
$397$			720.329i			1.81443i	0.420666	+	0.907216i	$0.361796\pi$
$397$			720.329i			1.81443i	−0.420666	+	0.907216i	$0.638204\pi$
$398$		0			0
$399$	−249.338	−	126.491i	−0.624908	−	0.317019i
$400$		0			0
$401$	697.176			1.73859			0.869296	−	0.494291i	$-0.164572\pi$
$401$	697.176			1.73859			0.869296	+	0.494291i	$0.164572\pi$
$402$		0			0
$403$	161.470			0.400670
$404$		0			0
$405$		−	53.2223i		−	0.131413i
$406$		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
$411$		−	287.113i		−	0.698571i
$412$		0			0
$413$	308.735	−	608.578i	0.747543	−	1.47355i
$414$		0			0
$415$	38.0589			0.0917081
$416$		0			0
$417$	−381.941			−0.915926
$418$		0			0
$419$			391.426i			0.934191i	0.884207	+	0.467095i	$0.154700\pi$
$419$			391.426i			0.934191i	−0.884207	+	0.467095i	$0.845300\pi$
$420$		0			0
$421$	−354.441			−0.841902			−0.420951	−	0.907083i	$-0.638304\pi$
$421$	−354.441			−0.841902			−0.420951	+	0.907083i	$0.638304\pi$
$422$		0			0
$423$			116.098i			0.274464i
$424$		0			0
$425$			234.112i			0.550851i
$426$		0			0
$427$	−52.9706	+	104.415i	−0.124053	+	0.244533i
$428$		0			0
$429$	57.0883			0.133073
$430$		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
$435$	307.279			0.706389
$436$		0			0
$437$		−	431.866i		−	0.988252i
$438$		0			0
$439$			392.513i			0.894106i	0.894507	+	0.447053i	$0.147527\pi$
$439$			392.513i			0.894106i	−0.894507	+	0.447053i	$0.852473\pi$
$440$		0			0
$441$	−86.8234	−	118.620i	−0.196878	−	0.268980i
$442$		0			0
$443$	814.742			1.83915			0.919574	−	0.392918i	$-0.128534\pi$
$443$	814.742			1.83915			0.919574	+	0.392918i	$0.128534\pi$
$444$		0			0
$445$	248.912			0.559352
$446$		0			0
$447$			365.208i			0.817020i
$448$		0			0
$449$	180.323			0.401610			0.200805	−	0.979631i	$-0.435644\pi$
$449$	180.323			0.401610			0.200805	+	0.979631i	$0.435644\pi$
$450$		0			0
$451$		−	72.7461i		−	0.161300i
$452$		0			0
$453$			125.344i			0.276698i
$454$		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
$459$	122.007			0.265811
$460$		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	$-0.335524\pi$
$463$	457.470			0.988056			0.494028	+	0.869446i	$0.335524\pi$
$464$		0			0
$465$			88.1816i			0.189638i
$466$		0			0
$467$		−	643.711i		−	1.37840i	−0.724573	−	0.689198i	$-0.757963\pi$
$467$		−	643.711i		−	1.37840i	0.724573	−	0.689198i	$-0.242037\pi$
$468$		0			0
$469$	−380.617	−	193.089i	−0.811551	−	0.411705i
$470$		0			0
$471$	404.735			0.859310
$472$		0			0
$473$	−18.3229			−0.0387377
$474$		0			0
$475$			229.921i			0.484045i
$476$		0			0
$477$	−111.088			−0.232890
$478$		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
$483$	102.728	−	202.497i	0.212687	−	0.419248i
$484$		0			0
$485$	−305.823			−0.630564
$486$		0			0
$487$	−2.77965			−0.00570771			−0.00285385	−	0.999996i	$-0.500908\pi$
$487$	−2.77965			−0.00570771			−0.00285385	+	0.999996i	$0.500908\pi$
$488$		0			0
$489$		−	126.593i		−	0.258881i
$490$		0			0
$491$	247.477			0.504027			0.252014	−	0.967724i	$-0.418907\pi$
$491$	247.477			0.504027			0.252014	+	0.967724i	$0.418907\pi$
$492$		0			0
$493$			704.409i			1.42882i
$494$		0			0
$495$			31.1769i			0.0629837i
$496$		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
$501$	68.2355			0.136199
$502$		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
$507$			316.557i			0.624374i
$508$		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
$513$	119.823			0.233574
$514$		0			0
$515$	−1040.03			−2.01947
$516$		0			0
$517$		−	68.0089i		−	0.131545i
$518$		0			0
$519$	41.2721			0.0795223
$520$		0			0
$521$		−	292.720i		−	0.561843i	−0.959731	−	0.280922i	$-0.909360\pi$
$521$		−	292.720i		−	0.561843i	0.959731	−	0.280922i	$-0.0906401\pi$
$522$		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
$525$	−54.6913	+	107.807i	−0.104174	+	0.205347i
$526$		0			0
$527$	−202.148			−0.383583
$528$		0			0
$529$	−178.265			−0.336985
$530$		0			0
$531$			292.462i			0.550775i
$532$		0			0
$533$	776.382			1.45663
$534$		0			0
$535$		−	273.460i		−	0.511140i
$536$		0			0
$537$		−	22.3513i		−	0.0416226i
$538$		0			0
$539$	50.8600	+	69.4860i	0.0943598	+	0.128916i
$540$		0			0
$541$	1037.85			1.91840			0.959198	−	0.282736i	$-0.0912421\pi$
$541$	1037.85			1.91840			0.959198	+	0.282736i	$0.0912421\pi$
$542$		0			0
$543$	113.220			0.208509
$544$		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
$549$		−	50.1785i		−	0.0913998i
$550$		0			0
$551$			691.801i			1.25554i
$552$		0			0
$553$	−435.882	−	221.126i	−0.788214	−	0.399866i
$554$		0			0
$555$	−726.323			−1.30869
$556$		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
$561$	−71.4701			−0.127398
$562$		0			0
$563$			829.295i			1.47299i	0.676441	+	0.736497i	$0.263521\pi$
$563$			829.295i			1.47299i	−0.676441	+	0.736497i	$0.736479\pi$
$564$		0			0
$565$			431.866i			0.764365i
$566$		0			0
$567$	56.1838	+	28.5024i	0.0990895	+	0.0502687i
$568$		0			0
$569$	706.971			1.24248			0.621240	−	0.783621i	$-0.286629\pi$
$569$	706.971			1.24248			0.621240	+	0.783621i	$0.286629\pi$
$570$		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
$573$			173.319i			0.302477i
$574$		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
$579$		−	136.679i		−	0.236061i
$580$		0			0
$581$	−20.3818	+	40.1766i	−0.0350806	+	0.0691507i
$582$		0			0
$583$	65.0740			0.111619
$584$		0			0
$585$	−332.735			−0.568778
$586$		0			0
$587$		−	702.499i		−	1.19676i	−0.801212	−	0.598381i	$-0.795811\pi$
$587$		−	702.499i		−	1.19676i	0.801212	−	0.598381i	$-0.204189\pi$
$588$		0			0
$589$	−198.530			−0.337063
$590$		0			0
$591$		−	318.595i		−	0.539078i
$592$		0			0
$593$		−	942.519i		−	1.58941i	−0.606997	−	0.794704i	$-0.707626\pi$
$593$		−	942.519i		−	1.58941i	0.606997	−	0.794704i	$-0.292374\pi$
$594$		0			0
$595$	−866.808	−	439.737i	−1.45682	−	0.739054i
$596$		0			0
$597$	−294.500			−0.493299
$598$		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
$603$	182.912			0.303336
$604$		0			0
$605$			697.282i			1.15253i
$606$		0			0
$607$			1006.34i			1.65789i	0.559332	+	0.828944i	$0.311058\pi$
$607$			1006.34i			1.65789i	−0.559332	+	0.828944i	$0.688942\pi$
$608$		0			0
$609$	−164.558	+	324.377i	−0.270211	+	0.532639i
$610$		0			0
$611$	725.823			1.18793
$612$		0			0
$613$	199.588			0.325592			0.162796	−	0.986660i	$-0.447949\pi$
$613$	199.588			0.325592			0.162796	+	0.986660i	$0.447949\pi$
$614$		0			0
$615$			423.996i			0.689424i
$616$		0			0
$617$	−353.294			−0.572599			−0.286299	−	0.958140i	$-0.592425\pi$
$617$	−353.294			−0.572599			−0.286299	+	0.958140i	$0.592425\pi$
$618$		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
$621$			97.3131i			0.156704i
$622$		0			0
$623$	−133.301	+	262.762i	−0.213966	+	0.421769i
$624$		0			0
$625$	−774.852			−1.23976
$626$		0			0
$627$	−70.1909			−0.111947
$628$		0			0
$629$		−	1665.03i		−	2.64710i
$630$		0			0
$631$	807.322			1.27943			0.639716	−	0.768611i	$-0.279052\pi$
$631$	807.322			1.27943			0.639716	+	0.768611i	$0.279052\pi$
$632$		0			0
$633$			37.3666i			0.0590309i
$634$		0			0
$635$		−	531.875i		−	0.837599i
$636$		0			0
$637$	−741.588	+	542.802i	−1.16419	+	0.852122i
$638$		0			0
$639$	331.831			0.519297
$640$		0			0
$641$	1016.35			1.58557			0.792786	−	0.609500i	$-0.208630\pi$
$641$	1016.35			1.58557			0.792786	+	0.609500i	$0.208630\pi$
$642$		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
$645$	106.794			0.165572
$646$		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
$651$	−93.0883	−	47.2243i	−0.142993	−	0.0725411i
$652$		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
$657$		−	170.276i		−	0.259171i
$658$		0			0
$659$	−904.316			−1.37225			−0.686127	−	0.727481i	$-0.740691\pi$
$659$	−904.316			−1.37225			−0.686127	+	0.727481i	$0.740691\pi$
$660$		0			0
$661$		−	335.881i		−	0.508141i	−0.967186	−	0.254070i	$-0.918231\pi$
$661$		−	335.881i		−	0.508141i	0.967186	−	0.254070i	$-0.0817695\pi$
$662$		0			0
$663$		−	762.764i		−	1.15047i
$664$		0			0
$665$	−851.294	−	431.866i	−1.28014	−	0.649423i
$666$		0			0
$667$	−561.838			−0.842335
$668$		0			0
$669$	206.735			0.309021
$670$		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
$675$		−	51.8086i		−	0.0767534i
$676$		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
$681$	−294.177			−0.431977
$682$		0			0
$683$	1233.89			1.80657			0.903287	−	0.429038i	$-0.141147\pi$
$683$	1233.89			1.80657			0.903287	+	0.429038i	$0.141147\pi$
$684$		0			0
$685$		−	980.264i		−	1.43104i
$686$		0			0
$687$	−190.544			−0.277357
$688$		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
$693$	−32.9117	−	16.6963i	−0.0474916	−	0.0240928i
$694$		0			0
$695$	−1304.03			−1.87630
$696$		0			0
$697$	−971.970			−1.39450
$698$		0			0
$699$		−	99.1858i		−	0.141897i
$700$		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
$705$			396.385i			0.562248i
$706$		0			0
$707$	−20.9331	+	41.2633i	−0.0296084	+	0.0583639i
$708$		0			0
$709$	−189.647			−0.267485			−0.133742	−	0.991016i	$-0.542699\pi$
$709$	−189.647			−0.267485			−0.133742	+	0.991016i	$0.542699\pi$
$710$		0			0
$711$	209.470			0.294613
$712$		0			0
$713$		−	161.234i		−	0.226134i
$714$		0			0
$715$	194.912			0.272604
$716$		0			0
$717$		−	487.610i		−	0.680070i
$718$		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
$723$	−291.515			−0.403202
$724$		0			0
$725$	299.117			0.412575
$726$		0			0
$727$			417.169i			0.573823i	0.957957	+	0.286911i	$0.0926286\pi$
$727$			417.169i			0.573823i	−0.957957	+	0.286911i	$0.907371\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			244.815i			0.334904i
$732$		0			0
$733$			1286.99i			1.75578i	0.478859	+	0.877892i	$0.341051\pi$
$733$			1286.99i			1.75578i	−0.478859	+	0.877892i	$0.658949\pi$
$734$		0			0
$735$	−296.434	−	404.994i	−0.403311	−	0.551012i
$736$		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
$741$		−	749.111i		−	1.01095i
$742$		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
$747$		−	19.3075i		−	0.0258467i
$748$		0			0
$749$	288.676	+	146.447i	0.385415	+	0.195524i
$750$		0			0
$751$	48.8385			0.0650313			0.0325157	−	0.999471i	$-0.489648\pi$
$751$	48.8385			0.0650313			0.0325157	+	0.999471i	$0.489648\pi$
$752$		0			0
$753$	183.765			0.244043
$754$		0			0
$755$			427.952i			0.566824i
$756$		0			0
$757$	1279.47			1.69019			0.845093	−	0.534620i	$-0.179545\pi$
$757$	1279.47			1.69019			0.845093	+	0.534620i	$0.179545\pi$
$758$		0			0
$759$		−	57.0047i		−	0.0751050i
$760$		0			0
$761$			1316.12i			1.72947i	0.502231	+	0.864734i	$0.332513\pi$
$761$			1316.12i			1.72947i	−0.502231	+	0.864734i	$0.667487\pi$
$762$		0			0
$763$	224.368	+	113.823i	0.294060	+	0.149178i
$764$		0			0
$765$	416.558			0.544521
$766$		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
$771$	503.095			0.652523
$772$		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
$777$	388.971	−	766.738i	0.500606	−	0.986792i
$778$		0			0
$779$	−954.573			−1.22538
$780$		0			0
$781$	−194.382			−0.248888
$782$		0			0
$783$		−	155.885i		−	0.199086i
$784$		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
$789$			154.191i			0.195426i
$790$		0			0
$791$	−455.897	−	231.279i	−0.576355	−	0.292388i
$792$		0			0
$793$	−313.706			−0.395593
$794$		0			0
$795$	−379.279			−0.477081
$796$		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
$801$		−	126.274i		−	0.157646i
$802$		0			0
$803$			99.7451i			0.124216i
$804$		0			0
$805$	350.735	−	691.368i	0.435696	−	0.858842i
$806$		0			0
$807$	331.684			0.411009
$808$		0			0
$809$	−113.147			−0.139861			−0.0699303	−	0.997552i	$-0.522278\pi$
$809$	−113.147			−0.139861			−0.0699303	+	0.997552i	$0.522278\pi$
$810$		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
$813$	376.617			0.463244
$814$		0			0
$815$		−	432.214i		−	0.530324i
$816$		0			0
$817$			240.433i			0.294288i
$818$		0			0
$819$	178.191	−	351.249i	0.217571	−	0.428876i
$820$		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	$-0.693912\pi$
$823$	−941.852			−1.14441			−0.572207	+	0.820110i	$0.693912\pi$
$824$		0			0
$825$			30.3488i			0.0367864i
$826$		0			0
$827$	−966.978			−1.16926			−0.584630	−	0.811300i	$-0.698760\pi$
$827$	−966.978			−1.16926			−0.584630	+	0.811300i	$0.698760\pi$
$828$		0			0
$829$		−	1185.53i		−	1.43007i	−0.699088	−	0.715035i	$-0.746411\pi$
$829$		−	1185.53i		−	1.43007i	0.699088	−	0.715035i	$-0.253589\pi$
$830$		0			0
$831$			503.466i			0.605856i
$832$		0			0
$833$	928.410	−	679.546i	1.11454	−	0.815781i
$834$		0			0
$835$	232.971			0.279007
$836$		0			0
$837$	44.7351			0.0534469
$838$		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
$843$		−	32.6540i		−	0.0387355i
$844$		0			0
$845$			1080.79i			1.27905i
$846$		0			0
$847$	−736.080	−	373.418i	−0.869044	−	0.440871i
$848$		0			0
$849$	−695.823			−0.819580
$850$		0			0
$851$	1328.03			1.56055
$852$		0			0
$853$		−	175.006i		−	0.205165i	−0.994725	−	0.102582i	$-0.967289\pi$
$853$		−	175.006i		−	0.205165i	0.994725	−	0.102582i	$-0.0327105\pi$
$854$		0			0
$855$	409.103			0.478483
$856$		0			0
$857$			61.7471i			0.0720503i	0.999351	+	0.0360252i	$0.0114696\pi$
$857$			61.7471i			0.0720503i	−0.999351	+	0.0360252i	$0.988530\pi$
$858$		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
$861$	−447.588	−	227.064i	−0.519847	−	0.263721i
$862$		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
$867$			454.358i			0.524058i
$868$		0			0
$869$	−122.705			−0.141202
$870$		0			0
$871$		−	1143.53i		−	1.31289i
$872$		0			0
$873$			155.146i			0.177716i
$874$		0			0
$875$	281.470	−	554.833i	0.321680	−	0.634095i
$876$		0			0
$877$	454.353			0.518077			0.259038	−	0.965867i	$-0.416594\pi$
$877$	454.353			0.518077			0.259038	+	0.965867i	$0.416594\pi$
$878$		0			0
$879$	−486.521			−0.553494
$880$		0			0
$881$		−	143.493i		−	0.162875i	−0.996678	−	0.0814375i	$-0.974049\pi$
$881$		−	143.493i		−	0.162875i	0.996678	−	0.0814375i	$-0.0259511\pi$
$882$		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
$885$			998.527i			1.12828i
$886$		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
$891$	15.8162			0.0177511
$892$		0			0
$893$	−892.410			−0.999340
$894$		0			0
$895$		−	76.3122i		−	0.0852650i
$896$		0			0
$897$	608.382			0.678241
$898$		0			0
$899$			258.278i			0.287295i
$900$		0			0
$901$		−	869.462i		−	0.964996i
$902$		0			0
$903$	−57.1918	+	112.736i	−0.0633353	+	0.124846i
$904$		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
$909$		−	19.8297i		−	0.0218149i
$910$		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
$915$		−	171.320i		−	0.187235i
$916$		0			0
$917$	−38.5584	+	76.0063i	−0.0420485	+	0.0828858i
$918$		0			0
$919$	1086.45			1.18221			0.591107	−	0.806593i	$-0.298691\pi$
$919$	1086.45			1.18221			0.591107	+	0.806593i	$0.298691\pi$
$920$		0			0
$921$	263.823			0.286453
$922$		0			0
$923$		−	2074.54i		−	2.24760i
$924$		0			0
$925$	−707.029			−0.764356
$926$		0			0
$927$			527.613i			0.569161i
$928$		0			0
$929$			1550.75i			1.66927i	0.550806	+	0.834633i	$0.314320\pi$
$929$			1550.75i			1.66927i	−0.550806	+	0.834633i	$0.685680\pi$
$930$		0			0
$931$	911.793	−	667.383i	0.979370	−	0.716845i
$932$		0			0
$933$	490.441			0.525660
$934$		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
$939$	−84.1463			−0.0896127
$940$		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
$945$	191.823	+	97.3131i	0.202988	+	0.102977i
$946$		0			0
$947$	−92.5370			−0.0977160			−0.0488580	−	0.998806i	$-0.515558\pi$
$947$	−92.5370			−0.0977160			−0.0488580	+	0.998806i	$0.515558\pi$
$948$		0			0
$949$	−1064.53			−1.12174
$950$		0			0
$951$		−	1001.17i		−	1.05276i
$952$		0			0
$953$	−1361.29			−1.42843			−0.714215	−	0.699927i	$-0.753216\pi$
$953$	−1361.29			−1.42843			−0.714215	+	0.699927i	$0.753216\pi$
$954$		0			0
$955$			591.750i			0.619633i
$956$		0			0
$957$			91.3151i			0.0954180i
$958$		0			0
$959$	1034.81	+	524.964i	1.07905	+	0.547408i
$960$		0			0
$961$	886.881			0.922873
$962$		0			0
$963$	−138.728			−0.144058
$964$		0			0
$965$		−	466.651i		−	0.483577i
$966$		0			0
$967$	481.677			0.498115			0.249057	−	0.968489i	$-0.419879\pi$
$967$	481.677			0.498115			0.249057	+	0.968489i	$0.419879\pi$
$968$		0			0
$969$			937.829i			0.967832i
$970$		0			0
$971$		−	1427.34i		−	1.46997i	−0.678082	−	0.734987i	$-0.737188\pi$
$971$		−	1427.34i		−	1.46997i	0.678082	−	0.734987i	$-0.262812\pi$
$972$		0			0
$973$	698.352	−	1376.59i	0.717730	−	1.41479i
$974$		0			0
$975$	−323.897			−0.332202
$976$		0			0
$977$	−1356.26			−1.38819			−0.694095	−	0.719883i	$-0.744195\pi$
$977$	−1356.26			−1.38819			−0.694095	+	0.719883i	$0.744195\pi$
$978$		0			0
$979$			73.9698i			0.0755565i
$980$		0			0
$981$	−107.823			−0.109912
$982$		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
$987$	−418.441	−	212.277i	−0.423952	−	0.215073i
$988$		0			0
$989$	−195.265			−0.197437
$990$		0			0
$991$	1043.28			1.05275			0.526377	−	0.850251i	$-0.323550\pi$
$991$	1043.28			1.05275			0.526377	+	0.850251i	$0.323550\pi$
$992$		0			0
$993$			575.677i			0.579736i
$994$		0			0
$995$	−1005.48			−1.01054
$996$		0			0
$997$			820.328i			0.822796i	0.911456	+	0.411398i	$0.134959\pi$
$997$			820.328i			0.822796i	−0.911456	+	0.411398i	$0.865041\pi$
$998$		0			0
$999$			368.468i			0.368837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.f.769.1		4
4.3	odd	2		1344.3.f.e.769.3		4
7.6	odd	2	inner	1344.3.f.f.769.4		4
8.3	odd	2		336.3.f.c.97.2		4
8.5	even	2		42.3.c.a.13.2	yes	4
24.5	odd	2		126.3.c.b.55.3		4
24.11	even	2		1008.3.f.g.433.1		4
28.27	even	2		1344.3.f.e.769.2		4
40.13	odd	4		1050.3.h.a.349.6		8
40.29	even	2		1050.3.f.a.601.3		4
40.37	odd	4		1050.3.h.a.349.3		8
56.5	odd	6		294.3.g.c.31.2		4
56.13	odd	2		42.3.c.a.13.1	✓	4
56.27	even	2		336.3.f.c.97.3		4
56.37	even	6		294.3.g.b.31.2		4
56.45	odd	6		294.3.g.b.19.2		4
56.53	even	6		294.3.g.c.19.2		4
168.5	even	6		882.3.n.d.325.1		4
168.53	odd	6		882.3.n.d.19.1		4
168.83	odd	2		1008.3.f.g.433.4		4
168.101	even	6		882.3.n.a.19.1		4
168.125	even	2		126.3.c.b.55.4		4
168.149	odd	6		882.3.n.a.325.1		4
280.13	even	4		1050.3.h.a.349.7		8
280.69	odd	2		1050.3.f.a.601.4		4
280.237	even	4		1050.3.h.a.349.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.1	✓	4	56.13	odd	2
42.3.c.a.13.2	yes	4	8.5	even	2
126.3.c.b.55.3		4	24.5	odd	2
126.3.c.b.55.4		4	168.125	even	2
294.3.g.b.19.2		4	56.45	odd	6
294.3.g.b.31.2		4	56.37	even	6
294.3.g.c.19.2		4	56.53	even	6
294.3.g.c.31.2		4	56.5	odd	6
336.3.f.c.97.2		4	8.3	odd	2
336.3.f.c.97.3		4	56.27	even	2
882.3.n.a.19.1		4	168.101	even	6
882.3.n.a.325.1		4	168.149	odd	6
882.3.n.d.19.1		4	168.53	odd	6
882.3.n.d.325.1		4	168.5	even	6
1008.3.f.g.433.1		4	24.11	even	2
1008.3.f.g.433.4		4	168.83	odd	2
1050.3.f.a.601.3		4	40.29	even	2
1050.3.f.a.601.4		4	280.69	odd	2
1050.3.h.a.349.2		8	280.237	even	4
1050.3.h.a.349.3		8	40.37	odd	4
1050.3.h.a.349.6		8	40.13	odd	4
1050.3.h.a.349.7		8	280.13	even	4
1344.3.f.e.769.2		4	28.27	even	2
1344.3.f.e.769.3		4	4.3	odd	2
1344.3.f.f.769.1		4	1.1	even	1	trivial
1344.3.f.f.769.4		4	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -5.91359i q^{5} +(6.24264 + 3.16693i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -5.91359i q^{5} +(6.24264 + 3.16693i) q^{7} -3.00000 q^{9} +1.75736 q^{11} +18.7554i q^{13} -10.2426 q^{15} -23.4803i q^{17} -23.0600i q^{19} +(5.48528 - 10.8126i) q^{21} +18.7279 q^{23} -9.97056 q^{25} +5.19615i q^{27} -30.0000 q^{29} -8.60927i q^{31} -3.04384i q^{33} +(18.7279 - 36.9164i) q^{35} +70.9117 q^{37} +32.4853 q^{39} -41.3951i q^{41} -10.4264 q^{43} +17.7408i q^{45} -38.6995i q^{47} +(28.9411 + 39.5400i) q^{49} -40.6690 q^{51} +37.0294 q^{53} -10.3923i q^{55} -39.9411 q^{57} -97.4872i q^{59} +16.7262i q^{61} +(-18.7279 - 9.50079i) q^{63} +110.912 q^{65} -60.9706 q^{67} -32.4377i q^{69} -110.610 q^{71} +56.7585i q^{73} +17.2695i q^{75} +(10.9706 + 5.56543i) q^{77} -69.8234 q^{79} +9.00000 q^{81} +6.43583i q^{83} -138.853 q^{85} +51.9615i q^{87} +42.0915i q^{89} +(-59.3970 + 117.083i) q^{91} -14.9117 q^{93} -136.368 q^{95} -51.7153i q^{97} -5.27208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^7 - 12 * q^9	\( 4 q + 8 q^{7} - 12 q^{9} + 24 q^{11} - 24 q^{15} - 12 q^{21} + 24 q^{23} + 28 q^{25} - 120 q^{29} + 24 q^{35} + 80 q^{37} + 96 q^{39} + 128 q^{43} - 20 q^{49} + 24 q^{51} + 216 q^{53} - 24 q^{57} - 24 q^{63} + 240 q^{65} - 176 q^{67} - 120 q^{71} - 24 q^{77} + 128 q^{79} + 36 q^{81} - 216 q^{85} + 144 q^{93} - 240 q^{95} - 72 q^{99}+O(q^{100}) \) 4 * q + 8 * q^7 - 12 * q^9 + 24 * q^11 - 24 * q^15 - 12 * q^21 + 24 * q^23 + 28 * q^25 - 120 * q^29 + 24 * q^35 + 80 * q^37 + 96 * q^39 + 128 * q^43 - 20 * q^49 + 24 * q^51 + 216 * q^53 - 24 * q^57 - 24 * q^63 + 240 * q^65 - 176 * q^67 - 120 * q^71 - 24 * q^77 + 128 * q^79 + 36 * q^81 - 216 * q^85 + 144 * q^93 - 240 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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