LMFDB - Embedded newform 48.11.e.d.17.3

Newspace parameters

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$30.4971481283$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{85})$
Defining polynomial:	$ x^{4} - 2x^{3} - 37x^{2} + 38x + 531 $ x^4 - 2x^3 - 37x^2 + 38*x + 531
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.3
Root			$-4.10977 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.11.e.d.17.4

$q$-expansion

$f(q)$	$=$	$q+(200.269 - 137.627i) q^{3} +3630.47i q^{5} +23226.5 q^{7} +(21166.4 - 55125.0i) q^{9} +O(q^{10})$	\(q+(200.269 - 137.627i) q^{3} +3630.47i q^{5} +23226.5 q^{7} +(21166.4 - 55125.0i) q^{9} +62442.7i q^{11} -170161. q^{13} +(499653. + 727072. i) q^{15} +2.66626e6i q^{17} -766825. q^{19} +(4.65156e6 - 3.19661e6i) q^{21} +1.40327e6i q^{23} -3.41471e6 q^{25} +(-3.34774e6 - 1.39529e7i) q^{27} -4.83245e6i q^{29} +4.18297e7 q^{31} +(8.59382e6 + 1.25053e7i) q^{33} +8.43233e7i q^{35} +5.01619e7 q^{37} +(-3.40779e7 + 2.34188e7i) q^{39} +1.49239e8i q^{41} +1.98719e8 q^{43} +(2.00130e8 + 7.68440e7i) q^{45} -1.55059e8i q^{47} +2.56996e8 q^{49} +(3.66951e8 + 5.33970e8i) q^{51} -4.21541e7i q^{53} -2.26696e8 q^{55} +(-1.53571e8 + 1.05536e8i) q^{57} +2.92026e8i q^{59} -5.30727e8 q^{61} +(4.91622e8 - 1.28036e9i) q^{63} -6.17764e8i q^{65} -5.22093e8 q^{67} +(1.93128e8 + 2.81031e8i) q^{69} +5.71364e8i q^{71} +2.18588e9 q^{73} +(-6.83861e8 + 4.69958e8i) q^{75} +1.45033e9i q^{77} -1.96592e9 q^{79} +(-2.59075e9 - 2.33360e9i) q^{81} -2.18558e9i q^{83} -9.67979e9 q^{85} +(-6.65078e8 - 9.67791e8i) q^{87} +2.38742e8i q^{89} -3.95224e9 q^{91} +(8.37720e9 - 5.75692e9i) q^{93} -2.78394e9i q^{95} -8.84112e9 q^{97} +(3.44215e9 + 1.32169e9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9}+O(q^{10}) $ 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9	$ 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + 275240 q^{13} + 1180800 q^{15} + 1568728 q^{19} + 9628008 q^{21} - 33732380 q^{25} - 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} - 71014168 q^{37} - 217287240 q^{39} + 470688664 q^{43} + 312318720 q^{45} - 50058420 q^{49} + 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} - 1184038744 q^{61} + 905007096 q^{63} + 297365848 q^{67} + 596268288 q^{69} + 6534269000 q^{73} + 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} - 12880512000 q^{85} - 210268800 q^{87} - 8317232080 q^{91} + 31744468392 q^{93} - 39176355064 q^{97} + 2626912512 q^{99}+O(q^{100}) $ 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9 + 275240 * q^13 + 1180800 * q^15 + 1568728 * q^19 + 9628008 * q^21 - 33732380 * q^25 - 34619508 * q^27 + 21785848 * q^31 + 25974144 * q^33 - 71014168 * q^37 - 217287240 * q^39 + 470688664 * q^43 + 312318720 * q^45 - 50058420 * q^49 + 708576768 * q^51 - 2701359360 * q^55 - 1058753208 * q^57 - 1184038744 * q^61 + 905007096 * q^63 + 297365848 * q^67 + 596268288 * q^69 + 6534269000 * q^73 + 5150031180 * q^75 - 199282568 * q^79 + 1458964548 * q^81 - 12880512000 * q^85 - 210268800 * q^87 - 8317232080 * q^91 + 31744468392 * q^93 - 39176355064 * q^97 + 2626912512 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	200.269	−	137.627i	0.824153	−	0.566368i
$3$	200.269	−	137.627i	0.824153	−	0.566368i
$4$		0			0
$5$			3630.47i			1.16175i	0.813992	+	0.580876i	$0.197290\pi$
$5$			3630.47i			1.16175i	−0.813992	+	0.580876i	$0.802710\pi$
$6$		0			0
$7$	23226.5			1.38196			0.690978	−	0.722876i	$-0.257180\pi$
$7$	23226.5			1.38196			0.690978	+	0.722876i	$0.257180\pi$
$8$		0			0
$9$	21166.4	−	55125.0i	0.358455	−	0.933547i
$10$		0			0
$11$			62442.7i			0.387720i	0.981029	+	0.193860i	$0.0621007\pi$
$11$			62442.7i			0.387720i	−0.981029	+	0.193860i	$0.937899\pi$
$12$		0			0
$13$	−170161.			−0.458292			−0.229146	−	0.973392i	$-0.573593\pi$
$13$	−170161.			−0.458292			−0.229146	+	0.973392i	$0.573593\pi$
$14$		0			0
$15$	499653.	+	727072.i	0.657979	+	0.957460i
$16$		0			0
$17$			2.66626e6i			1.87784i	0.344139	+	0.938919i	$0.388171\pi$
$17$			2.66626e6i			1.87784i	−0.344139	+	0.938919i	$0.611829\pi$
$18$		0			0
$19$	−766825.			−0.309691			−0.154845	−	0.987939i	$-0.549488\pi$
$19$	−766825.			−0.309691			−0.154845	+	0.987939i	$0.549488\pi$
$20$		0			0
$21$	4.65156e6	−	3.19661e6i	1.13894	−	0.782695i
$22$		0			0
$23$			1.40327e6i			0.218022i	0.994041	+	0.109011i	$0.0347684\pi$
$23$			1.40327e6i			0.218022i	−0.994041	+	0.109011i	$0.965232\pi$
$24$		0			0
$25$	−3.41471e6			−0.349667
$26$		0			0
$27$	−3.34774e6	−	1.39529e7i	−0.233310	−	0.972402i
$28$		0			0
$29$		−	4.83245e6i		−	0.235601i	−0.993037	−	0.117801i	$-0.962416\pi$
$29$		−	4.83245e6i		−	0.235601i	0.993037	−	0.117801i	$-0.0375844\pi$
$30$		0			0
$31$	4.18297e7			1.46109			0.730544	−	0.682865i	$-0.239266\pi$
$31$	4.18297e7			1.46109			0.730544	+	0.682865i	$0.239266\pi$
$32$		0			0
$33$	8.59382e6	+	1.25053e7i	0.219592	+	0.319540i
$34$		0			0
$35$			8.43233e7i			1.60549i
$36$		0			0
$37$	5.01619e7			0.723378			0.361689	−	0.932299i	$-0.382200\pi$
$37$	5.01619e7			0.723378			0.361689	+	0.932299i	$0.382200\pi$
$38$		0			0
$39$	−3.40779e7	+	2.34188e7i	−0.377702	+	0.259562i
$40$		0			0
$41$			1.49239e8i			1.28814i	0.764967	+	0.644069i	$0.222755\pi$
$41$			1.49239e8i			1.28814i	−0.764967	+	0.644069i	$0.777245\pi$
$42$		0			0
$43$	1.98719e8			1.35175			0.675876	−	0.737015i	$-0.263765\pi$
$43$	1.98719e8			1.35175			0.675876	+	0.737015i	$0.263765\pi$
$44$		0			0
$45$	2.00130e8	+	7.68440e7i	1.08455	+	0.416435i
$46$		0			0
$47$		−	1.55059e8i		−	0.676093i	−0.941129	−	0.338047i	$-0.890234\pi$
$47$		−	1.55059e8i		−	0.676093i	0.941129	−	0.338047i	$-0.109766\pi$
$48$		0			0
$49$	2.56996e8			0.909802
$50$		0			0
$51$	3.66951e8	+	5.33970e8i	1.06355	+	1.54762i
$52$		0			0
$53$		−	4.21541e7i		−	0.100800i	−0.998729	−	0.0503999i	$-0.983950\pi$
$53$		−	4.21541e7i		−	0.100800i	0.998729	−	0.0503999i	$-0.0160496\pi$
$54$		0			0
$55$	−2.26696e8			−0.450434
$56$		0			0
$57$	−1.53571e8	+	1.05536e8i	−0.255233	+	0.175399i
$58$		0			0
$59$			2.92026e8i			0.408471i	0.978922	+	0.204236i	$0.0654708\pi$
$59$			2.92026e8i			0.408471i	−0.978922	+	0.204236i	$0.934529\pi$
$60$		0			0
$61$	−5.30727e8			−0.628379			−0.314190	−	0.949360i	$-0.601733\pi$
$61$	−5.30727e8			−0.628379			−0.314190	+	0.949360i	$0.601733\pi$
$62$		0			0
$63$	4.91622e8	−	1.28036e9i	0.495369	−	1.29012i
$64$		0			0
$65$		−	6.17764e8i		−	0.532421i
$66$		0			0
$67$	−5.22093e8			−0.386700			−0.193350	−	0.981130i	$-0.561935\pi$
$67$	−5.22093e8			−0.386700			−0.193350	+	0.981130i	$0.561935\pi$
$68$		0			0
$69$	1.93128e8	+	2.81031e8i	0.123481	+	0.179684i
$70$		0			0
$71$			5.71364e8i			0.316681i	0.987385	+	0.158340i	$0.0506143\pi$
$71$			5.71364e8i			0.316681i	−0.987385	+	0.158340i	$0.949386\pi$
$72$		0			0
$73$	2.18588e9			1.05441			0.527207	−	0.849737i	$-0.323239\pi$
$73$	2.18588e9			1.05441			0.527207	+	0.849737i	$0.323239\pi$
$74$		0			0
$75$	−6.83861e8	+	4.69958e8i	−0.288179	+	0.198040i
$76$		0			0
$77$			1.45033e9i			0.535812i
$78$		0			0
$79$	−1.96592e9			−0.638897			−0.319449	−	0.947604i	$-0.603498\pi$
$79$	−1.96592e9			−0.638897			−0.319449	+	0.947604i	$0.603498\pi$
$80$		0			0
$81$	−2.59075e9	−	2.33360e9i	−0.743020	−	0.669269i
$82$		0			0
$83$		−	2.18558e9i		−	0.554850i	−0.960747	−	0.277425i	$-0.910519\pi$
$83$		−	2.18558e9i		−	0.554850i	0.960747	−	0.277425i	$-0.0894810\pi$
$84$		0			0
$85$	−9.67979e9			−2.18158
$86$		0			0
$87$	−6.65078e8	−	9.67791e8i	−0.133437	−	0.194171i
$88$		0			0
$89$			2.38742e8i			0.0427542i	0.999771	+	0.0213771i	$0.00680506\pi$
$89$			2.38742e8i			0.0427542i	−0.999771	+	0.0213771i	$0.993195\pi$
$90$		0			0
$91$	−3.95224e9			−0.633339
$92$		0			0
$93$	8.37720e9	−	5.75692e9i	1.20416	−	0.827514i
$94$		0			0
$95$		−	2.78394e9i		−	0.359784i
$96$		0			0
$97$	−8.84112e9			−1.02955			−0.514776	−	0.857324i	$-0.672125\pi$
$97$	−8.84112e9			−1.02955			−0.514776	+	0.857324i	$0.672125\pi$
$98$		0			0
$99$	3.44215e9	+	1.32169e9i	0.361955	+	0.138980i
$100$		0			0
$101$		−	1.67928e10i		−	1.59778i	−0.601477	−	0.798890i	$-0.705421\pi$
$101$		−	1.67928e10i		−	1.59778i	0.601477	−	0.798890i	$-0.294579\pi$
$102$		0			0
$103$	−8.36865e9			−0.721887			−0.360944	−	0.932588i	$-0.617545\pi$
$103$	−8.36865e9			−0.721887			−0.360944	+	0.932588i	$0.617545\pi$
$104$		0			0
$105$	1.16052e10	+	1.68873e10i	0.909297	+	1.32317i
$106$		0			0
$107$		−	1.43555e10i		−	1.02353i	−0.859126	−	0.511764i	$-0.828992\pi$
$107$		−	1.43555e10i		−	1.02353i	0.859126	−	0.511764i	$-0.171008\pi$
$108$		0			0
$109$	−4.72564e9			−0.307134			−0.153567	−	0.988138i	$-0.549076\pi$
$109$	−4.72564e9			−0.307134			−0.153567	+	0.988138i	$0.549076\pi$
$110$		0			0
$111$	1.00459e10	−	6.90365e9i	0.596174	−	0.409698i
$112$		0			0
$113$			1.32158e10i			0.717303i	0.933472	+	0.358651i	$0.116763\pi$
$113$			1.32158e10i			0.717303i	−0.933472	+	0.358651i	$0.883237\pi$
$114$		0			0
$115$	−5.09452e9			−0.253288
$116$		0			0
$117$	−3.60169e9	+	9.38011e9i	−0.164277	+	0.427837i
$118$		0			0
$119$			6.19280e10i			2.59509i
$120$		0			0
$121$	2.20383e10			0.849673
$122$		0			0
$123$	2.05393e10	+	2.98879e10i	0.729560	+	1.06162i
$124$		0			0
$125$			2.30568e10i			0.755526i
$126$		0			0
$127$	3.17814e10			0.961953			0.480976	−	0.876733i	$-0.340282\pi$
$127$	3.17814e10			0.961953			0.480976	+	0.876733i	$0.340282\pi$
$128$		0			0
$129$	3.97973e10	−	2.73492e10i	1.11405	−	0.765589i
$130$		0			0
$131$		−	5.01094e10i		−	1.29886i	−0.760421	−	0.649430i	$-0.775007\pi$
$131$		−	5.01094e10i		−	1.29886i	0.760421	−	0.649430i	$-0.224993\pi$
$132$		0			0
$133$	−1.78107e10			−0.427979
$134$		0			0
$135$	5.06557e10	−	1.21539e10i	1.12969	−	0.271048i
$136$		0			0
$137$			2.74744e10i			0.569279i	0.958635	+	0.284639i	$0.0918738\pi$
$137$			2.74744e10i			0.569279i	−0.958635	+	0.284639i	$0.908126\pi$
$138$		0			0
$139$	−6.32488e10			−1.21893			−0.609464	−	0.792814i	$-0.708615\pi$
$139$	−6.32488e10			−1.21893			−0.609464	+	0.792814i	$0.708615\pi$
$140$		0			0
$141$	−2.13403e10	−	3.10534e10i	−0.382917	−	0.557204i
$142$		0			0
$143$		−	1.06253e10i		−	0.177689i
$144$		0			0
$145$	1.75441e10			0.273710
$146$		0			0
$147$	5.14684e10	−	3.53697e10i	0.749815	−	0.515282i
$148$		0			0
$149$		−	3.38479e10i		−	0.460893i	−0.973085	−	0.230446i	$-0.925981\pi$
$149$		−	3.38479e10i		−	0.460893i	0.973085	−	0.230446i	$-0.0740186\pi$
$150$		0			0
$151$	−1.22386e11			−1.55901			−0.779503	−	0.626399i	$-0.784528\pi$
$151$	−1.22386e11			−1.55901			−0.779503	+	0.626399i	$0.784528\pi$
$152$		0			0
$153$	1.46978e11	+	5.64351e10i	1.75305	+	0.673120i
$154$		0			0
$155$			1.51862e11i			1.69742i
$156$		0			0
$157$	−1.58971e11			−1.66656			−0.833279	−	0.552853i	$-0.813539\pi$
$157$	−1.58971e11			−1.66656			−0.833279	+	0.552853i	$0.813539\pi$
$158$		0			0
$159$	−5.80155e9	−	8.44215e9i	−0.0570898	−	0.0830745i
$160$		0			0
$161$			3.25930e10i			0.301297i
$162$		0			0
$163$	9.20831e10			0.800280			0.400140	−	0.916454i	$-0.368962\pi$
$163$	9.20831e10			0.800280			0.400140	+	0.916454i	$0.368962\pi$
$164$		0			0
$165$	−4.54003e10	+	3.11996e10i	−0.371226	+	0.255111i
$166$		0			0
$167$		−	2.12416e11i		−	1.63533i	−0.575697	−	0.817663i	$-0.695269\pi$
$167$		−	2.12416e11i		−	1.63533i	0.575697	−	0.817663i	$-0.304731\pi$
$168$		0			0
$169$	−1.08904e11			−0.789968
$170$		0			0
$171$	−1.62309e10	+	4.22713e10i	−0.111010	+	0.289111i
$172$		0			0
$173$		−	3.04132e11i		−	1.96260i	−0.192482	−	0.981300i	$-0.561654\pi$
$173$		−	3.04132e11i		−	1.96260i	0.192482	−	0.981300i	$-0.438346\pi$
$174$		0			0
$175$	−7.93119e10			−0.483224
$176$		0			0
$177$	4.01908e10	+	5.84837e10i	0.231345	+	0.336642i
$178$		0			0
$179$			7.07114e9i			0.0384791i	0.999815	+	0.0192395i	$0.00612451\pi$
$179$			7.07114e9i			0.0384791i	−0.999815	+	0.0192395i	$0.993875\pi$
$180$		0			0
$181$	−1.27309e11			−0.655337			−0.327669	−	0.944793i	$-0.606263\pi$
$181$	−1.27309e11			−0.655337			−0.327669	+	0.944793i	$0.606263\pi$
$182$		0			0
$183$	−1.06288e11	+	7.30425e10i	−0.517880	+	0.355894i
$184$		0			0
$185$			1.82111e11i			0.840386i
$186$		0			0
$187$	−1.66488e11			−0.728075
$188$		0			0
$189$	−7.77563e10	−	3.24078e11i	−0.322424	−	1.34382i
$190$		0			0
$191$		−	2.06460e11i		−	0.812209i	−0.913827	−	0.406105i	$-0.866887\pi$
$191$		−	2.06460e11i		−	0.812209i	0.913827	−	0.406105i	$-0.133113\pi$
$192$		0			0
$193$	3.11606e11			1.16364			0.581821	−	0.813317i	$-0.302340\pi$
$193$	3.11606e11			1.16364			0.581821	+	0.813317i	$0.302340\pi$
$194$		0			0
$195$	−8.50212e10	−	1.23719e11i	−0.301546	−	0.438796i
$196$		0			0
$197$			1.51694e11i			0.511255i	0.966775	+	0.255628i	$0.0822821\pi$
$197$			1.51694e11i			0.511255i	−0.966775	+	0.255628i	$0.917718\pi$
$198$		0			0
$199$	1.11692e11			0.357895			0.178947	−	0.983859i	$-0.442731\pi$
$199$	1.11692e11			0.357895			0.178947	+	0.983859i	$0.442731\pi$
$200$		0			0
$201$	−1.04559e11	+	7.18543e10i	−0.318700	+	0.219014i
$202$		0			0
$203$		−	1.12241e11i		−	0.325591i
$204$		0			0
$205$	−5.41807e11			−1.49650
$206$		0			0
$207$	7.73551e10	+	2.97021e10i	0.203534	+	0.0781512i
$208$		0			0
$209$		−	4.78826e10i		−	0.120073i
$210$		0			0
$211$	−1.53066e11			−0.365987			−0.182994	−	0.983114i	$-0.558579\pi$
$211$	−1.53066e11			−0.365987			−0.182994	+	0.983114i	$0.558579\pi$
$212$		0			0
$213$	7.86354e10	+	1.14427e11i	0.179358	+	0.260993i
$214$		0			0
$215$			7.21444e11i			1.57040i
$216$		0			0
$217$	9.71560e11			2.01916
$218$		0			0
$219$	4.37763e11	−	3.00836e11i	0.868998	−	0.597186i
$220$		0			0
$221$		−	4.53693e11i		−	0.860598i
$222$		0			0
$223$	6.05759e11			1.09844			0.549219	−	0.835678i	$-0.314925\pi$
$223$	6.05759e11			1.09844			0.549219	+	0.835678i	$0.314925\pi$
$224$		0			0
$225$	−7.22772e10	+	1.88236e11i	−0.125340	+	0.326430i
$226$		0			0
$227$			6.70588e11i			1.11257i	0.830992	+	0.556284i	$0.187773\pi$
$227$			6.70588e11i			1.11257i	−0.830992	+	0.556284i	$0.812227\pi$
$228$		0			0
$229$	3.53356e11			0.561094			0.280547	−	0.959840i	$-0.409484\pi$
$229$	3.53356e11			0.561094			0.280547	+	0.959840i	$0.409484\pi$
$230$		0			0
$231$	1.99605e11	+	2.90456e11i	0.303467	+	0.441591i
$232$		0			0
$233$			1.12390e12i			1.63662i	0.574779	+	0.818309i	$0.305088\pi$
$233$			1.12390e12i			1.63662i	−0.574779	+	0.818309i	$0.694912\pi$
$234$		0			0
$235$	5.62936e11			0.785452
$236$		0			0
$237$	−3.93714e11	+	2.70565e11i	−0.526549	+	0.361851i
$238$		0			0
$239$		−	2.64603e11i		−	0.339317i	−0.985503	−	0.169658i	$-0.945734\pi$
$239$		−	2.64603e11i		−	0.339317i	0.985503	−	0.169658i	$-0.0542665\pi$
$240$		0			0
$241$	8.86768e11			1.09075			0.545375	−	0.838192i	$-0.316387\pi$
$241$	8.86768e11			1.09075			0.545375	+	0.838192i	$0.316387\pi$
$242$		0			0
$243$	−8.40014e11	−	1.10789e11i	−0.991414	−	0.130757i
$244$		0			0
$245$			9.33019e11i			1.05696i
$246$		0			0
$247$	1.30483e11			0.141929
$248$		0			0
$249$	−3.00795e11	−	4.37704e11i	−0.314249	−	0.457281i
$250$		0			0
$251$		−	8.59494e11i		−	0.862729i	−0.902178	−	0.431364i	$-0.858032\pi$
$251$		−	8.59494e11i		−	0.862729i	0.902178	−	0.431364i	$-0.141968\pi$
$252$		0			0
$253$	−8.76237e10			−0.0845316
$254$		0			0
$255$	−1.93856e12	+	1.33220e12i	−1.79796	+	1.23558i
$256$		0			0
$257$		−	2.17900e12i		−	1.94353i	−0.235954	−	0.971764i	$-0.575821\pi$
$257$		−	2.17900e12i		−	1.94353i	0.235954	−	0.971764i	$-0.424179\pi$
$258$		0			0
$259$	1.16509e12			0.999677
$260$		0			0
$261$	−2.66389e11	−	1.02286e11i	−0.219945	−	0.0844524i
$262$		0			0
$263$		−	1.37942e12i		−	1.09627i	−0.836391	−	0.548134i	$-0.815339\pi$
$263$		−	1.37942e12i		−	1.09627i	0.836391	−	0.548134i	$-0.184661\pi$
$264$		0			0
$265$	1.53039e11			0.117104
$266$		0			0
$267$	3.28574e10	+	4.78126e10i	0.0242146	+	0.0352360i
$268$		0			0
$269$		−	1.18129e12i		−	0.838680i	−0.907829	−	0.419340i	$-0.862262\pi$
$269$		−	1.18129e12i		−	0.838680i	0.907829	−	0.419340i	$-0.137738\pi$
$270$		0			0
$271$	1.42251e12			0.973213			0.486606	−	0.873621i	$-0.338235\pi$
$271$	1.42251e12			0.973213			0.486606	+	0.873621i	$0.338235\pi$
$272$		0			0
$273$	−7.91511e11	+	5.43937e11i	−0.521968	+	0.358703i
$274$		0			0
$275$		−	2.13224e11i		−	0.135573i
$276$		0			0
$277$	5.29960e11			0.324971			0.162485	−	0.986711i	$-0.448049\pi$
$277$	5.29960e11			0.324971			0.162485	+	0.986711i	$0.448049\pi$
$278$		0			0
$279$	8.85385e11	−	2.30587e12i	0.523734	−	1.36400i
$280$		0			0
$281$			2.28441e12i			1.30389i	0.758265	+	0.651946i	$0.226047\pi$
$281$			2.28441e12i			1.30389i	−0.758265	+	0.651946i	$0.773953\pi$
$282$		0			0
$283$	−2.31274e12			−1.27408			−0.637038	−	0.770833i	$-0.719840\pi$
$283$	−2.31274e12			−1.27408			−0.637038	+	0.770833i	$0.719840\pi$
$284$		0			0
$285$	−3.83146e11	−	5.57537e11i	−0.203770	−	0.296517i
$286$		0			0
$287$			3.46630e12i			1.78015i
$288$		0			0
$289$	−5.09295e12			−2.52627
$290$		0			0
$291$	−1.77060e12	+	1.21678e12i	−0.848509	+	0.583106i
$292$		0			0
$293$		−	1.05989e12i		−	0.490822i	−0.969419	−	0.245411i	$-0.921077\pi$
$293$		−	1.05989e12i		−	0.490822i	0.969419	−	0.245411i	$-0.0789229\pi$
$294$		0			0
$295$	−1.06019e12			−0.474542
$296$		0			0
$297$	8.71257e11	−	2.09042e11i	0.377020	−	0.0904588i
$298$		0			0
$299$		−	2.38781e11i		−	0.0999179i
$300$		0			0
$301$	4.61555e12			1.86806
$302$		0			0
$303$	−2.31115e12	−	3.36308e12i	−0.904931	−	1.31681i
$304$		0			0
$305$		−	1.92679e12i		−	0.730020i
$306$		0			0
$307$	−6.89209e11			−0.252731			−0.126366	−	0.991984i	$-0.540331\pi$
$307$	−6.89209e11			−0.252731			−0.126366	+	0.991984i	$0.540331\pi$
$308$		0			0
$309$	−1.67598e12	+	1.15176e12i	−0.594945	+	0.408854i
$310$		0			0
$311$		−	1.95674e12i		−	0.672559i	−0.941762	−	0.336280i	$-0.890831\pi$
$311$		−	1.95674e12i		−	0.672559i	0.941762	−	0.336280i	$-0.109169\pi$
$312$		0			0
$313$	−2.48821e11			−0.0828256			−0.0414128	−	0.999142i	$-0.513186\pi$
$313$	−2.48821e11			−0.0828256			−0.0414128	+	0.999142i	$0.513186\pi$
$314$		0			0
$315$	4.64832e12	+	1.78482e12i	1.49880	+	0.575495i
$316$		0			0
$317$			4.63675e12i			1.44850i	0.689539	+	0.724249i	$0.257813\pi$
$317$			4.63675e12i			1.44850i	−0.689539	+	0.724249i	$0.742187\pi$
$318$		0			0
$319$	3.01751e11			0.0913473
$320$		0			0
$321$	−1.97571e12	−	2.87497e12i	−0.579694	−	0.843544i
$322$		0			0
$323$		−	2.04456e12i		−	0.581549i
$324$		0			0
$325$	5.81050e11			0.160249
$326$		0			0
$327$	−9.46400e11	+	6.50378e11i	−0.253125	+	0.173951i
$328$		0			0
$329$		−	3.60147e12i		−	0.934331i
$330$		0			0
$331$	1.41443e12			0.355994			0.177997	−	0.984031i	$-0.443038\pi$
$331$	1.41443e12			0.355994			0.177997	+	0.984031i	$0.443038\pi$
$332$		0			0
$333$	1.06175e12	−	2.76518e12i	0.259298	−	0.675308i
$334$		0			0
$335$		−	1.89545e12i		−	0.449249i
$336$		0			0
$337$	8.67755e11			0.199640			0.0998201	−	0.995006i	$-0.468173\pi$
$337$	8.67755e11			0.199640			0.0998201	+	0.995006i	$0.468173\pi$
$338$		0			0
$339$	1.81886e12	+	2.64672e12i	0.406257	+	0.591167i
$340$		0			0
$341$			2.61196e12i			0.566493i
$342$		0			0
$343$	−5.91785e11			−0.124650
$344$		0			0
$345$	−1.02028e12	+	7.01146e11i	−0.208748	+	0.143454i
$346$		0			0
$347$			5.20712e12i			1.03502i	0.855676	+	0.517512i	$0.173142\pi$
$347$			5.20712e12i			1.03502i	−0.855676	+	0.517512i	$0.826858\pi$
$348$		0			0
$349$	1.11953e12			0.216226			0.108113	−	0.994139i	$-0.465519\pi$
$349$	1.11953e12			0.216226			0.108113	+	0.994139i	$0.465519\pi$
$350$		0			0
$351$	5.69653e11	+	2.37424e12i	0.106924	+	0.445644i
$352$		0			0
$353$		−	3.73952e12i		−	0.682248i	−0.940018	−	0.341124i	$-0.889192\pi$
$353$		−	3.73952e12i		−	0.682248i	0.940018	−	0.341124i	$-0.110808\pi$
$354$		0			0
$355$	−2.07432e12			−0.367904
$356$		0			0
$357$	8.52299e12	+	1.24023e13i	1.46977	+	2.13875i
$358$		0			0
$359$			3.54489e12i			0.594471i	0.954804	+	0.297235i	$0.0960646\pi$
$359$			3.54489e12i			0.594471i	−0.954804	+	0.297235i	$0.903935\pi$
$360$		0			0
$361$	−5.54305e12			−0.904092
$362$		0			0
$363$	4.41360e12	−	3.03308e12i	0.700260	−	0.481228i
$364$		0			0
$365$			7.93577e12i			1.22497i
$366$		0			0
$367$	1.01161e13			1.51944			0.759722	−	0.650248i	$-0.225335\pi$
$367$	1.01161e13			1.51944			0.759722	+	0.650248i	$0.225335\pi$
$368$		0			0
$369$	8.22679e12	+	3.15885e12i	1.20254	+	0.461739i
$370$		0			0
$371$		−	9.79093e11i		−	0.139301i
$372$		0			0
$373$	3.81055e12			0.527769			0.263885	−	0.964554i	$-0.414996\pi$
$373$	3.81055e12			0.527769			0.263885	+	0.964554i	$0.414996\pi$
$374$		0			0
$375$	3.17325e12	+	4.61757e12i	0.427906	+	0.622669i
$376$		0			0
$377$			8.22293e11i			0.107974i
$378$		0			0
$379$	−1.11052e13			−1.42014			−0.710069	−	0.704132i	$-0.751336\pi$
$379$	−1.11052e13			−1.42014			−0.710069	+	0.704132i	$0.751336\pi$
$380$		0			0
$381$	6.36482e12	−	4.37399e12i	0.792796	−	0.544819i
$382$		0			0
$383$			4.29169e12i			0.520756i	0.965507	+	0.260378i	$0.0838472\pi$
$383$			4.29169e12i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$384$		0			0
$385$	−5.26537e12			−0.622480
$386$		0			0
$387$	4.20617e12	−	1.09544e13i	0.484542	−	1.26192i
$388$		0			0
$389$		−	1.50554e13i		−	1.69023i	−0.534588	−	0.845113i	$-0.679533\pi$
$389$		−	1.50554e13i		−	1.69023i	0.534588	−	0.845113i	$-0.320467\pi$
$390$		0			0
$391$	−3.74148e12			−0.409411
$392$		0			0
$393$	−6.89642e12	−	1.00354e13i	−0.735633	−	1.07046i
$394$		0			0
$395$		−	7.13723e12i		−	0.742240i
$396$		0			0
$397$	−1.11669e13			−1.13235			−0.566174	−	0.824286i	$-0.691577\pi$
$397$	−1.11669e13			−1.13235			−0.566174	+	0.824286i	$0.691577\pi$
$398$		0			0
$399$	−3.56693e12	+	2.45124e12i	−0.352720	+	0.242394i
$400$		0			0
$401$			1.15685e13i			1.11572i	0.829935	+	0.557861i	$0.188378\pi$
$401$			1.15685e13i			1.11572i	−0.829935	+	0.557861i	$0.811622\pi$
$402$		0			0
$403$	−7.11777e12			−0.669605
$404$		0			0
$405$	8.47206e12	−	9.40565e12i	0.777524	−	0.863205i
$406$		0			0
$407$			3.13224e12i			0.280468i
$408$		0			0
$409$	−1.07411e13			−0.938498			−0.469249	−	0.883066i	$-0.655475\pi$
$409$	−1.07411e13			−0.938498			−0.469249	+	0.883066i	$0.655475\pi$
$410$		0			0
$411$	3.78123e12	+	5.50227e12i	0.322421	+	0.469172i
$412$		0			0
$413$			6.78275e12i			0.564489i
$414$		0			0
$415$	7.93468e12			0.644598
$416$		0			0
$417$	−1.26668e13	+	8.70477e12i	−1.00458	+	0.690362i
$418$		0			0
$419$		−	8.81717e12i		−	0.682746i	−0.939928	−	0.341373i	$-0.889108\pi$
$419$		−	8.81717e12i		−	0.682746i	0.939928	−	0.341373i	$-0.110892\pi$
$420$		0			0
$421$	−5.69741e12			−0.430792			−0.215396	−	0.976527i	$-0.569104\pi$
$421$	−5.69741e12			−0.430792			−0.215396	+	0.976527i	$0.569104\pi$
$422$		0			0
$423$	−8.54761e12	−	3.28203e12i	−0.631165	−	0.242349i
$424$		0			0
$425$		−	9.10451e12i		−	0.656617i
$426$		0			0
$427$	−1.23269e13			−0.868392
$428$		0			0
$429$	−1.46233e12	−	2.12792e12i	−0.100637	−	0.146443i
$430$		0			0
$431$		−	1.80323e13i		−	1.21245i	−0.795293	−	0.606226i	$-0.792683\pi$
$431$		−	1.80323e13i		−	1.21245i	0.795293	−	0.606226i	$-0.207317\pi$
$432$		0			0
$433$	2.06339e13			1.35563			0.677816	−	0.735232i	$-0.262927\pi$
$433$	2.06339e13			1.35563			0.677816	+	0.735232i	$0.262927\pi$
$434$		0			0
$435$	3.51354e12	−	2.41455e12i	0.225579	−	0.155021i
$436$		0			0
$437$		−	1.07606e12i		−	0.0675195i
$438$		0			0
$439$	−1.95313e13			−1.19787			−0.598935	−	0.800798i	$-0.704409\pi$
$439$	−1.95313e13			−1.19787			−0.598935	+	0.800798i	$0.704409\pi$
$440$		0			0
$441$	5.43969e12	−	1.41669e13i	0.326123	−	0.849343i
$442$		0			0
$443$			2.97469e13i			1.74351i	0.489946	+	0.871753i	$0.337016\pi$
$443$			2.97469e13i			1.74351i	−0.489946	+	0.871753i	$0.662984\pi$
$444$		0			0
$445$	−8.66747e11			−0.0496698
$446$		0			0
$447$	−4.65839e12	−	6.77868e12i	−0.261035	−	0.379846i
$448$		0			0
$449$		−	1.37505e13i		−	0.753509i	−0.926313	−	0.376754i	$-0.877040\pi$
$449$		−	1.37505e13i		−	0.753509i	0.926313	−	0.376754i	$-0.122960\pi$
$450$		0			0
$451$	−9.31887e12			−0.499437
$452$		0			0
$453$	−2.45102e13	+	1.68437e13i	−1.28486	+	0.882971i
$454$		0			0
$455$		−	1.43485e13i		−	0.735783i
$456$		0			0
$457$	4.10224e11			0.0205798			0.0102899	−	0.999947i	$-0.496725\pi$
$457$	4.10224e11			0.0205798			0.0102899	+	0.999947i	$0.496725\pi$
$458$		0			0
$459$	3.72021e13	−	8.92594e12i	1.82601	−	0.438118i
$460$		0			0
$461$		−	2.10914e11i		−	0.0101298i	−0.999987	−	0.00506490i	$-0.998388\pi$
$461$		−	2.10914e11i		−	0.0101298i	0.999987	−	0.00506490i	$-0.00161221\pi$
$462$		0			0
$463$	1.96738e13			0.924660			0.462330	−	0.886708i	$-0.347013\pi$
$463$	1.96738e13			0.924660			0.462330	+	0.886708i	$0.347013\pi$
$464$		0			0
$465$	2.09003e13	+	3.04132e13i	0.961365	+	1.39893i
$466$		0			0
$467$			1.21902e13i			0.548815i	0.961614	+	0.274407i	$0.0884817\pi$
$467$			1.21902e13i			0.548815i	−0.961614	+	0.274407i	$0.911518\pi$
$468$		0			0
$469$	−1.21264e13			−0.534402
$470$		0			0
$471$	−3.18370e13	+	2.18788e13i	−1.37350	+	0.943885i
$472$		0			0
$473$			1.24085e13i			0.524101i
$474$		0			0
$475$	2.61849e12			0.108289
$476$		0			0
$477$	−2.32374e12	−	8.92250e11i	−0.0941014	−	0.0361322i
$478$		0			0
$479$		−	5.00886e12i		−	0.198637i	−0.995056	−	0.0993187i	$-0.968334\pi$
$479$		−	5.00886e12i		−	0.198637i	0.995056	−	0.0993187i	$-0.0316663\pi$
$480$		0			0
$481$	−8.53558e12			−0.331518
$482$		0			0
$483$	4.48569e12	+	6.52737e12i	0.170645	+	0.248315i
$484$		0			0
$485$		−	3.20975e13i		−	1.19608i
$486$		0			0
$487$	4.17000e13			1.52227			0.761134	−	0.648595i	$-0.224643\pi$
$487$	4.17000e13			1.52227			0.761134	+	0.648595i	$0.224643\pi$
$488$		0			0
$489$	1.84414e13	−	1.26732e13i	0.659553	−	0.453253i
$490$		0			0
$491$			2.10340e12i			0.0737081i	0.999321	+	0.0368540i	$0.0117337\pi$
$491$			2.10340e12i			0.0737081i	−0.999321	+	0.0368540i	$0.988266\pi$
$492$		0			0
$493$	1.28846e13			0.442421
$494$		0			0
$495$	−4.79835e12	+	1.24966e13i	−0.161460	+	0.420501i
$496$		0			0
$497$			1.32708e13i			0.437639i
$498$		0			0
$499$	−2.74482e13			−0.887179			−0.443589	−	0.896230i	$-0.646295\pi$
$499$	−2.74482e13			−0.887179			−0.443589	+	0.896230i	$0.646295\pi$
$500$		0			0
$501$	−2.92342e13	−	4.25403e13i	−0.926197	−	1.34776i
$502$		0			0
$503$		−	2.26985e13i		−	0.704949i	−0.935822	−	0.352474i	$-0.885340\pi$
$503$		−	2.26985e13i		−	0.704949i	0.935822	−	0.352474i	$-0.114660\pi$
$504$		0			0
$505$	6.09659e13			1.85622
$506$		0			0
$507$	−2.18101e13	+	1.49882e13i	−0.651055	+	0.447413i
$508$		0			0
$509$			4.21460e13i			1.23358i	0.787127	+	0.616791i	$0.211567\pi$
$509$			4.21460e13i			1.23358i	−0.787127	+	0.616791i	$0.788433\pi$
$510$		0			0
$511$	5.07703e13			1.45715
$512$		0			0
$513$	2.56713e12	+	1.06994e13i	0.0722539	+	0.301144i
$514$		0			0
$515$		−	3.03822e13i		−	0.838653i
$516$		0			0
$517$	9.68227e12			0.262135
$518$		0			0
$519$	−4.18569e13	−	6.09083e13i	−1.11155	−	1.61748i
$520$		0			0
$521$			6.54420e13i			1.70478i	0.522907	+	0.852390i	$0.324848\pi$
$521$			6.54420e13i			1.70478i	−0.522907	+	0.852390i	$0.675152\pi$
$522$		0			0
$523$	6.14531e13			1.57049			0.785245	−	0.619185i	$-0.212537\pi$
$523$	6.14531e13			1.57049			0.785245	+	0.619185i	$0.212537\pi$
$524$		0			0
$525$	−1.58837e13	+	1.09155e13i	−0.398250	+	0.273682i
$526$		0			0
$527$			1.11529e14i			2.74369i
$528$		0			0
$529$	3.94574e13			0.952466
$530$		0			0
$531$	1.60979e13	+	6.18114e12i	0.381327	+	0.146418i
$532$		0			0
$533$		−	2.53946e13i		−	0.590343i
$534$		0			0
$535$	5.21173e13			1.18909
$536$		0			0
$537$	9.73183e11	+	1.41613e12i	0.0217933	+	0.0317126i
$538$		0			0
$539$			1.60475e13i			0.352748i
$540$		0			0
$541$	1.50207e13			0.324118			0.162059	−	0.986781i	$-0.448187\pi$
$541$	1.50207e13			0.324118			0.162059	+	0.986781i	$0.448187\pi$
$542$		0			0
$543$	−2.54960e13	+	1.75211e13i	−0.540098	+	0.371162i
$544$		0			0
$545$		−	1.71563e13i		−	0.356814i
$546$		0			0
$547$	−7.15097e13			−1.46025			−0.730127	−	0.683312i	$-0.760539\pi$
$547$	−7.15097e13			−1.46025			−0.730127	+	0.683312i	$0.760539\pi$
$548$		0			0
$549$	−1.12336e13	+	2.92563e13i	−0.225245	+	0.586621i
$550$		0			0
$551$			3.70565e12i			0.0729636i
$552$		0			0
$553$	−4.56616e13			−0.882928
$554$		0			0
$555$	2.50635e13	+	3.64713e13i	0.475967	+	0.692606i
$556$		0			0
$557$		−	7.34107e13i		−	1.36925i	−0.728895	−	0.684626i	$-0.759966\pi$
$557$		−	7.34107e13i		−	1.36925i	0.728895	−	0.684626i	$-0.240034\pi$
$558$		0			0
$559$	−3.38141e13			−0.619497
$560$		0			0
$561$	−3.33425e13	+	2.29134e13i	−0.600045	+	0.412358i
$562$		0			0
$563$		−	4.77347e13i		−	0.843902i	−0.906619	−	0.421951i	$-0.861345\pi$
$563$		−	4.77347e13i		−	0.843902i	0.906619	−	0.421951i	$-0.138655\pi$
$564$		0			0
$565$	−4.79797e13			−0.833327
$566$		0			0
$567$	−6.01742e13	−	5.42013e13i	−1.02682	−	0.924900i
$568$		0			0
$569$			1.90951e13i			0.320156i	0.987104	+	0.160078i	$0.0511745\pi$
$569$			1.90951e13i			0.320156i	−0.987104	+	0.160078i	$0.948825\pi$
$570$		0			0
$571$	2.09448e13			0.345060			0.172530	−	0.985004i	$-0.444806\pi$
$571$	2.09448e13			0.345060			0.172530	+	0.985004i	$0.444806\pi$
$572$		0			0
$573$	−2.84145e13	−	4.13475e13i	−0.460009	−	0.669385i
$574$		0			0
$575$		−	4.79175e12i		−	0.0762351i
$576$		0			0
$577$	3.56688e13			0.557711			0.278855	−	0.960333i	$-0.410045\pi$
$577$	3.56688e13			0.557711			0.278855	+	0.960333i	$0.410045\pi$
$578$		0			0
$579$	6.24051e13	−	4.28856e13i	0.959019	−	0.659050i
$580$		0			0
$581$		−	5.07634e13i		−	0.766778i
$582$		0			0
$583$	2.63221e12			0.0390821
$584$		0			0
$585$	−3.40542e13	−	1.30758e13i	−0.497040	−	0.190849i
$586$		0			0
$587$		−	1.29418e14i		−	1.85697i	−0.371375	−	0.928483i	$-0.621114\pi$
$587$		−	1.29418e14i		−	1.85697i	0.371375	−	0.928483i	$-0.378886\pi$
$588$		0			0
$589$	−3.20761e13			−0.452486
$590$		0			0
$591$	2.08773e13	+	3.03797e13i	0.289559	+	0.421352i
$592$		0			0
$593$		−	3.96596e13i		−	0.540848i	−0.962741	−	0.270424i	$-0.912836\pi$
$593$		−	3.96596e13i		−	0.540848i	0.962741	−	0.270424i	$-0.0871639\pi$
$594$		0			0
$595$	−2.24828e14			−3.01485
$596$		0			0
$597$	2.23684e13	−	1.53718e13i	0.294960	−	0.202700i
$598$		0			0
$599$		−	4.57555e13i		−	0.593347i	−0.954979	−	0.296674i	$-0.904123\pi$
$599$		−	4.57555e13i		−	0.593347i	0.954979	−	0.296674i	$-0.0958774\pi$
$600$		0			0
$601$	1.06033e14			1.35229			0.676143	−	0.736770i	$-0.263650\pi$
$601$	1.06033e14			1.35229			0.676143	+	0.736770i	$0.263650\pi$
$602$		0			0
$603$	−1.10508e13	+	2.87804e13i	−0.138614	+	0.361003i
$604$		0			0
$605$			8.00096e13i			0.987109i
$606$		0			0
$607$	9.55067e13			1.15902			0.579509	−	0.814966i	$-0.303244\pi$
$607$	9.55067e13			1.15902			0.579509	+	0.814966i	$0.303244\pi$
$608$		0			0
$609$	−1.54475e13	−	2.24784e13i	−0.184404	−	0.268336i
$610$		0			0
$611$			2.63849e13i			0.309848i
$612$		0			0
$613$	−5.18411e13			−0.598924			−0.299462	−	0.954108i	$-0.596807\pi$
$613$	−5.18411e13			−0.598924			−0.299462	+	0.954108i	$0.596807\pi$
$614$		0			0
$615$	−1.08507e14	+	7.45675e13i	−1.23334	+	0.847567i
$616$		0			0
$617$		−	8.05443e13i		−	0.900760i	−0.892837	−	0.450380i	$-0.851289\pi$
$617$		−	8.05443e13i		−	0.900760i	0.892837	−	0.450380i	$-0.148711\pi$
$618$		0			0
$619$	5.14946e13			0.566642			0.283321	−	0.959025i	$-0.408564\pi$
$619$	5.14946e13			0.566642			0.283321	+	0.959025i	$0.408564\pi$
$620$		0			0
$621$	1.95797e13	−	4.69777e12i	0.212005	−	0.0508667i
$622$		0			0
$623$			5.54515e12i			0.0590844i
$624$		0			0
$625$	−1.17054e14			−1.22740
$626$		0			0
$627$	−6.58996e12	−	9.58941e12i	−0.0680057	−	0.0989587i
$628$		0			0
$629$			1.33745e14i			1.35839i
$630$		0			0
$631$	1.02922e13			0.102887			0.0514435	−	0.998676i	$-0.483618\pi$
$631$	1.02922e13			0.102887			0.0514435	+	0.998676i	$0.483618\pi$
$632$		0			0
$633$	−3.06543e13	+	2.10660e13i	−0.301629	+	0.207283i
$634$		0			0
$635$			1.15381e14i			1.11755i
$636$		0			0
$637$	−4.37307e13			−0.416955
$638$		0			0
$639$	3.14965e13	+	1.20937e13i	0.295636	+	0.113516i
$640$		0			0
$641$			4.62451e13i			0.427342i	0.976906	+	0.213671i	$0.0685420\pi$
$641$			4.62451e13i			0.427342i	−0.976906	+	0.213671i	$0.931458\pi$
$642$		0			0
$643$	−2.73102e13			−0.248468			−0.124234	−	0.992253i	$-0.539647\pi$
$643$	−2.73102e13			−0.248468			−0.124234	+	0.992253i	$0.539647\pi$
$644$		0			0
$645$	9.92905e13	+	1.44483e14i	0.889424	+	1.29425i
$646$		0			0
$647$			1.34984e13i			0.119059i	0.998227	+	0.0595293i	$0.0189600\pi$
$647$			1.34984e13i			0.119059i	−0.998227	+	0.0595293i	$0.981040\pi$
$648$		0			0
$649$	−1.82349e13			−0.158372
$650$		0			0
$651$	1.94573e14	−	1.33713e14i	1.66410	−	1.14359i
$652$		0			0
$653$		−	8.41840e13i		−	0.709029i	−0.935051	−	0.354514i	$-0.884646\pi$
$653$		−	8.41840e13i		−	0.709029i	0.935051	−	0.354514i	$-0.115354\pi$
$654$		0			0
$655$	1.81921e14			1.50895
$656$		0			0
$657$	4.62671e13	−	1.20496e14i	0.377960	−	0.984345i
$658$		0			0
$659$			1.65464e14i			1.33130i	0.746264	+	0.665650i	$0.231846\pi$
$659$			1.65464e14i			1.33130i	−0.746264	+	0.665650i	$0.768154\pi$
$660$		0			0
$661$	2.01681e14			1.59830			0.799148	−	0.601135i	$-0.205284\pi$
$661$	2.01681e14			1.59830			0.799148	+	0.601135i	$0.205284\pi$
$662$		0			0
$663$	−6.24405e13	−	9.08606e13i	−0.487415	−	0.709264i
$664$		0			0
$665$		−	6.46612e13i		−	0.497205i
$666$		0			0
$667$	6.78122e12			0.0513664
$668$		0			0
$669$	1.21315e14	−	8.33691e13i	0.905281	−	0.622120i
$670$		0			0
$671$		−	3.31400e13i		−	0.243635i
$672$		0			0
$673$	6.04368e12			0.0437750			0.0218875	−	0.999760i	$-0.493032\pi$
$673$	6.04368e12			0.0437750			0.0218875	+	0.999760i	$0.493032\pi$
$674$		0			0
$675$	1.14316e13	+	4.76452e13i	0.0815806	+	0.340017i
$676$		0			0
$677$			1.21474e14i			0.854164i	0.904213	+	0.427082i	$0.140458\pi$
$677$			1.21474e14i			0.854164i	−0.904213	+	0.427082i	$0.859542\pi$
$678$		0			0
$679$	−2.05349e14			−1.42280
$680$		0			0
$681$	9.22913e13	+	1.34298e14i	0.630123	+	0.916926i
$682$		0			0
$683$			1.51629e14i			1.02019i	0.860119	+	0.510094i	$0.170389\pi$
$683$			1.51629e14i			1.02019i	−0.860119	+	0.510094i	$0.829611\pi$
$684$		0			0
$685$	−9.97450e13			−0.661360
$686$		0			0
$687$	7.07663e13	−	4.86315e13i	0.462427	−	0.317785i
$688$		0			0
$689$			7.17296e12i			0.0461958i
$690$		0			0
$691$	−6.62249e13			−0.420369			−0.210185	−	0.977662i	$-0.567407\pi$
$691$	−6.62249e13			−0.420369			−0.210185	+	0.977662i	$0.567407\pi$
$692$		0			0
$693$	7.99493e13	+	3.06982e13i	0.500205	+	0.192064i
$694$		0			0
$695$		−	2.29623e14i		−	1.41609i
$696$		0			0
$697$	−3.97909e14			−2.41891
$698$		0			0
$699$	1.54679e14	+	2.25082e14i	0.926928	+	1.34882i
$700$		0			0
$701$			5.78990e13i			0.342043i	0.985267	+	0.171022i	$0.0547068\pi$
$701$			5.78990e13i			0.342043i	−0.985267	+	0.171022i	$0.945293\pi$
$702$		0			0
$703$	−3.84654e13			−0.224024
$704$		0			0
$705$	1.12739e14	−	7.74754e13i	0.647332	−	0.444855i
$706$		0			0
$707$		−	3.90039e14i		−	2.20806i
$708$		0			0
$709$	2.70077e14			1.50750			0.753750	−	0.657162i	$-0.228243\pi$
$709$	2.70077e14			1.50750			0.753750	+	0.657162i	$0.228243\pi$
$710$		0			0
$711$	−4.16115e13	+	1.08372e14i	−0.229016	+	0.596441i
$712$		0			0
$713$			5.86983e13i			0.318550i
$714$		0			0
$715$	3.85748e13			0.206430
$716$		0			0
$717$	−3.64166e13	−	5.29918e13i	−0.192178	−	0.279649i
$718$		0			0
$719$			1.48102e14i			0.770755i	0.922759	+	0.385377i	$0.125929\pi$
$719$			1.48102e14i			0.770755i	−0.922759	+	0.385377i	$0.874071\pi$
$720$		0			0
$721$	−1.94375e14			−0.997616
$722$		0			0
$723$	1.77592e14	−	1.22044e14i	0.898944	−	0.617765i
$724$		0			0
$725$			1.65014e13i			0.0823819i
$726$		0			0
$727$	−5.21602e13			−0.256842			−0.128421	−	0.991720i	$-0.540991\pi$
$727$	−5.21602e13			−0.256842			−0.128421	+	0.991720i	$0.540991\pi$
$728$		0			0
$729$	−1.83476e14	+	9.34214e13i	−0.891133	+	0.453742i
$730$		0			0
$731$			5.29837e14i			2.53837i
$732$		0			0
$733$	3.21782e14			1.52069			0.760346	−	0.649518i	$-0.225029\pi$
$733$	3.21782e14			1.52069			0.760346	+	0.649518i	$0.225029\pi$
$734$		0			0
$735$	1.28409e14	+	1.86855e14i	0.598630	+	0.871099i
$736$		0			0
$737$		−	3.26009e13i		−	0.149931i
$738$		0			0
$739$	−1.66542e14			−0.755617			−0.377809	−	0.925884i	$-0.623322\pi$
$739$	−1.66542e14			−0.755617			−0.377809	+	0.925884i	$0.623322\pi$
$740$		0			0
$741$	2.61318e13	−	1.79581e13i	0.116971	−	0.0803839i
$742$		0			0
$743$		−	1.13631e14i		−	0.501825i	−0.968010	−	0.250912i	$-0.919269\pi$
$743$		−	1.13631e14i		−	0.501825i	0.968010	−	0.250912i	$-0.0807306\pi$
$744$		0			0
$745$	1.22884e14			0.535443
$746$		0			0
$747$	−1.20480e14	−	4.62608e13i	−0.517979	−	0.198889i
$748$		0			0
$749$		−	3.33429e14i		−	1.41447i
$750$		0			0
$751$	1.64956e14			0.690508			0.345254	−	0.938509i	$-0.387793\pi$
$751$	1.64956e14			0.690508			0.345254	+	0.938509i	$0.387793\pi$
$752$		0			0
$753$	−1.18290e14	−	1.72130e14i	−0.488622	−	0.711020i
$754$		0			0
$755$		−	4.44320e14i		−	1.81118i
$756$		0			0
$757$	−2.67990e14			−1.07805			−0.539026	−	0.842289i	$-0.681207\pi$
$757$	−2.67990e14			−1.07805			−0.539026	+	0.842289i	$0.681207\pi$
$758$		0			0
$759$	−1.75483e13	+	1.20594e13i	−0.0696669	+	0.0478760i
$760$		0			0
$761$		−	1.29795e14i		−	0.508550i	−0.967132	−	0.254275i	$-0.918163\pi$
$761$		−	1.29795e14i		−	0.508550i	0.967132	−	0.254275i	$-0.0818368\pi$
$762$		0			0
$763$	−1.09760e14			−0.424446
$764$		0			0
$765$	−2.04886e14	+	5.33599e14i	−0.781998	+	2.03661i
$766$		0			0
$767$		−	4.96913e13i		−	0.187199i
$768$		0			0
$769$	−2.19185e14			−0.815041			−0.407521	−	0.913196i	$-0.633607\pi$
$769$	−2.19185e14			−0.815041			−0.407521	+	0.913196i	$0.633607\pi$
$770$		0			0
$771$	−2.99890e14	−	4.36386e14i	−1.10075	−	1.60176i
$772$		0			0
$773$			4.05042e14i			1.46758i	0.679374	+	0.733792i	$0.262251\pi$
$773$			4.05042e14i			1.46758i	−0.679374	+	0.733792i	$0.737749\pi$
$774$		0			0
$775$	−1.42837e14			−0.510894
$776$		0			0
$777$	2.33331e14	−	1.60348e14i	0.823886	−	0.566185i
$778$		0			0
$779$		−	1.14440e14i		−	0.398925i
$780$		0			0
$781$	−3.56775e13			−0.122783
$782$		0			0
$783$	−6.74268e13	+	1.61778e13i	−0.229099	+	0.0549681i
$784$		0			0
$785$		−	5.77141e14i

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(200.269 - 137.627i) q^{3} +3630.47i q^{5} +23226.5 q^{7} +(21166.4 - 55125.0i) q^{9} +O(q^{10})\)	\(q+(200.269 - 137.627i) q^{3} +3630.47i q^{5} +23226.5 q^{7} +(21166.4 - 55125.0i) q^{9} +62442.7i q^{11} -170161. q^{13} +(499653. + 727072. i) q^{15} +2.66626e6i q^{17} -766825. q^{19} +(4.65156e6 - 3.19661e6i) q^{21} +1.40327e6i q^{23} -3.41471e6 q^{25} +(-3.34774e6 - 1.39529e7i) q^{27} -4.83245e6i q^{29} +4.18297e7 q^{31} +(8.59382e6 + 1.25053e7i) q^{33} +8.43233e7i q^{35} +5.01619e7 q^{37} +(-3.40779e7 + 2.34188e7i) q^{39} +1.49239e8i q^{41} +1.98719e8 q^{43} +(2.00130e8 + 7.68440e7i) q^{45} -1.55059e8i q^{47} +2.56996e8 q^{49} +(3.66951e8 + 5.33970e8i) q^{51} -4.21541e7i q^{53} -2.26696e8 q^{55} +(-1.53571e8 + 1.05536e8i) q^{57} +2.92026e8i q^{59} -5.30727e8 q^{61} +(4.91622e8 - 1.28036e9i) q^{63} -6.17764e8i q^{65} -5.22093e8 q^{67} +(1.93128e8 + 2.81031e8i) q^{69} +5.71364e8i q^{71} +2.18588e9 q^{73} +(-6.83861e8 + 4.69958e8i) q^{75} +1.45033e9i q^{77} -1.96592e9 q^{79} +(-2.59075e9 - 2.33360e9i) q^{81} -2.18558e9i q^{83} -9.67979e9 q^{85} +(-6.65078e8 - 9.67791e8i) q^{87} +2.38742e8i q^{89} -3.95224e9 q^{91} +(8.37720e9 - 5.75692e9i) q^{93} -2.78394e9i q^{95} -8.84112e9 q^{97} +(3.44215e9 + 1.32169e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9}+O(q^{10}) \) 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9	\( 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + 275240 q^{13} + 1180800 q^{15} + 1568728 q^{19} + 9628008 q^{21} - 33732380 q^{25} - 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} - 71014168 q^{37} - 217287240 q^{39} + 470688664 q^{43} + 312318720 q^{45} - 50058420 q^{49} + 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} - 1184038744 q^{61} + 905007096 q^{63} + 297365848 q^{67} + 596268288 q^{69} + 6534269000 q^{73} + 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} - 12880512000 q^{85} - 210268800 q^{87} - 8317232080 q^{91} + 31744468392 q^{93} - 39176355064 q^{97} + 2626912512 q^{99}+O(q^{100}) \) 4 * q - 84 * q^3 + 45112 * q^7 + 159012 * q^9 + 275240 * q^13 + 1180800 * q^15 + 1568728 * q^19 + 9628008 * q^21 - 33732380 * q^25 - 34619508 * q^27 + 21785848 * q^31 + 25974144 * q^33 - 71014168 * q^37 - 217287240 * q^39 + 470688664 * q^43 + 312318720 * q^45 - 50058420 * q^49 + 708576768 * q^51 - 2701359360 * q^55 - 1058753208 * q^57 - 1184038744 * q^61 + 905007096 * q^63 + 297365848 * q^67 + 596268288 * q^69 + 6534269000 * q^73 + 5150031180 * q^75 - 199282568 * q^79 + 1458964548 * q^81 - 12880512000 * q^85 - 210268800 * q^87 - 8317232080 * q^91 + 31744468392 * q^93 - 39176355064 * q^97 + 2626912512 * q^99

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