LMFDB - Embedded newform 33.6.d.b.32.15

Newspace parameters

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	33.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.29266605383$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 6 x^{15} - 195 x^{14} - 642 x^{13} + 89670 x^{12} + 53946 x^{11} + 91115757 x^{10} - 2121785838 x^{9} + 37710373995 x^{8} - 835758339660 x^{7} + 12972600642204 x^{6} - 129499271268696 x^{5} + 2168293345395660 x^{4} - 17336133272224368 x^{3} + 169639595563975056 x^{2} - 1075523563426213440 x + 9241272870780234240$
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			32.15
Root			$18.0754 + 11.0840i$ of defining polynomial
Character	$\chi$	$=$	33.32
Dual form			33.6.d.b.32.16

$q$-expansion

$f(q)$	$=$	$q+10.1143 q^{2} +(-10.9611 - 11.0840i) q^{3} +70.2982 q^{4} -88.5196i q^{5} +(-110.863 - 112.106i) q^{6} +126.220i q^{7} +387.358 q^{8} +(-2.70883 + 242.985i) q^{9} +O(q^{10})$	\(q+10.1143 q^{2} +(-10.9611 - 11.0840i) q^{3} +70.2982 q^{4} -88.5196i q^{5} +(-110.863 - 112.106i) q^{6} +126.220i q^{7} +387.358 q^{8} +(-2.70883 + 242.985i) q^{9} -895.310i q^{10} +(398.959 + 43.3939i) q^{11} +(-770.545 - 779.183i) q^{12} +523.100i q^{13} +1276.62i q^{14} +(-981.149 + 970.272i) q^{15} +1668.30 q^{16} -1039.45 q^{17} +(-27.3978 + 2457.61i) q^{18} -842.227i q^{19} -6222.77i q^{20} +(1399.02 - 1383.51i) q^{21} +(4035.17 + 438.897i) q^{22} +2472.99i q^{23} +(-4245.87 - 4293.47i) q^{24} -4710.72 q^{25} +5290.77i q^{26} +(2722.93 - 2633.36i) q^{27} +8873.04i q^{28} +3569.99 q^{29} +(-9923.59 + 9813.58i) q^{30} -215.057 q^{31} +4478.13 q^{32} +(-3892.05 - 4897.69i) q^{33} -10513.3 q^{34} +11172.9 q^{35} +(-190.426 + 17081.4i) q^{36} -3626.51 q^{37} -8518.50i q^{38} +(5798.03 - 5733.75i) q^{39} -34288.8i q^{40} -8845.46 q^{41} +(14150.0 - 13993.2i) q^{42} -5262.04i q^{43} +(28046.1 + 3050.51i) q^{44} +(21508.9 + 239.785i) q^{45} +25012.4i q^{46} -9059.89i q^{47} +(-18286.4 - 18491.4i) q^{48} +875.528 q^{49} -47645.5 q^{50} +(11393.5 + 11521.2i) q^{51} +36773.0i q^{52} -897.191i q^{53} +(27540.4 - 26634.4i) q^{54} +(3841.21 - 35315.7i) q^{55} +48892.3i q^{56} +(-9335.22 + 9231.73i) q^{57} +36107.8 q^{58} -36226.0i q^{59} +(-68973.0 + 68208.4i) q^{60} +15319.3i q^{61} -2175.14 q^{62} +(-30669.5 - 341.908i) q^{63} -8092.58 q^{64} +46304.6 q^{65} +(-39365.2 - 49536.5i) q^{66} -53380.7 q^{67} -73071.6 q^{68} +(27410.5 - 27106.6i) q^{69} +113006. q^{70} +33693.5i q^{71} +(-1049.29 + 94122.2i) q^{72} +76140.5i q^{73} -36679.4 q^{74} +(51634.7 + 52213.5i) q^{75} -59207.1i q^{76} +(-5477.18 + 50356.5i) q^{77} +(58642.7 - 57992.6i) q^{78} -106783. i q^{79} -147677. i q^{80} +(-59034.3 - 1316.41i) q^{81} -89465.2 q^{82} +20385.5 q^{83} +(98348.5 - 97258.2i) q^{84} +92011.8i q^{85} -53221.6i q^{86} +(-39131.0 - 39569.7i) q^{87} +(154540. + 16809.0i) q^{88} +11850.9i q^{89} +(217547. + 2425.24i) q^{90} -66025.7 q^{91} +173847. i q^{92} +(2357.26 + 2383.68i) q^{93} -91634.1i q^{94} -74553.6 q^{95} +(-49085.2 - 49635.4i) q^{96} -67716.0 q^{97} +8855.32 q^{98} +(-11624.8 + 96823.4i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 54q^{3} + 316q^{4} - 222q^{9} + O(q^{10}) $	$ 16q - 54q^{3} + 316q^{4} - 222q^{9} - 552q^{12} - 1674q^{15} + 1684q^{16} + 7932q^{22} - 1356q^{25} - 3240q^{27} - 11980q^{31} - 5106q^{33} - 34032q^{34} + 14016q^{36} + 9356q^{37} + 45912q^{42} + 77430q^{45} - 78012q^{48} - 1136q^{49} + 117308q^{55} + 31848q^{58} - 220548q^{60} + 5860q^{64} - 164796q^{66} - 364132q^{67} + 113790q^{69} + 231144q^{70} + 320364q^{75} + 296088q^{78} - 251334q^{81} + 4824q^{82} + 586836q^{88} - 209184q^{91} - 521046q^{93} + 119852q^{97} - 243894q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$-1$	$-1$

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 $

$2$	10.1143			1.78797			0.893983	−	0.448102i	$-0.147900\pi$
$2$	10.1143			1.78797			0.893983	+	0.448102i	$0.147900\pi$
$3$	−10.9611	−	11.0840i	−0.703155	−	0.711037i
$3$	−10.9611	−	11.0840i	−0.703155	−	0.711037i
$4$	70.2982			2.19682
$5$		−	88.5196i		−	1.58349i	−0.610854	−	0.791744i	$-0.709174\pi$
$5$		−	88.5196i		−	1.58349i	0.610854	−	0.791744i	$-0.290826\pi$
$6$	−110.863	−	112.106i	−1.25722	−	1.27131i
$7$			126.220i			0.973605i	0.873512	+	0.486803i	$0.161837\pi$
$7$			126.220i			0.973605i	−0.873512	+	0.486803i	$0.838163\pi$
$8$	387.358			2.13987
$9$	−2.70883	+	242.985i	−0.0111475	+	0.999938i
$10$		−	895.310i		−	2.83122i
$11$	398.959	+	43.3939i	0.994137	+	0.108130i
$11$	398.959	+	43.3939i	0.994137	+	0.108130i
$12$	−770.545	−	779.183i	−1.54470	−	1.56202i
$13$			523.100i			0.858473i	0.903192	+	0.429236i	$0.141217\pi$
$13$			523.100i			0.858473i	−0.903192	+	0.429236i	$0.858783\pi$
$14$			1276.62i			1.74077i
$15$	−981.149	+	970.272i	−1.12592	+	1.11344i
$16$	1668.30			1.62920
$17$	−1039.45			−0.872332			−0.436166	−	0.899866i	$-0.643664\pi$
$17$	−1039.45			−0.872332			−0.436166	+	0.899866i	$0.643664\pi$
$18$	−27.3978	+	2457.61i	−0.0199313	+	1.78785i
$19$		−	842.227i		−	0.535236i	−0.963525	−	0.267618i	$-0.913764\pi$
$19$		−	842.227i		−	0.535236i	0.963525	−	0.267618i	$-0.0862365\pi$
$20$		−	6222.77i		−	3.47864i
$21$	1399.02	−	1383.51i	0.692269	−	0.684595i
$22$	4035.17	+	438.897i	1.77748	+	0.193333i
$23$			2472.99i			0.974770i	0.873187	+	0.487385i	$0.162049\pi$
$23$			2472.99i			0.974770i	−0.873187	+	0.487385i	$0.837951\pi$
$24$	−4245.87	−	4293.47i	−1.50466	−	1.52153i
$25$	−4710.72			−1.50743
$26$			5290.77i			1.53492i
$27$	2722.93	−	2633.36i	0.718831	−	0.695185i
$28$			8873.04i			2.13883i
$29$	3569.99			0.788265			0.394133	−	0.919054i	$-0.371045\pi$
$29$	3569.99			0.788265			0.394133	+	0.919054i	$0.371045\pi$
$30$	−9923.59	+	9813.58i	−2.01310	+	1.99078i
$31$	−215.057			−0.0401929			−0.0200964	−	0.999798i	$-0.506397\pi$
$31$	−215.057			−0.0401929			−0.0200964	+	0.999798i	$0.506397\pi$
$32$	4478.13			0.773074
$33$	−3892.05	−	4897.69i	−0.622147	−	0.782900i
$34$	−10513.3			−1.55970
$35$	11172.9			1.54169
$36$	−190.426	+	17081.4i	−0.0244889	+	2.19668i
$37$	−3626.51			−0.435496			−0.217748	−	0.976005i	$-0.569871\pi$
$37$	−3626.51			−0.435496			−0.217748	+	0.976005i	$0.569871\pi$
$38$		−	8518.50i		−	0.956983i
$39$	5798.03	−	5733.75i	0.610406	−	0.603639i
$40$		−	34288.8i		−	3.38846i
$41$	−8845.46			−0.821790			−0.410895	−	0.911683i	$-0.634784\pi$
$41$	−8845.46			−0.821790			−0.410895	+	0.911683i	$0.634784\pi$
$42$	14150.0	−	13993.2i	1.23775	−	1.22403i
$43$		−	5262.04i		−	0.433993i	−0.976172	−	0.216996i	$-0.930374\pi$
$43$		−	5262.04i		−	0.433993i	0.976172	−	0.216996i	$-0.0696260\pi$
$44$	28046.1	+	3050.51i	2.18394	+	0.237543i
$45$	21508.9	+	239.785i	1.58339	+	0.0176518i
$46$			25012.4i			1.74285i
$47$		−	9059.89i		−	0.598244i	−0.954215	−	0.299122i	$-0.903306\pi$
$47$		−	9059.89i		−	0.598244i	0.954215	−	0.299122i	$-0.0966938\pi$
$48$	−18286.4	−	18491.4i	−1.14558	−	1.15842i
$49$	875.528			0.0520931
$50$	−47645.5			−2.69523
$51$	11393.5	+	11521.2i	0.613384	+	0.620260i
$52$			36773.0i			1.88591i
$53$		−	897.191i		−	0.0438728i	−0.999759	−	0.0219364i	$-0.993017\pi$
$53$		−	897.191i		−	0.0438728i	0.999759	−	0.0219364i	$-0.00698313\pi$
$54$	27540.4	−	26634.4i	1.28525	−	1.24297i
$55$	3841.21	−	35315.7i	0.171223	−	1.57420i
$56$			48892.3i			2.08339i
$57$	−9335.22	+	9231.73i	−0.380572	+	0.376353i
$58$	36107.8			1.40939
$59$		−	36226.0i		−	1.35485i	−0.735593	−	0.677424i	$-0.763096\pi$
$59$		−	36226.0i		−	1.35485i	0.735593	−	0.677424i	$-0.236904\pi$
$60$	−68973.0	+	68208.4i	−2.47344	+	2.44602i
$61$			15319.3i			0.527125i	0.964642	+	0.263563i	$0.0848976\pi$
$61$			15319.3i			0.527125i	−0.964642	+	0.263563i	$0.915102\pi$
$62$	−2175.14			−0.0718634
$63$	−30669.5	−	341.908i	−0.973545	−	0.0108532i
$64$	−8092.58			−0.246966
$65$	46304.6			1.35938
$66$	−39365.2	−	49536.5i	−1.11238	−	1.39980i
$67$	−53380.7			−1.45277			−0.726385	−	0.687288i	$-0.758801\pi$
$67$	−53380.7			−1.45277			−0.726385	+	0.687288i	$0.758801\pi$
$68$	−73071.6			−1.91636
$69$	27410.5	−	27106.6i	0.693098	−	0.685414i
$70$	113006.			2.75649
$71$			33693.5i			0.793233i	0.917984	+	0.396616i	$0.129816\pi$
$71$			33693.5i			0.793233i	−0.917984	+	0.396616i	$0.870184\pi$
$72$	−1049.29	+	94122.2i	−0.0238541	+	2.13974i
$73$			76140.5i			1.67228i	0.548518	+	0.836139i	$0.315192\pi$
$73$			76140.5i			1.67228i	−0.548518	+	0.836139i	$0.684808\pi$
$74$	−36679.4			−0.778652
$75$	51634.7	+	52213.5i	1.05996	+	1.07184i
$76$		−	59207.1i		−	1.17582i
$77$	−5477.18	+	50356.5i	−0.105276	+	0.967897i
$78$	58642.7	−	57992.6i	1.09138	−	1.07929i
$79$		−	106783.i		−	1.92502i	−0.271251	−	0.962509i	$-0.587437\pi$
$79$		−	106783.i		−	1.92502i	0.271251	−	0.962509i	$-0.412563\pi$
$80$		−	147677.i		−	2.57981i
$81$	−59034.3	−	1316.41i	−0.999751	−	0.0222935i
$82$	−89465.2			−1.46933
$83$	20385.5			0.324808			0.162404	−	0.986724i	$-0.448075\pi$
$83$	20385.5			0.324808			0.162404	+	0.986724i	$0.448075\pi$
$84$	98348.5	−	97258.2i	1.52079	−	1.50393i
$85$			92011.8i			1.38133i
$86$		−	53221.6i		−	0.775964i
$87$	−39131.0	−	39569.7i	−0.554272	−	0.560486i
$88$	154540.	+	16809.0i	2.12732	+	0.231385i
$89$			11850.9i			0.158591i	0.996851	+	0.0792953i	$0.0252670\pi$
$89$			11850.9i			0.158591i	−0.996851	+	0.0792953i	$0.974733\pi$
$90$	217547.	+	2425.24i	2.83104	+	0.0315609i
$91$	−66025.7			−0.835813
$92$			173847.i			2.14139i
$93$	2357.26	+	2383.68i	0.0282618	+	0.0285786i
$94$		−	91634.1i		−	1.06964i
$95$	−74553.6			−0.847539
$96$	−49085.2	−	49635.4i	−0.543591	−	0.549685i
$97$	−67716.0			−0.730738			−0.365369	−	0.930863i	$-0.619057\pi$
$97$	−67716.0			−0.730738			−0.365369	+	0.930863i	$0.619057\pi$
$98$	8855.32			0.0931406
$99$	−11624.8	+	96823.4i	−0.119206	+	0.992870i
$100$	−331155.			−3.31155
$101$	−90346.0			−0.881263			−0.440631	−	0.897688i	$-0.645245\pi$
$101$	−90346.0			−0.881263			−0.440631	+	0.897688i	$0.645245\pi$
$102$	115237.	+	116529.i	1.09671	+	1.10900i
$103$	−11291.0			−0.104867			−0.0524335	−	0.998624i	$-0.516698\pi$
$103$	−11291.0			−0.104867			−0.0524335	+	0.998624i	$0.516698\pi$
$104$			202627.i			1.83702i
$105$	−122468.	−	123841.i	−1.08405	−	1.09620i
$106$		−	9074.42i		−	0.0784430i
$107$	187842.			1.58611			0.793054	−	0.609151i	$-0.208490\pi$
$107$	187842.			1.58611			0.793054	+	0.609151i	$0.208490\pi$
$108$	191417.	−	185120.i	1.57914	−	1.52719i
$109$			85380.2i			0.688321i	0.938911	+	0.344161i	$0.111836\pi$
$109$			85380.2i			0.688321i	−0.938911	+	0.344161i	$0.888164\pi$
$110$	38851.0	−	357192.i	0.306140	−	2.81462i
$111$	39750.5	+	40196.1i	0.306221	+	0.309654i
$112$			210572.i			1.58619i
$113$		−	72617.4i		−	0.534989i	−0.963559	−	0.267494i	$-0.913804\pi$
$113$		−	72617.4i		−	0.534989i	0.963559	−	0.267494i	$-0.0861957\pi$
$114$	−94418.8	+	93372.1i	−0.680450	+	0.672907i
$115$	218908.			1.54354
$116$	250964.			1.73168
$117$	−127105.	−	1416.99i	−0.858419	−	0.00956978i
$118$		−	366399.i		−	2.42242i
$119$		−	131199.i		−	0.849307i
$120$	−380056.	+	375843.i	−2.40932	+	2.38261i
$121$	157285.	+	34624.8i	0.976616	+	0.214992i
$122$			154943.i			0.942482i
$123$	96955.9	+	98042.8i	0.577845	+	0.584323i
$124$	−15118.1			−0.0882965
$125$			140368.i			0.803511i
$126$	−310200.	−	3458.15i	−1.74066	−	0.0194052i
$127$			151435.i			0.833139i	0.909104	+	0.416569i	$0.136768\pi$
$127$			151435.i			0.833139i	−0.909104	+	0.416569i	$0.863232\pi$
$128$	−225150.			−1.21464
$129$	−58324.3	+	57677.7i	−0.308585	+	0.305164i
$130$	468337.			2.43052
$131$	206332.			1.05048			0.525242	−	0.850953i	$-0.323975\pi$
$131$	206332.			1.05048			0.525242	+	0.850953i	$0.323975\pi$
$132$	−273604.	−	344299.i	−1.36674	−	1.71989i
$133$	106306.			0.521108
$134$	−539906.			−2.59750
$135$	−233104.	−	241033.i	−1.10082	−	1.13826i
$136$	−402640.			−1.86668
$137$		−	275648.i		−	1.25474i	−0.778721	−	0.627371i	$-0.784131\pi$
$137$		−	275648.i		−	1.25474i	0.778721	−	0.627371i	$-0.215869\pi$
$138$	277237.	−	274164.i	1.23923	−	1.22550i
$139$			81387.0i			0.357288i	0.983914	+	0.178644i	$0.0571710\pi$
$139$			81387.0i			0.357288i	−0.983914	+	0.178644i	$0.942829\pi$
$140$	785438.			3.38682
$141$	−100420.	+	99306.4i	−0.425374	+	0.420658i
$142$			340785.i			1.41827i
$143$	−22699.4	+	208695.i	−0.0928268	+	0.853439i
$144$	−4519.13	+	405371.i	−0.0181614	+	1.62909i
$145$		−	316014.i		−	1.24821i
$146$			770104.i			2.98997i
$147$	−9596.74	−	9704.33i	−0.0366295	−	0.0370401i
$148$	−254937.			−0.956706
$149$	−60929.5			−0.224834			−0.112417	−	0.993661i	$-0.535859\pi$
$149$	−60929.5			−0.224834			−0.112417	+	0.993661i	$0.535859\pi$
$150$	522246.	+	528101.i	1.89517	+	1.91641i
$151$		−	438115.i		−	1.56367i	−0.623483	−	0.781837i	$-0.714283\pi$
$151$		−	438115.i		−	1.56367i	0.623483	−	0.781837i	$-0.285717\pi$
$152$		−	326243.i		−	1.14534i
$153$	2815.70	−	252571.i	0.00972428	−	0.872278i
$154$	−55397.6	+	509319.i	−0.188230	+	1.73057i
$155$			19036.7i			0.0636449i
$156$	407591.	−	403072.i	1.34095	−	1.32609i
$157$	345267.			1.11791			0.558954	−	0.829198i	$-0.311203\pi$
$157$	345267.			1.11791			0.558954	+	0.829198i	$0.311203\pi$
$158$		−	1.08003e6i		−	3.44186i
$159$	−9944.43	+	9834.19i	−0.0311952	+	0.0308493i
$160$		−	396402.i		−	1.22415i
$161$	−312140.			−0.949041
$162$	−597088.	−	13314.5i	−1.78752	−	0.0398600i
$163$	65897.0			0.194266			0.0971330	−	0.995271i	$-0.469033\pi$
$163$	65897.0			0.194266			0.0971330	+	0.995271i	$0.469033\pi$
$164$	−621820.			−1.80532
$165$	−433542.	+	344522.i	−1.23971	+	0.985162i
$166$	206185.			0.580746
$167$	685804.			1.90287			0.951434	−	0.307852i	$-0.0996101\pi$
$167$	685804.			1.90287			0.951434	+	0.307852i	$0.0996101\pi$
$168$	541921.	−	535913.i	1.48137	−	1.46494i
$169$	97659.3			0.263025
$170$			930631.i			2.46976i
$171$	204648.	+	2281.45i	0.535203	+	0.00596652i
$172$		−	369912.i		−	0.953404i
$173$	331628.			0.842434			0.421217	−	0.906960i	$-0.361603\pi$
$173$	331628.			0.842434			0.421217	+	0.906960i	$0.361603\pi$
$174$	−395781.	−	400218.i	−0.991019	−	1.00213i
$175$		−	594587.i		−	1.46764i
$176$	665581.	+	72393.9i	1.61964	+	0.176165i
$177$	−401528.	+	397077.i	−0.963347	+	0.952668i
$178$			119863.i			0.283554i
$179$		−	502450.i		−	1.17209i	−0.810279	−	0.586044i	$-0.800685\pi$
$179$		−	502450.i		−	1.17209i	0.810279	−	0.586044i	$-0.199315\pi$
$180$	1.51204e6	+	16856.4i	3.47842	+	0.0387779i
$181$	−815453.			−1.85013			−0.925066	−	0.379808i	$-0.875990\pi$
$181$	−815453.			−1.85013			−0.925066	+	0.379808i	$0.875990\pi$
$182$	−667801.			−1.49440
$183$	169799.	−	167916.i	0.374806	−	0.370651i
$184$			957931.i			2.08588i
$185$			321017.i			0.689602i
$186$	23841.9	+	24109.2i	0.0505311	+	0.0510976i
$187$	−414698.	−	45105.9i	−0.867217	−	0.0943255i
$188$		−	636894.i		−	1.31423i
$189$	332382.	+	343688.i	0.676835	+	0.699858i
$190$	−754054.			−1.51537
$191$			527471.i			1.04620i	0.852271	+	0.523100i	$0.175225\pi$
$191$			527471.i			1.04620i	−0.852271	+	0.523100i	$0.824775\pi$
$192$	88703.6	+	89697.9i	0.173655	+	0.175602i
$193$			572950.i			1.10719i	0.832785	+	0.553597i	$0.186745\pi$
$193$			572950.i			1.10719i	−0.832785	+	0.553597i	$0.813255\pi$
$194$	−684897.			−1.30653
$195$	−507549.	−	513239.i	−0.955854	−	0.966570i
$196$	61548.1			0.114439
$197$	284594.			0.522468			0.261234	−	0.965276i	$-0.415871\pi$
$197$	284594.			0.522468			0.261234	+	0.965276i	$0.415871\pi$
$198$	−117576.	+	979297.i	−0.213135	+	1.77522i
$199$	425383.			0.761461			0.380730	−	0.924686i	$-0.375673\pi$
$199$	425383.			0.761461			0.380730	+	0.924686i	$0.375673\pi$
$200$	−1.82474e6			−3.22571
$201$	585110.	+	591670.i	1.02152	+	1.03297i
$202$	−913783.			−1.57567
$203$			450604.i			0.767459i
$204$	800944.	+	809923.i	1.34749	+	1.36260i
$205$			782997.i			1.30129i
$206$	−114200.			−0.187499
$207$	−600898.	−	6698.90i	−0.974709	−	0.0108662i
$208$			872686.i			1.39862i
$209$	36547.5	−	336014.i	0.0578752	−	0.532098i
$210$	−1.23867e6	−	1.25256e6i	−1.93824	−	1.95997i
$211$			120728.i			0.186682i	0.995634	+	0.0933410i	$0.0297547\pi$
$211$			120728.i			0.186682i	−0.995634	+	0.0933410i	$0.970245\pi$
$212$		−	63070.9i		−	0.0963805i
$213$	373458.	−	369318.i	0.564018	−	0.557765i
$214$	1.89988e6			2.83591
$215$	−465793.			−0.687222
$216$	1.05475e6	−	1.02005e6i	1.53821	−	1.48761i
$217$		−	27144.5i		−	0.0391320i
$218$			863558.i			1.23069i
$219$	843939.	−	834583.i	1.18905	−	1.17587i
$220$	270030.	−	2.48263e6i	0.376146	−	3.45824i
$221$		−	543737.i		−	0.748873i
$222$	402047.	+	406554.i	0.547512	+	0.553650i
$223$	307332.			0.413853			0.206926	−	0.978357i	$-0.433654\pi$
$223$	307332.			0.413853			0.206926	+	0.978357i	$0.433654\pi$
$224$			565229.i			0.752669i
$225$	12760.6	−	1.14463e6i	0.0168040	−	1.50734i
$226$		−	734471.i		−	0.956542i
$227$	−458186.			−0.590170			−0.295085	−	0.955471i	$-0.595348\pi$
$227$	−458186.			−0.590170			−0.295085	+	0.955471i	$0.595348\pi$
$228$	−656249.	+	648974.i	−0.836049	+	0.826781i
$229$	246534.			0.310662			0.155331	−	0.987862i	$-0.450356\pi$
$229$	246534.			0.310662			0.155331	+	0.987862i	$0.450356\pi$
$230$	2.21409e6			2.75979
$231$	618186.	−	491254.i	0.762236	−	0.605726i
$232$	1.38287e6			1.68679
$233$	−599273.			−0.723161			−0.361580	−	0.932341i	$-0.617763\pi$
$233$	−599273.			−0.723161			−0.361580	+	0.932341i	$0.617763\pi$
$234$	−1.28558e6	−	14331.8i	−1.53482	−	0.0171104i
$235$	−801978.			−0.947312
$236$		−	2.54663e6i		−	2.97636i
$237$	−1.18358e6	+	1.17046e6i	−1.36876	+	1.35358i
$238$		−	1.32699e6i		−	1.51853i
$239$	−357849.			−0.405234			−0.202617	−	0.979258i	$-0.564945\pi$
$239$	−357849.			−0.405234			−0.202617	+	0.979258i	$0.564945\pi$
$240$	−1.63685e6	+	1.61870e6i	−1.83434	+	1.81401i
$241$		−	570630.i		−	0.632867i	−0.948615	−	0.316433i	$-0.897515\pi$
$241$		−	570630.i		−	0.632867i	0.948615	−	0.316433i	$-0.102485\pi$
$242$	1.59082e6	+	350204.i	1.74615	+	0.384399i
$243$	632490.	+	668764.i	0.687128	+	0.726536i
$244$			1.07692e6i			1.15800i
$245$		−	77501.4i		−	0.0824887i
$246$	980637.	+	991630.i	1.03317	+	1.04475i
$247$	440569.			0.459485
$248$	−83304.0			−0.0860076
$249$	−223448.	−	225953.i	−0.228390	−	0.230951i
$250$			1.41971e6i			1.43665i
$251$		−	823625.i		−	0.825173i	−0.910918	−	0.412587i	$-0.864625\pi$
$251$		−	823625.i		−	0.825173i	0.910918	−	0.412587i	$-0.135375\pi$
$252$	−2.15601e6	−	24035.6i	−2.13870	−	0.0238426i
$253$	−107313.	+	986619.i	−0.105402	+	0.969055i
$254$			1.53165e6i			1.48962i
$255$	1.01986e6	−	1.00855e6i	0.982174	−	0.971286i
$256$	−2.01827e6			−1.92477
$257$			1.66449e6i			1.57198i	0.618238	+	0.785991i	$0.287847\pi$
$257$			1.66449e6i			1.57198i	−0.618238	+	0.785991i	$0.712153\pi$
$258$	−589907.	+	583367.i	−0.551739	+	0.545623i
$259$		−	457737.i		−	0.424001i
$260$	3.25513e6			2.98631
$261$	−9670.51	+	867454.i	−0.00878715	+	0.788216i
$262$	2.08690e6			1.87823
$263$	612387.			0.545929			0.272965	−	0.962024i	$-0.411996\pi$
$263$	612387.			0.545929			0.272965	+	0.962024i	$0.411996\pi$
$264$	−1.50762e6	−	1.89716e6i	−1.33131	−	1.67531i
$265$	−79419.0			−0.0694720
$266$	1.07520e6			0.931723
$267$	131355.	−	129899.i	0.112764	−	0.111514i
$268$	−3.75257e6			−3.19147
$269$		−	662606.i		−	0.558309i	−0.960246	−	0.279155i	$-0.909946\pi$
$269$		−	662606.i		−	0.558309i	0.960246	−	0.279155i	$-0.0900542\pi$
$270$	−2.35767e6	−	2.43787e6i	−1.96822	−	2.03517i
$271$			1.01210e6i			0.837144i	0.908184	+	0.418572i	$0.137469\pi$
$271$			1.01210e6i			0.837144i	−0.908184	+	0.418572i	$0.862531\pi$
$272$	−1.73411e6			−1.42120
$273$	723713.	+	731826.i	0.587706	+	0.594294i
$274$		−	2.78798e6i		−	2.24343i
$275$	−1.87938e6	−	204417.i	−1.49859	−	0.162999i
$276$	1.92691e6	−	1.90555e6i	1.52261	−	1.50573i
$277$		−	2.02312e6i		−	1.58425i	−0.610360	−	0.792124i	$-0.708975\pi$
$277$		−	2.02312e6i		−	1.58425i	0.610360	−	0.792124i	$-0.291025\pi$
$278$			823169.i			0.638818i
$279$	582.553	−	52255.6i	0.000448048	−	0.0401904i
$280$	4.32793e6			3.29902
$281$	818601.			0.618453			0.309226	−	0.950988i	$-0.399930\pi$
$281$	818601.			0.618453			0.309226	+	0.950988i	$0.399930\pi$
$282$	−1.01567e6	+	1.00441e6i	−0.760554	+	0.752122i
$283$			1.73555e6i			1.28816i	0.764957	+	0.644081i	$0.222760\pi$
$283$			1.73555e6i			1.28816i	−0.764957	+	0.644081i	$0.777240\pi$
$284$			2.36860e6i			1.74259i
$285$	817189.	+	826350.i	0.595951	+	0.602632i
$286$	−229587.	+	2.11080e6i	−0.165971	+	1.52592i
$287$		−	1.11647e6i		−	0.800099i
$288$	−12130.5	+	1.08812e6i	−0.00861781	+	0.773026i
$289$	−339398.			−0.239037
$290$		−	3.19625e6i		−	2.23175i
$291$	742241.	+	750562.i	0.513822	+	0.519582i
$292$			5.35254e6i			3.67369i
$293$	−2.61963e6			−1.78267			−0.891336	−	0.453344i	$-0.850231\pi$
$293$	−2.61963e6			−1.78267			−0.891336	+	0.453344i	$0.850231\pi$
$294$	−97063.9	−	98152.1i	−0.0654922	−	0.0662264i
$295$	−3.20671e6			−2.14538
$296$	−1.40476e6			−0.931905
$297$	1.20061e6	−	932441.i	0.789787	−	0.613381i
$298$	−616257.			−0.401995
$299$	−1.29362e6			−0.836813
$300$	3.62983e6	+	3.67052e6i	2.32853	+	2.35464i
$301$	664174.			0.422538
$302$		−	4.43121e6i		−	2.79579i
$303$	990291.	+	1.00139e6i	0.619664	+	0.626610i
$304$		−	1.40508e6i		−	0.872004i
$305$	1.35606e6			0.834696
$306$	28478.7	−	2.55457e6i	0.0173867	−	1.55960i
$307$		−	103438.i		−	0.0626377i	−0.999509	−	0.0313188i	$-0.990029\pi$
$307$		−	103438.i		−	0.0626377i	0.999509	−	0.0313188i	$-0.00997072\pi$
$308$	−385036.	+	3.53997e6i	−0.231273	+	2.12629i
$309$	123762.	+	125149.i	0.0737377	+	0.0745643i
$310$			192543.i			0.113795i
$311$		−	1.12357e6i		−	0.658716i	−0.944205	−	0.329358i	$-0.893168\pi$
$311$		−	1.12357e6i		−	0.658716i	0.944205	−	0.329358i	$-0.106832\pi$
$312$	2.24591e6	−	2.22101e6i	1.30619	−	1.29171i
$313$	−1.99753e6			−1.15248			−0.576239	−	0.817281i	$-0.695480\pi$
$313$	−1.99753e6			−1.15248			−0.576239	+	0.817281i	$0.695480\pi$
$314$	3.49212e6			1.99878
$315$	−30265.6	+	2.71486e6i	−0.0171859	+	1.54160i
$316$		−	7.50666e6i		−	4.22892i
$317$			2.55991e6i			1.43079i	0.698720	+	0.715395i	$0.253753\pi$
$317$			2.55991e6i			1.43079i	−0.698720	+	0.715395i	$0.746247\pi$
$318$	−100581.	+	99465.5i	−0.0557759	+	0.0551575i
$319$	1.42428e6	+	154916.i	0.783643	+	0.0852353i
$320$			716352.i			0.391067i
$321$	−2.05895e6	−	2.08203e6i	−1.11528	−	1.12778i
$322$	−3.15707e6			−1.69685
$323$			875454.i			0.466903i
$324$	−4.15001e6	−	92541.3i	−2.19627	−	0.0489748i
$325$		−	2.46418e6i		−	1.29409i
$326$	666499.			0.347341
$327$	946352.	−	935861.i	0.489422	−	0.483996i
$328$	−3.42636e6			−1.75852
$329$	1.14354e6			0.582454
$330$	−4.38495e6	+	3.48459e6i	−2.21656	+	1.76144i
$331$	−537352.			−0.269581			−0.134790	−	0.990874i	$-0.543036\pi$
$331$	−537352.			−0.269581			−0.134790	+	0.990874i	$0.543036\pi$
$332$	1.43307e6			0.713545
$333$	9823.59	−	881186.i	0.00485467	−	0.435469i
$334$	6.93640e6			3.40226
$335$			4.72524e6i			2.30044i
$336$	2.33398e6	−	2.30810e6i	1.12784	−	1.11534i
$337$			156793.i			0.0752058i	0.999293	+	0.0376029i	$0.0119722\pi$
$337$			156793.i			0.0752058i	−0.999293	+	0.0376029i	$0.988028\pi$
$338$	987752.			0.470279
$339$	−804889.	+	795966.i	−0.380397	+	0.376180i
$340$			6.46827e6i			3.03453i
$341$	−85798.8	−	9332.16i	−0.0399572	−	0.00434606i
$342$	2.06987e6	+	23075.2i	0.956923	+	0.0106679i
$343$			2.23189e6i			1.02432i
$344$		−	2.03829e6i		−	0.928689i
$345$	−2.39947e6	−	2.42637e6i	−1.08534	−	1.09751i
$346$	3.35417e6			1.50624
$347$	−1.90705e6			−0.850235			−0.425117	−	0.905138i	$-0.639767\pi$
$347$	−1.90705e6			−0.850235			−0.425117	+	0.905138i	$0.639767\pi$
$348$	−2.75084e6	−	2.78168e6i	−1.21764	−	1.23129i
$349$		−	444271.i		−	0.195247i	−0.995223	−	0.0976236i	$-0.968876\pi$
$349$		−	444271.i		−	0.195247i	0.995223	−	0.0976236i	$-0.0311241\pi$
$350$		−	6.01381e6i		−	2.62409i
$351$	1.37751e6	+	1.42436e6i	0.596797	+	0.617097i
$352$	1.78659e6	+	194323.i	0.768542	+	0.0835927i
$353$		−	3.40494e6i		−	1.45436i	−0.686444	−	0.727182i	$-0.740829\pi$
$353$		−	3.40494e6i		−	1.45436i	0.686444	−	0.727182i	$-0.259171\pi$
$354$	−4.06116e6	+	4.01614e6i	−1.72243	+	1.70334i
$355$	2.98254e6			1.25607
$356$			833099.i			0.348395i
$357$	−1.45421e6	+	1.43809e6i	−0.603889	+	0.597194i
$358$		−	5.08191e6i		−	2.09565i
$359$	585269.			0.239673			0.119837	−	0.992794i	$-0.461763\pi$
$359$	585269.			0.239673			0.119837	+	0.992794i	$0.461763\pi$
$360$	8.33166e6	+	92882.5i	3.38825	+	0.0377727i
$361$	1.76675e6			0.713523
$362$	−8.24770e6			−3.30797
$363$	−1.34024e6	−	2.12287e6i	−0.533844	−	0.845583i
$364$	−4.64149e6			−1.83613
$365$	6.73993e6			2.64803
$366$	1.71739e6	−	1.69835e6i	0.670140	−	0.662710i
$367$	−314950.			−0.122061			−0.0610304	−	0.998136i	$-0.519439\pi$
$367$	−314950.			−0.122061			−0.0610304	+	0.998136i	$0.519439\pi$
$368$			4.12568e6i			1.58809i
$369$	23960.9	−	2.14931e6i	0.00916086	−	0.821739i
$370$			3.24685e6i			1.23298i
$371$	113243.			0.0427148
$372$	165711.	+	167569.i	0.0620861	+	0.0627821i
$373$		−	2.27086e6i		−	0.845121i	−0.906335	−	0.422561i	$-0.861131\pi$
$373$		−	2.27086e6i		−	0.845121i	0.906335	−	0.422561i	$-0.138869\pi$
$374$	−4.19436e6	−	456212.i	−1.55055	−	0.168651i
$375$	1.55583e6	−	1.53858e6i	0.571326	−	0.564992i
$376$		−	3.50942e6i		−	1.28017i
$377$			1.86746e6i			0.676704i
$378$	3.36180e6	+	3.47615e6i	1.21016	+	1.25132i
$379$	3.88525e6			1.38938			0.694691	−	0.719309i	$-0.255541\pi$
$379$	3.88525e6			1.38938			0.694691	+	0.719309i	$0.255541\pi$
$380$	−5.24099e6			−1.86189
$381$	1.67850e6	−	1.65989e6i	0.592392	−	0.585825i
$382$			5.33498e6i			1.87057i
$383$		−	2.31613e6i		−	0.806801i	−0.915023	−	0.403401i	$-0.867828\pi$
$383$		−	2.31613e6i		−	0.806801i	0.915023	−	0.403401i	$-0.132172\pi$
$384$	2.46790e6	+	2.49556e6i	0.854080	+	0.863655i
$385$	4.45754e6	+	484838.i	1.53265	+	0.166703i
$386$			5.79497e6i			1.97962i
$387$	1.27860e6	+	14254.0i	0.433966	+	0.00483792i
$388$	−4.76031e6			−1.60530
$389$		−	3.97451e6i		−	1.33171i	−0.746081	−	0.665855i	$-0.768067\pi$
$389$		−	3.97451e6i		−	1.33171i	0.746081	−	0.665855i	$-0.231933\pi$
$390$	−5.13348e6	−	5.19103e6i	−1.70903	−	1.72819i
$391$		−	2.57055e6i		−	0.850323i
$392$	339143.			0.111472
$393$	−2.26163e6	−	2.28698e6i	−0.738652	−	0.746933i
$394$	2.87845e6			0.934154
$395$	−9.45240e6			−3.04824
$396$	−817201.	+	6.80651e6i	−0.261873	+	2.18116i
$397$	−2.41352e6			−0.768554			−0.384277	−	0.923218i	$-0.625549\pi$
$397$	−2.41352e6			−0.768554			−0.384277	+	0.923218i	$0.625549\pi$
$398$	4.30243e6			1.36146
$399$	−1.16523e6	−	1.17829e6i	−0.366420	−	0.370527i
$400$	−7.85888e6			−2.45590
$401$		−	2.99915e6i		−	0.931402i	−0.884942	−	0.465701i	$-0.845802\pi$
$401$		−	2.99915e6i		−	0.931402i	0.884942	−	0.465701i	$-0.154198\pi$
$402$	5.91796e6	+	5.98430e6i	1.82645	+	1.84692i
$403$		−	112496.i		−	0.0345045i
$404$	−6.35116e6			−1.93598
$405$	−116528.	+	5.22570e6i	−0.0353015	+	1.58309i
$406$			4.55753e6i			1.37219i
$407$	−1.44683e6	−	157368.i	−0.432943	−	0.0470903i
$408$	4.41337e6	+	4.46285e6i	1.31256	+	1.32728i
$409$			3.05264e6i			0.902335i	0.892439	+	0.451167i	$0.148992\pi$
$409$			3.05264e6i			0.902335i	−0.892439	+	0.451167i	$0.851008\pi$
$410$			7.91943e6i			2.32667i
$411$	−3.05528e6	+	3.02141e6i	−0.892168	+	0.882277i
$412$	−793736.			−0.230374
$413$	4.57245e6			1.31909
$414$	−6.07764e6	−	67754.4i	−1.74275	−	0.0194284i
$415$		−	1.80452e6i		−	0.514330i
$416$			2.34251e6i			0.663663i
$417$	902091.	−	892090.i	0.254045	−	0.251228i
$418$	369651.	−	3.39853e6i	0.103479	−	0.951372i
$419$		−	1.96679e6i		−	0.547296i	−0.961830	−	0.273648i	$-0.911770\pi$
$419$		−	1.96679e6i		−	0.547296i	0.961830	−	0.273648i	$-0.0882304\pi$
$420$	−8.60926e6	−	8.70577e6i	−2.38146	−	2.40815i
$421$	3.03122e6			0.833514			0.416757	−	0.909018i	$-0.363167\pi$
$421$	3.03122e6			0.833514			0.416757	+	0.909018i	$0.363167\pi$
$422$			1.22108e6i			0.333781i
$423$	2.20142e6	+	24541.7i	0.598207	+	0.00666890i
$424$		−	347534.i		−	0.0938821i
$425$	4.89657e6			1.31498
$426$	3.77725e6	−	3.73538e6i	1.00844	−	0.997265i
$427$	−1.93360e6			−0.513212
$428$	1.32049e7			3.48439
$429$	2.56198e6	−	2.03593e6i	0.672098	−	0.534096i
$430$	−4.71116e6			−1.22873
$431$	−1.36368e6			−0.353605			−0.176802	−	0.984246i	$-0.556575\pi$
$431$	−1.36368e6			−0.353605			−0.176802	+	0.984246i	$0.556575\pi$
$432$	4.54265e6	−	4.39322e6i	1.17112	−	1.13259i
$433$	−5.36351e6			−1.37477			−0.687384	−	0.726295i	$-0.741241\pi$
$433$	−5.36351e6			−1.37477			−0.687384	+	0.726295i	$0.741241\pi$
$434$		−	274546.i		−	0.0699666i
$435$	−3.50269e6	+	3.46386e6i	−0.887522	+	0.877683i
$436$			6.00208e6i			1.51212i
$437$	2.08282e6			0.521732
$438$	8.53581e6	−	8.44119e6i	2.12598	−	2.10241i
$439$		−	406895.i		−	0.100768i	−0.998730	−	0.0503839i	$-0.983956\pi$
$439$		−	406895.i		−	0.100768i	0.998730	−	0.0503839i	$-0.0160445\pi$
$440$	1.48792e6	−	1.36798e7i	0.366395	−	3.36859i
$441$	−2371.66	+	212740.i	−0.000580705	+	0.0520898i
$442$		−	5.49950e6i		−	1.33896i
$443$		−	2.29937e6i		−	0.556673i	−0.960484	−	0.278337i	$-0.910217\pi$
$443$		−	2.29937e6i		−	0.556673i	0.960484	−	0.278337i	$-0.0897831\pi$
$444$	2.79439e6	+	2.82571e6i	0.672712	+	0.680253i
$445$	1.04904e6			0.251126
$446$	3.10844e6			0.739954
$447$	667854.	+	675341.i	0.158093	+	0.159865i
$448$		−	1.02145e6i		−	0.240447i
$449$			6.52698e6i			1.52791i	0.645272	+	0.763953i	$0.276744\pi$
$449$			6.52698e6i			1.52791i	−0.645272	+	0.763953i	$0.723256\pi$
$450$	129064.	−	1.15771e7i	0.0300450	−	2.69507i
$451$	−3.52897e6	−	383839.i	−0.816971	−	0.0888603i
$452$		−	5.10488e6i		−	1.17527i
$453$	−4.85606e6	+	4.80222e6i	−1.11183	+	1.09950i
$454$	−4.63421e6			−1.05520
$455$			5.84457e6i			1.32350i
$456$	−3.61607e6	+	3.57598e6i	−0.814376	+	0.805348i
$457$			7.61992e6i			1.70671i	0.521330	+	0.853355i	$0.325436\pi$
$457$			7.61992e6i			1.70671i	−0.521330	+	0.853355i	$0.674564\pi$
$458$	2.49351e6			0.555453
$459$	−2.83035e6	+	2.73725e6i	−0.627060	+	0.606432i
$460$	1.53888e7			3.39087
$461$	−1.84657e6			−0.404682			−0.202341	−	0.979315i	$-0.564855\pi$
$461$	−1.84657e6			−0.404682			−0.202341	+	0.979315i	$0.564855\pi$
$462$	6.25249e6	−	4.96867e6i	1.36285	−	1.08302i
$463$	8.90164e6			1.92982			0.964912	−	0.262573i	$-0.0845711\pi$
$463$	8.90164e6			1.92982			0.964912	+	0.262573i	$0.0845711\pi$
$464$	5.95581e6			1.28424
$465$	211003.	−	208664.i	0.0452539	−	0.0447522i
$466$	−6.06120e6			−1.29299
$467$			3.83609e6i			0.813947i	0.913440	+	0.406974i	$0.133416\pi$
$467$			3.83609e6i			0.813947i	−0.913440	+	0.406974i	$0.866584\pi$
$468$	−8.93528e6	−	99611.9i	−1.88579	−	0.0210231i
$469$		−	6.73770e6i		−	1.41442i
$470$	−8.11142e6			−1.69376
$471$	−3.78451e6	−	3.82693e6i	−0.786063	−	0.794875i
$472$		−	1.40324e7i		−	2.89920i
$473$	228340.	−	2.09933e6i	0.0469278	−	0.431448i
$474$	−1.19710e7	+	1.18383e7i	−2.44729	+	2.42016i
$475$			3.96750e6i			0.806831i
$476$		−	9.22309e6i		−	1.86577i
$477$	218004.	+	2430.34i	0.0438700	+	0.000489070i
$478$	−3.61938e6			−0.724544
$479$	4.28323e6			0.852968			0.426484	−	0.904495i	$-0.359752\pi$
$479$	4.28323e6			0.852968			0.426484	+	0.904495i	$0.359752\pi$
$480$	−4.39371e6	+	4.34500e6i	−0.870418	+	0.860769i
$481$		−	1.89703e6i		−	0.373861i
$482$		−	5.77150e6i		−	1.13154i
$483$	3.42140e6	+	3.45975e6i	0.667323	+	0.674803i
$484$	1.10569e7	+	2.43406e6i	2.14545	+	0.472300i
$485$			5.99419e6i			1.15711i
$486$	6.39716e6	+	6.76405e6i	1.22856	+	1.29902i
$487$	−7.16028e6			−1.36807			−0.684034	−	0.729450i	$-0.739776\pi$
$487$	−7.16028e6			−1.36807			−0.684034	+	0.729450i	$0.739776\pi$
$488$			5.93405e6i			1.12798i
$489$	−722303.	−	730400.i	−0.136599	−	0.138130i
$490$		−	783869.i		−	0.147487i
$491$	−2.57405e6			−0.481851			−0.240925	−	0.970544i	$-0.577451\pi$
$491$	−2.57405e6			−0.481851			−0.240925	+	0.970544i	$0.577451\pi$
$492$	6.81583e6	+	6.89223e6i	1.26942	+	1.28365i
$493$	−3.71083e6			−0.687629
$494$	4.45603e6			0.821544
$495$	8.57077e6	+	1.02902e6i	1.57220	+	0.188761i
$496$	−358779.			−0.0654821
$497$	−4.25280e6			−0.772296
$498$	−2.26001e6	−	2.28534e6i	−0.408354	−	0.412932i
$499$	3.30406e6			0.594014			0.297007	−	0.954875i	$-0.404012\pi$
$499$	3.30406e6			0.594014			0.297007	+	0.954875i	$0.404012\pi$
$500$			9.86759e6i			1.76517i
$501$	−7.51716e6	−	7.60143e6i	−1.33801	−	1.35301i
$502$		−	8.33035e6i		−	1.47538i
$503$	−934368.			−0.164664			−0.0823318	−	0.996605i	$-0.526237\pi$
$503$	−934368.			−0.164664			−0.0823318	+	0.996605i	$0.526237\pi$
$504$	−1.18801e7	−	132441.i	−2.08326	−	0.0232245i
$505$			7.99739e6i			1.39547i
$506$	−1.08539e6	+	9.97892e6i	−0.188455	+	1.73264i
$507$	−1.07045e6	−	1.08245e6i	−0.184947	−	0.187020i
$508$			1.06456e7i			1.83026i
$509$		−	2.42079e6i		−	0.414155i	−0.978325	−	0.207077i	$-0.933605\pi$
$509$		−	2.42079e6i		−	0.414155i	0.978325	−	0.207077i	$-0.0663952\pi$
$510$	1.03151e7	−	1.02007e7i	1.75609	−	1.73663i
$511$	−9.61045e6			−1.62814
$512$	−1.32085e7			−2.22678
$513$	−2.21788e6	−	2.29332e6i	−0.372088	−	0.384744i
$514$			1.68350e7i			2.81065i
$515$			999474.i			0.166056i
$516$	−4.10009e6	+	4.05464e6i	−0.677906	+	0.670390i
$517$	393144.	−	3.61452e6i	0.0646883	−	0.594737i
$518$		−	4.62967e6i		−	0.758099i
$519$	−3.63500e6	−	3.67575e6i	−0.592361	−	0.599002i
$520$	1.79365e7			2.90890
$521$		−	5.66823e6i		−	0.914858i	−0.889246	−	0.457429i	$-0.848770\pi$
$521$		−	5.66823e6i		−	0.914858i	0.889246	−	0.457429i	$-0.151230\pi$
$522$	−97810.0	+	8.77366e6i	−0.0157111	+	1.40930i
$523$			7.10021e6i			1.13506i	0.823354	+	0.567528i	$0.192100\pi$
$523$			7.10021e6i			1.13506i	−0.823354	+	0.567528i	$0.807900\pi$
$524$	1.45048e7			2.30772
$525$	−6.59039e6	+	6.51733e6i	−1.04355	+	1.03198i
$526$	6.19384e6			0.976102
$527$	223541.			0.0350615
$528$	−6.49309e6	−	8.17080e6i	−1.01360	−	1.27550i
$529$	320681.			0.0498234
$530$	−803264.			−0.124213
$531$	8.80238e6	+	98130.2i	1.35476	+	0.0151031i
$532$	7.47311e6			1.14478
$533$		−	4.62706e6i		−	0.705484i
$534$	1.32856e6	−	1.31383e6i	0.201618	−	0.199383i
$535$		−	1.66277e7i		−	2.51158i
$536$	−2.06774e7			−3.10874
$537$	−5.56914e6	+	5.50740e6i	−0.833398	+	0.824159i
$538$		−	6.70177e6i		−	0.998237i
$539$	349299.	+	37992.6i	0.0517876	+	0.00563283i
$540$	−1.63868e7	−	1.69442e7i	−2.41829	−	2.50055i
$541$		−	4.83518e6i		−	0.710263i	−0.934816	−	0.355132i	$-0.884436\pi$
$541$		−	4.83518e6i		−	0.710263i	0.934816	−	0.355132i	$-0.115564\pi$
$542$			1.02366e7i			1.49678i
$543$	8.93826e6	+	9.03846e6i	1.30093	+	1.31551i
$544$	−4.65479e6			−0.674378
$545$	7.55783e6			1.08995
$546$	7.31982e6	+	7.40188e6i	1.05080	+	1.06258i
$547$			5.54075e6i			0.791773i	0.918300	+	0.395886i	$0.129563\pi$
$547$			5.54075e6i			0.791773i	−0.918300	+	0.395886i	$0.870437\pi$
$548$		−	1.93776e7i		−	2.75644i
$549$	−3.72236e6	−	41497.4i	−0.527093	−	0.00587611i
$550$	−1.90086e7	−	2.06752e6i	−2.67943	−	0.291436i
$551$		−	3.00674e6i		−	0.421908i
$552$	1.06177e7	−	1.05000e7i	1.48314	−	1.46670i
$553$	1.34782e7			1.87421
$554$		−	2.04624e7i		−	2.83258i
$555$	3.55814e6	−	3.51870e6i	0.490333	−	0.484897i
$556$			5.72136e6i			0.784896i
$557$	7.00403e6			0.956555			0.478278	−	0.878209i	$-0.341261\pi$
$557$	7.00403e6			0.956555			0.478278	+	0.878209i	$0.341261\pi$
$558$	5892.09	−	528526.i	0.000801094	−	0.0718590i
$559$	2.75257e6			0.372571
$560$	1.86398e7			2.51172
$561$	4.04559e6	+	5.09091e6i	0.542719	+	0.682949i
$562$	8.27954e6			1.10577
$563$	−7.81431e6			−1.03901			−0.519505	−	0.854468i	$-0.673883\pi$
$563$	−7.81431e6			−1.03901			−0.519505	+	0.854468i	$0.673883\pi$
$564$	−7.05932e6	+	6.98106e6i	−0.934469	+	0.924110i
$565$	−6.42807e6			−0.847148
$566$			1.75538e7i			2.30319i
$567$	166157.	−	7.45131e6i	0.0217051	−	0.973363i
$568$			1.30515e7i			1.69742i
$569$	1.22848e7			1.59069			0.795347	−	0.606154i	$-0.207288\pi$
$569$	1.22848e7			1.59069			0.795347	+	0.606154i	$0.207288\pi$
$570$	8.26526e6	+	8.35792e6i	1.06554	+	1.07748i
$571$			1.28754e7i			1.65262i	0.563218	+	0.826308i	$0.309563\pi$
$571$			1.28754e7i			1.65262i	−0.563218	+	0.826308i	$0.690437\pi$
$572$	−1.59572e6	+	1.46709e7i	−0.203924	+	1.87485i
$573$	5.84647e6	−	5.78166e6i	0.743887	−	0.735641i
$574$		−	1.12923e7i		−	1.43055i
$575$		−	1.16496e7i		−	1.46940i
$576$	21921.4	−	1.96638e6i	0.00275304	−	0.246951i
$577$	−9.45884e6			−1.18276			−0.591382	−	0.806391i	$-0.701418\pi$
$577$	−9.45884e6			−1.18276			−0.591382	+	0.806391i	$0.701418\pi$
$578$	−3.43276e6			−0.427389
$579$	6.35056e6	−	6.28016e6i	0.787256	−	0.778528i
$580$		−	2.22153e7i		−	2.74209i
$581$			2.57306e6i			0.316235i
$582$	7.50722e6	+	7.59138e6i	0.918696	+	0.928994i
$583$	38932.6	−	357942.i	0.00474397	−	0.0436155i
$584$			2.94936e7i			3.57846i
$585$	−125431.	+	1.12513e7i	−0.0151536	+	1.35930i
$586$	−2.64956e7			−3.18735
$587$			1.02802e7i			1.23142i	0.787972	+	0.615711i	$0.211131\pi$
$587$			1.02802e7i			1.23142i	−0.787972	+	0.615711i	$0.788869\pi$
$588$	−674634.	−	682197.i	−0.0804683	−	0.0813704i
$589$			181127.i			0.0215127i
$590$	−3.24335e7			−3.83587
$591$	−3.11946e6	−	3.15443e6i	−0.367376	−	0.371494i
$592$	−6.05009e6			−0.709508
$593$	−1.25800e7			−1.46907			−0.734536	−	0.678569i	$-0.762600\pi$
$593$	−1.25800e7			−1.46907			−0.734536	+	0.678569i	$0.762600\pi$
$594$	1.21433e7	−	9.43095e6i	1.41211	−	1.09670i
$595$	−1.16137e7			−1.34487
$596$	−4.28324e6			−0.493920
$597$	−4.66266e6	−	4.71493e6i	−0.535424	−	0.541427i
$598$	−1.30840e7			−1.49619
$599$		−	6.21604e6i		−	0.707858i	−0.935272	−	0.353929i	$-0.884845\pi$
$599$		−	6.21604e6i		−	0.707858i	0.935272	−	0.353929i	$-0.115155\pi$
$600$	2.00011e7	+	2.02253e7i	2.26817	+	2.29360i
$601$			1.28321e7i			1.44914i	0.689199	+	0.724572i	$0.257962\pi$
$601$			1.28321e7i			1.44914i	−0.689199	+	0.724572i	$0.742038\pi$
$602$	6.71763e6			0.755483
$603$	144599.	−	1.29707e7i	0.0161947	−	1.45268i
$604$		−	3.07987e7i		−	3.43511i
$605$	3.06497e6	−	1.39228e7i	0.340438	−	1.54646i
$606$	1.00161e7	+	1.01283e7i	1.10794	+	1.12036i
$607$		−	3.31925e6i		−	0.365652i	−0.983145	−	0.182826i	$-0.941476\pi$
$607$		−	3.31925e6i		−	0.365652i	0.983145	−	0.182826i	$-0.0585245\pi$
$608$		−	3.77160e6i		−	0.413777i
$609$	4.99448e6	−	4.93912e6i	0.545692	−	0.539642i
$610$	1.37155e7			1.49241
$611$	4.73923e6			0.513576
$612$	197939.	−	1.77553e7i	0.0213625	−	1.91624i
$613$		−	5.25789e6i		−	0.565145i	−0.959246	−	0.282573i	$-0.908812\pi$
$613$		−	5.25789e6i		−	0.565145i	0.959246	−	0.282573i	$-0.0911878\pi$
$614$		−	1.04620e6i		−	0.111994i
$615$	8.67871e6	−	8.58250e6i	0.925268	−	0.915010i
$616$	−2.12163e6	+	1.95060e7i	−0.225277	+	2.07117i
$617$			8.52105e6i			0.901115i	0.892748	+	0.450557i	$0.148775\pi$
$617$			8.52105e6i			0.901115i	−0.892748	+	0.450557i	$0.851225\pi$
$618$	1.25176e6	+	1.26579e6i	0.131840	+	0.133318i
$619$	1.37799e7			1.44550			0.722751	−	0.691108i	$-0.242877\pi$
$619$	1.37799e7			1.44550			0.722751	+	0.691108i	$0.242877\pi$
$620$			1.33825e6i			0.139816i
$621$	6.51225e6	+	6.73377e6i	0.677645	+	0.700695i
$622$		−	1.13641e7i		−	1.17776i
$623$	−1.49582e6			−0.154405
$624$	9.67283e6	−	9.56559e6i	0.994471	−	0.983446i
$625$	−2.29572e6			−0.235082
$626$	−2.02036e7			−2.06059
$627$	−4.12497e6	+	3.27799e6i	−0.419036	+	0.332995i
$628$	2.42717e7			2.45584
$629$	3.76958e6			0.379897
$630$	−306114.	+	2.74588e7i	−0.0307278	+	2.75632i
$631$	−2.97282e6			−0.297231			−0.148616	−	0.988895i	$-0.547482\pi$
$631$	−2.97282e6			−0.297231			−0.148616	+	0.988895i	$0.547482\pi$
$632$		−	4.13633e7i		−	4.11929i
$633$	1.33815e6	−	1.32331e6i	0.132738	−	0.131266i
$634$			2.58916e7i			2.55820i
$635$	1.34050e7			1.31926
$636$	−699076.	+	691326.i	−0.0685301	+	0.0677704i
$637$			457989.i			0.0447205i
$638$	1.44055e7	+	1.56686e6i	1.40113	+	0.152398i
$639$	−8.18702e6	−	91270.1i	−0.793184	−	0.00884253i
$640$			1.99302e7i			1.92337i
$641$			1.03887e7i			0.998658i	0.866413	+	0.499329i	$0.166420\pi$
$641$			1.03887e7i			0.998658i	−0.866413	+	0.499329i	$0.833580\pi$
$642$	−2.08248e7	−	2.10582e7i	−1.99408	−	2.01644i
$643$	2.23793e6			0.213462			0.106731	−	0.994288i	$-0.465962\pi$
$643$	2.23793e6			0.213462			0.106731	+	0.994288i	$0.465962\pi$
$644$	−2.19429e7			−2.08487
$645$	5.10561e6	+	5.16284e6i	0.483223	+	0.488641i
$646$			8.85457e6i			0.834807i
$647$			9.54481e6i			0.896409i	0.893931	+	0.448205i	$0.147936\pi$
$647$			9.54481e6i			0.896409i	−0.893931	+	0.448205i	$0.852064\pi$
$648$	−2.28674e7	−	509922.i	−2.13934	−	0.0477053i
$649$	1.57199e6	−	1.44527e7i	0.146500	−	1.34690i
$650$		−	2.49233e7i		−	2.31378i
$651$	−300868.	+	297533.i	−0.0278243	+	0.0275158i
$652$	4.63244e6			0.426767
$653$			1.60519e7i			1.47314i	0.676360	+	0.736571i	$0.263556\pi$
$653$			1.60519e7i			1.47314i	−0.676360	+	0.736571i	$0.736444\pi$
$654$	9.57165e6	−	9.46554e6i	0.875069	−	0.865368i
$655$		−	1.82645e7i		−	1.66343i
$656$	−1.47568e7			−1.33886
$657$	−1.85010e7	−	206252.i	−1.67217	−	0.0186416i
$658$	1.15661e7			1.04141
$659$	−1.21513e7			−1.08996			−0.544979	−	0.838450i	$-0.683462\pi$
$659$	−1.21513e7			−1.08996			−0.544979	+	0.838450i	$0.683462\pi$
$660$	−3.04772e7	+	2.42193e7i	−2.72342	+	2.16422i
$661$	8.00011e6			0.712184			0.356092	−	0.934451i	$-0.384109\pi$
$661$	8.00011e6			0.712184			0.356092	+	0.934451i	$0.384109\pi$
$662$	−5.43492e6			−0.482001
$663$	−6.02677e6	+	5.95995e6i	−0.532477	+	0.526574i
$664$	7.89651e6			0.695048
$665$		−	9.41015e6i		−	0.825168i
$666$	99358.3	−	8.91255e6i	0.00867998	−	0.778603i
$667$			8.82854e6i			0.768377i
$668$	4.82108e7			4.18026
$669$	−3.36870e6	−	3.40646e6i	−0.291002	−	0.294265i
$670$			4.77923e7i			4.11311i
$671$	−664764.	+	6.11176e6i	−0.0569982	+	0.524035i
$672$	6.26498e6	−	6.19552e6i	0.535176	−	0.529243i
$673$		−	7.19089e6i		−	0.611991i	−0.952033	−	0.305995i	$-0.901011\pi$
$673$		−	7.19089e6i		−	0.611991i	0.952033	−	0.305995i	$-0.0989892\pi$
$674$			1.58584e6i			0.134465i
$675$	−1.28270e7	+	1.24050e7i	−1.08359	+	1.04794i
$676$	6.86528e6			0.577818
$677$	2.50078e6			0.209703			0.104851	−	0.994488i	$-0.466563\pi$
$677$	2.50078e6			0.209703			0.104851	+	0.994488i	$0.466563\pi$
$678$	−8.14086e6	+	8.05061e6i	−0.680136	+	0.672596i
$679$		−	8.54711e6i		−	0.711450i
$680$			3.56415e7i			2.95586i
$681$	5.02222e6	+	5.07852e6i	0.414981	+	0.419633i
$682$	−867791.	−	94387.8i	−0.0714421	−	0.00777061i
$683$		−	1.69294e6i		−	0.138864i	−0.997587	−	0.0694319i	$-0.977881\pi$
$683$		−	1.69294e6i		−	0.138864i	0.997587	−	0.0694319i	$-0.0221187\pi$
$684$	1.43864e7	+	160382.i	1.17574	+	0.0131074i
$685$	−2.44003e7			−1.98687
$686$			2.25739e7i			1.83145i
$687$	−2.70228e6	−	2.73258e6i	−0.218443	−	0.220892i
$688$		−	8.77864e6i		−	0.707060i
$689$	469320.			0.0376636
$690$	−2.42689e7	−	2.45409e7i	−1.94056	−	1.96231i
$691$	−8.84676e6			−0.704838			−0.352419	−	0.935842i	$-0.614641\pi$
$691$	−8.84676e6			−0.704838			−0.352419	+	0.935842i	$0.614641\pi$
$692$	2.33128e7			1.85067
$693$	−1.22210e7	−	1.46728e6i	−0.966663	−	0.116059i
$694$	−1.92884e7			−1.52019
$695$	7.20434e6			0.565760
$696$	−1.51577e7	−	1.53276e7i	−1.18607	−	1.19937i
$697$	9.19442e6			0.716874
$698$		−	4.49348e6i		−	0.349095i
$699$	6.56869e6	+	6.64232e6i	0.508494	+	0.514194i
$700$		−	4.17984e7i		−	3.22415i
$701$	−1.16656e6			−0.0896631			−0.0448315	−	0.998995i	$-0.514275\pi$
$701$	−1.16656e6			−0.0896631			−0.0448315	+	0.998995i	$0.514275\pi$
$702$	1.39325e7	+	1.44064e7i	1.06705	+	1.10335i
$703$			3.05434e6i			0.233093i
$704$	−3.22861e6	−	351169.i	−0.245518	−	0.0267045i
$705$	8.79056e6	+	8.88911e6i	0.666107	+	0.673574i
$706$		−	3.44385e7i		−	2.60035i
$707$		−	1.14035e7i		−	0.858002i
$708$	−2.82267e7	+	2.79138e7i	−2.11630	+	2.09284i
$709$	9.21961e6			0.688806			0.344403	−	0.938822i	$-0.388081\pi$
$709$	9.21961e6			0.688806			0.344403	+	0.938822i	$0.388081\pi$
$710$	3.01662e7			2.24582
$711$	2.59467e7	+	289257.i	1.92490	+	0.0214590i
$712$			4.59055e6i			0.339363i
$713$		−	531833.i		−	0.0391788i
$714$	−1.47083e7	+	1.45452e7i	−1.07973	+	1.06776i
$715$	1.84736e7	+	2.00934e6i	1.35141	+	0.146990i
$716$		−	3.53213e7i		−	2.57486i
$717$	3.92242e6	+	3.96639e6i	0.284942	+	0.288136i
$718$	5.91956e6			0.428527
$719$		−	2.18224e7i		−	1.57427i	−0.616779	−	0.787136i	$-0.711563\pi$
$719$		−	2.18224e7i		−	1.57427i	0.616779	−	0.787136i	$-0.288437\pi$
$720$	3.58833e7	+	400032.i	2.57965	+	0.0287583i
$721$		−	1.42515e6i		−	0.102099i
$722$	1.78694e7			1.27575
$723$	−6.32485e6	+	6.25473e6i	−0.449992	+	0.445003i
$724$	−5.73249e7			−4.06440
$725$	−1.68172e7			−1.18826
$726$	−1.35555e7	−	2.14712e7i	−0.954495	−	1.51187i
$727$	−741663.			−0.0520440			−0.0260220	−	0.999661i	$-0.508284\pi$
$727$	−741663.			−0.0520440			−0.0260220	+	0.999661i	$0.508284\pi$
$728$	−2.55756e7			−1.78853
$729$	479782.	−	1.43409e7i	0.0334368	−	0.999441i
$730$	6.81693e7			4.73459
$731$			5.46963e6i			0.378586i
$732$	1.19365e7	−	1.18042e7i	0.823380	−	0.814252i
$733$			6.80074e6i			0.467516i	0.972295	+	0.233758i	$0.0751023\pi$
$733$			6.80074e6i			0.467516i	−0.972295	+	0.233758i	$0.924898\pi$
$734$	−3.18548e6			−0.218240
$735$	−859023.	+	849500.i	−0.0586525	+	0.0580023i
$736$			1.10743e7i			0.753570i
$737$	−2.12967e7	−	2.31640e6i	−1.44425	−	0.157088i
$738$	242346.	−	2.17387e7i	0.0163793	−	1.46924i
$739$			9.15287e6i			0.616519i	0.951302	+	0.308259i	$0.0997465\pi$
$739$			9.15287e6i			0.616519i	−0.951302	+	0.308259i	$0.900254\pi$
$740$			2.25669e7i			1.51493i
$741$	−4.82912e6	−	4.88325e6i	−0.323089	−	0.326711i
$742$	1.14537e6			0.0763725
$743$	1.25723e7			0.835490			0.417745	−	0.908564i	$-0.362820\pi$
$743$	1.25723e7			0.835490			0.417745	+	0.908564i	$0.362820\pi$
$744$	913103.	+	923339.i	0.0604766	+	0.0611546i
$745$			5.39346e6i			0.356022i
$746$		−	2.29681e7i		−	1.51105i
$747$	−55221.0	+	4.95338e6i	−0.00362078	+	0.324788i
$748$	−2.91525e7	−	3.17086e6i	−1.90512	−	0.207216i
$749$			2.37094e7i			1.54424i
$750$	1.57361e7	−	1.55616e7i	1.02151	−	1.01019i
$751$	1.79363e7			1.16047			0.580235	−	0.814449i	$-0.302961\pi$
$751$	1.79363e7			1.16047			0.580235	+	0.814449i	$0.302961\pi$
$752$		−	1.51146e7i		−	0.974657i
$753$	−9.12903e6	+	9.02783e6i	−0.586729	+	0.580224i
$754$			1.88880e7i			1.20992i
$755$	−3.87818e7			−2.47606
$756$	2.33659e7	+	2.41607e7i	1.48688	+	1.53746i
$757$	1.91439e7			1.21420			0.607099	−	0.794626i	$-0.292333\pi$
$757$	1.91439e7			1.21420			0.607099	+	0.794626i	$0.292333\pi$
$758$	3.92965e7			2.48417
$759$	1.21119e7	−	9.62498e6i	0.763148	−	0.606450i
$760$	−2.88789e7			−1.81362
$761$	−5.64221e6			−0.353173			−0.176586	−	0.984285i	$-0.556506\pi$
$761$	−5.64221e6			−0.353173			−0.176586	+	0.984285i	$0.556506\pi$
$762$	1.69768e7	−	1.67886e7i	1.05918	−	1.04744i
$763$	−1.07767e7			−0.670153
$764$			3.70803e7i			2.29831i
$765$	−2.23575e7	−	249245.i	−1.38124	−	0.0153983i
$766$		−	2.34260e7i		−	1.44253i
$767$	1.89498e7			1.16310
$768$	2.21224e7	+	2.23704e7i	1.35341	+	1.36858i
$769$			5.21462e6i			0.317985i	0.987280	+	0.158992i	$0.0508246\pi$
$769$			5.21462e6i			0.317985i	−0.987280	+	0.158992i	$0.949175\pi$
$770$	4.50847e7	+	4.90377e6i	2.74033	+	0.298060i
$771$	1.84491e7	−	1.82446e7i	1.11774	−	1.10535i
$772$			4.02774e7i			2.43230i
$773$		−	6.01544e6i		−	0.362092i	−0.983475	−	0.181046i	$-0.942052\pi$
$773$		−	6.01544e6i		−	0.362092i	0.983475	−	0.181046i	$-0.0579483\pi$
$774$	1.29320e7	+	144168.i	0.775916	+	0.00865003i
$775$	1.01307e6			0.0605880
$776$

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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+10.1143 q^{2} +(-10.9611 - 11.0840i) q^{3} +70.2982 q^{4} -88.5196i q^{5} +(-110.863 - 112.106i) q^{6} +126.220i q^{7} +387.358 q^{8} +(-2.70883 + 242.985i) q^{9} +O(q^{10})\)	\(q+10.1143 q^{2} +(-10.9611 - 11.0840i) q^{3} +70.2982 q^{4} -88.5196i q^{5} +(-110.863 - 112.106i) q^{6} +126.220i q^{7} +387.358 q^{8} +(-2.70883 + 242.985i) q^{9} -895.310i q^{10} +(398.959 + 43.3939i) q^{11} +(-770.545 - 779.183i) q^{12} +523.100i q^{13} +1276.62i q^{14} +(-981.149 + 970.272i) q^{15} +1668.30 q^{16} -1039.45 q^{17} +(-27.3978 + 2457.61i) q^{18} -842.227i q^{19} -6222.77i q^{20} +(1399.02 - 1383.51i) q^{21} +(4035.17 + 438.897i) q^{22} +2472.99i q^{23} +(-4245.87 - 4293.47i) q^{24} -4710.72 q^{25} +5290.77i q^{26} +(2722.93 - 2633.36i) q^{27} +8873.04i q^{28} +3569.99 q^{29} +(-9923.59 + 9813.58i) q^{30} -215.057 q^{31} +4478.13 q^{32} +(-3892.05 - 4897.69i) q^{33} -10513.3 q^{34} +11172.9 q^{35} +(-190.426 + 17081.4i) q^{36} -3626.51 q^{37} -8518.50i q^{38} +(5798.03 - 5733.75i) q^{39} -34288.8i q^{40} -8845.46 q^{41} +(14150.0 - 13993.2i) q^{42} -5262.04i q^{43} +(28046.1 + 3050.51i) q^{44} +(21508.9 + 239.785i) q^{45} +25012.4i q^{46} -9059.89i q^{47} +(-18286.4 - 18491.4i) q^{48} +875.528 q^{49} -47645.5 q^{50} +(11393.5 + 11521.2i) q^{51} +36773.0i q^{52} -897.191i q^{53} +(27540.4 - 26634.4i) q^{54} +(3841.21 - 35315.7i) q^{55} +48892.3i q^{56} +(-9335.22 + 9231.73i) q^{57} +36107.8 q^{58} -36226.0i q^{59} +(-68973.0 + 68208.4i) q^{60} +15319.3i q^{61} -2175.14 q^{62} +(-30669.5 - 341.908i) q^{63} -8092.58 q^{64} +46304.6 q^{65} +(-39365.2 - 49536.5i) q^{66} -53380.7 q^{67} -73071.6 q^{68} +(27410.5 - 27106.6i) q^{69} +113006. q^{70} +33693.5i q^{71} +(-1049.29 + 94122.2i) q^{72} +76140.5i q^{73} -36679.4 q^{74} +(51634.7 + 52213.5i) q^{75} -59207.1i q^{76} +(-5477.18 + 50356.5i) q^{77} +(58642.7 - 57992.6i) q^{78} -106783. i q^{79} -147677. i q^{80} +(-59034.3 - 1316.41i) q^{81} -89465.2 q^{82} +20385.5 q^{83} +(98348.5 - 97258.2i) q^{84} +92011.8i q^{85} -53221.6i q^{86} +(-39131.0 - 39569.7i) q^{87} +(154540. + 16809.0i) q^{88} +11850.9i q^{89} +(217547. + 2425.24i) q^{90} -66025.7 q^{91} +173847. i q^{92} +(2357.26 + 2383.68i) q^{93} -91634.1i q^{94} -74553.6 q^{95} +(-49085.2 - 49635.4i) q^{96} -67716.0 q^{97} +8855.32 q^{98} +(-11624.8 + 96823.4i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 54q^{3} + 316q^{4} - 222q^{9} + O(q^{10}) \)	\( 16q - 54q^{3} + 316q^{4} - 222q^{9} - 552q^{12} - 1674q^{15} + 1684q^{16} + 7932q^{22} - 1356q^{25} - 3240q^{27} - 11980q^{31} - 5106q^{33} - 34032q^{34} + 14016q^{36} + 9356q^{37} + 45912q^{42} + 77430q^{45} - 78012q^{48} - 1136q^{49} + 117308q^{55} + 31848q^{58} - 220548q^{60} + 5860q^{64} - 164796q^{66} - 364132q^{67} + 113790q^{69} + 231144q^{70} + 320364q^{75} + 296088q^{78} - 251334q^{81} + 4824q^{82} + 586836q^{88} - 209184q^{91} - 521046q^{93} + 119852q^{97} - 243894q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about