LMFDB - Embedded newform 405.4.e.s.136.2

Newspace parameters

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.148347072.2
Defining polynomial:	$ x^{6} + 29x^{4} + 223x^{2} + 243 $ x^6 + 29x^4 + 223x^2 + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.2
Root			$-3.41374i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.4.e.s.271.2

$q$-expansion

$f(q)$	$=$	$q+(-0.663404 + 1.14905i) q^{2} +(3.11979 + 5.40363i) q^{4} +(2.50000 + 4.33013i) q^{5} +(12.0521 - 20.8749i) q^{7} -18.8932 q^{8} +O(q^{10})$	\(q+(-0.663404 + 1.14905i) q^{2} +(3.11979 + 5.40363i) q^{4} +(2.50000 + 4.33013i) q^{5} +(12.0521 - 20.8749i) q^{7} -18.8932 q^{8} -6.63404 q^{10} +(4.13810 - 7.16739i) q^{11} +(-43.5573 - 75.4434i) q^{13} +(15.9909 + 27.6970i) q^{14} +(-12.4245 + 21.5198i) q^{16} -51.9166 q^{17} -88.5107 q^{19} +(-15.5989 + 27.0182i) q^{20} +(5.49046 + 9.50976i) q^{22} +(-64.6224 - 111.929i) q^{23} +(-12.5000 + 21.6506i) q^{25} +115.584 q^{26} +150.400 q^{28} +(-135.554 + 234.787i) q^{29} +(-112.273 - 194.463i) q^{31} +(-92.0577 - 159.449i) q^{32} +(34.4417 - 59.6548i) q^{34} +120.521 q^{35} -70.5268 q^{37} +(58.7184 - 101.703i) q^{38} +(-47.2330 - 81.8099i) q^{40} +(183.469 + 317.778i) q^{41} +(97.7736 - 169.349i) q^{43} +51.6399 q^{44} +171.483 q^{46} +(-179.596 + 311.069i) q^{47} +(-119.008 - 206.128i) q^{49} +(-16.5851 - 28.7262i) q^{50} +(271.779 - 470.735i) q^{52} -29.4890 q^{53} +41.3810 q^{55} +(-227.703 + 394.394i) q^{56} +(-179.855 - 311.517i) q^{58} +(-429.052 - 743.140i) q^{59} +(278.406 - 482.213i) q^{61} +297.931 q^{62} +45.4941 q^{64} +(217.786 - 377.217i) q^{65} +(20.9493 + 36.2853i) q^{67} +(-161.969 - 280.538i) q^{68} +(-79.9544 + 138.485i) q^{70} +549.163 q^{71} -185.505 q^{73} +(46.7878 - 81.0388i) q^{74} +(-276.135 - 478.279i) q^{76} +(-99.7458 - 172.765i) q^{77} +(-40.2456 + 69.7075i) q^{79} -124.245 q^{80} -486.857 q^{82} +(-288.377 + 499.483i) q^{83} +(-129.792 - 224.806i) q^{85} +(129.727 + 224.694i) q^{86} +(-78.1818 + 135.415i) q^{88} -224.516 q^{89} -2099.83 q^{91} +(403.216 - 698.391i) q^{92} +(-238.290 - 412.730i) q^{94} +(-221.277 - 383.263i) q^{95} +(277.508 - 480.658i) q^{97} +315.801 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8}+O(q^{10}) $ 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8	$ 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8} - 10 q^{10} - 58 q^{11} + 47 q^{13} - 159 q^{14} + 127 q^{16} + 68 q^{17} - 10 q^{19} + 25 q^{20} - 260 q^{22} + 51 q^{23} - 75 q^{25} + 506 q^{26} + 166 q^{28} - 350 q^{29} - 638 q^{31} - 245 q^{32} + 154 q^{34} + 250 q^{35} - 828 q^{37} - 397 q^{38} - 135 q^{40} - 179 q^{41} + 836 q^{43} + 664 q^{44} + 522 q^{46} - 235 q^{47} - 892 q^{49} - 25 q^{50} + 1335 q^{52} + 1010 q^{53} - 580 q^{55} - 15 q^{56} - 1876 q^{58} - 535 q^{59} + 104 q^{61} + 696 q^{62} - 606 q^{64} - 235 q^{65} + 40 q^{67} - 830 q^{68} + 795 q^{70} + 904 q^{71} - 1420 q^{73} - 1394 q^{74} - 849 q^{76} - 2148 q^{77} + 634 q^{79} + 1270 q^{80} + 1226 q^{82} - 1734 q^{83} + 170 q^{85} - 460 q^{86} + 768 q^{88} - 1704 q^{89} - 2458 q^{91} + 1839 q^{92} - 1751 q^{94} - 25 q^{95} + 76 q^{98}+O(q^{100}) $ 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8 - 10 * q^10 - 58 * q^11 + 47 * q^13 - 159 * q^14 + 127 * q^16 + 68 * q^17 - 10 * q^19 + 25 * q^20 - 260 * q^22 + 51 * q^23 - 75 * q^25 + 506 * q^26 + 166 * q^28 - 350 * q^29 - 638 * q^31 - 245 * q^32 + 154 * q^34 + 250 * q^35 - 828 * q^37 - 397 * q^38 - 135 * q^40 - 179 * q^41 + 836 * q^43 + 664 * q^44 + 522 * q^46 - 235 * q^47 - 892 * q^49 - 25 * q^50 + 1335 * q^52 + 1010 * q^53 - 580 * q^55 - 15 * q^56 - 1876 * q^58 - 535 * q^59 + 104 * q^61 + 696 * q^62 - 606 * q^64 - 235 * q^65 + 40 * q^67 - 830 * q^68 + 795 * q^70 + 904 * q^71 - 1420 * q^73 - 1394 * q^74 - 849 * q^76 - 2148 * q^77 + 634 * q^79 + 1270 * q^80 + 1226 * q^82 - 1734 * q^83 + 170 * q^85 - 460 * q^86 + 768 * q^88 - 1704 * q^89 - 2458 * q^91 + 1839 * q^92 - 1751 * q^94 - 25 * q^95 + 76 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.663404	+	1.14905i	−0.234549	+	0.406251i	−0.959141	−	0.282927i	$-0.908695\pi$
$2$	−0.663404	+	1.14905i	−0.234549	+	0.406251i	0.724593	+	0.689177i	$0.242028\pi$
$3$		0			0
$3$		0			0
$4$	3.11979	+	5.40363i	0.389974	+	0.675454i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	12.0521	−	20.8749i	0.650754	−	1.12714i	−0.332186	−	0.943214i	$-0.607786\pi$
$7$	12.0521	−	20.8749i	0.650754	−	1.12714i	0.982940	−	0.183925i	$-0.0588804\pi$
$8$	−18.8932			−0.834969
$9$		0			0
$10$	−6.63404			−0.209787
$11$	4.13810	−	7.16739i	0.113426	−	0.196459i	−0.803724	−	0.595003i	$-0.797151\pi$
$11$	4.13810	−	7.16739i	0.113426	−	0.196459i	0.917149	+	0.398544i	$0.130484\pi$
$12$		0			0
$13$	−43.5573	−	75.4434i	−0.929278	−	1.60956i	−0.784533	−	0.620087i	$-0.787097\pi$
$13$	−43.5573	−	75.4434i	−0.929278	−	1.60956i	−0.144745	−	0.989469i	$-0.546236\pi$
$14$	15.9909	+	27.6970i	0.305267	+	0.528738i
$15$		0			0
$16$	−12.4245	+	21.5198i	−0.194133	+	0.336248i
$17$	−51.9166			−0.740684			−0.370342	−	0.928895i	$-0.620760\pi$
$17$	−51.9166			−0.740684			−0.370342	+	0.928895i	$0.620760\pi$
$18$		0			0
$19$	−88.5107			−1.06872			−0.534362	−	0.845256i	$-0.679448\pi$
$19$	−88.5107			−1.06872			−0.534362	+	0.845256i	$0.679448\pi$
$20$	−15.5989	+	27.0182i	−0.174402	+	0.302072i
$21$		0			0
$22$	5.49046	+	9.50976i	0.0532077	+	0.0921585i
$23$	−64.6224	−	111.929i	−0.585856	−	1.01473i	−0.994768	−	0.102159i	$-0.967425\pi$
$23$	−64.6224	−	111.929i	−0.585856	−	1.01473i	0.408912	−	0.912574i	$-0.365908\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	115.584			0.871844
$27$		0			0
$28$	150.400			1.01511
$29$	−135.554	+	234.787i	−0.867993	+	1.50341i	−0.00394885	+	0.999992i	$0.501257\pi$
$29$	−135.554	+	234.787i	−0.867993	+	1.50341i	−0.864044	+	0.503416i	$0.832076\pi$
$30$		0			0
$31$	−112.273	−	194.463i	−0.650481	−	1.12667i	−0.983006	−	0.183571i	$-0.941234\pi$
$31$	−112.273	−	194.463i	−0.650481	−	1.12667i	0.332526	−	0.943094i	$-0.392099\pi$
$32$	−92.0577	−	159.449i	−0.508552	−	0.880837i
$33$		0			0
$34$	34.4417	−	59.6548i	0.173727	−	0.300903i
$35$	120.521			0.582052
$36$		0			0
$37$	−70.5268			−0.313366			−0.156683	−	0.987649i	$-0.550080\pi$
$37$	−70.5268			−0.313366			−0.156683	+	0.987649i	$0.550080\pi$
$38$	58.7184	−	101.703i	0.250668	−	0.434169i
$39$		0			0
$40$	−47.2330	−	81.8099i	−0.186705	−	0.323382i
$41$	183.469	+	317.778i	0.698855	+	1.21045i	0.968864	+	0.247594i	$0.0796401\pi$
$41$	183.469	+	317.778i	0.698855	+	1.21045i	−0.270009	+	0.962858i	$0.587027\pi$
$42$		0			0
$43$	97.7736	−	169.349i	0.346752	−	0.600592i	−0.638918	−	0.769274i	$-0.720618\pi$
$43$	97.7736	−	169.349i	0.346752	−	0.600592i	0.985670	+	0.168682i	$0.0539512\pi$
$44$	51.6399			0.176932
$45$		0			0
$46$	171.483			0.549648
$47$	−179.596	+	311.069i	−0.557378	+	0.965407i	0.440336	+	0.897833i	$0.354859\pi$
$47$	−179.596	+	311.069i	−0.557378	+	0.965407i	−0.997714	+	0.0675743i	$0.978474\pi$
$48$		0			0
$49$	−119.008	−	206.128i	−0.346962	−	0.600955i
$50$	−16.5851	−	28.7262i	−0.0469098	−	0.0812501i
$51$		0			0
$52$	271.779	−	470.735i	0.724788	−	1.25537i
$53$	−29.4890			−0.0764270			−0.0382135	−	0.999270i	$-0.512167\pi$
$53$	−29.4890			−0.0764270			−0.0382135	+	0.999270i	$0.512167\pi$
$54$		0			0
$55$	41.3810			0.101451
$56$	−227.703	+	394.394i	−0.543360	+	0.941126i
$57$		0			0
$58$	−179.855	−	311.517i	−0.407174	−	0.705245i
$59$	−429.052	−	743.140i	−0.946743	−	1.63981i	−0.752224	−	0.658908i	$-0.771019\pi$
$59$	−429.052	−	743.140i	−0.946743	−	1.63981i	−0.194519	−	0.980899i	$-0.562315\pi$
$60$		0			0
$61$	278.406	−	482.213i	0.584364	−	1.01215i	−0.410591	−	0.911820i	$-0.634677\pi$
$61$	278.406	−	482.213i	0.584364	−	1.01215i	0.994954	−	0.100328i	$-0.0319892\pi$
$62$	297.931			0.610278
$63$		0			0
$64$	45.4941			0.0888557
$65$	217.786	−	377.217i	0.415586	−	0.719815i
$66$		0			0
$67$	20.9493	+	36.2853i	0.0381995	+	0.0661635i	0.884493	−	0.466553i	$-0.154504\pi$
$67$	20.9493	+	36.2853i	0.0381995	+	0.0661635i	−0.846294	+	0.532717i	$0.821171\pi$
$68$	−161.969	−	280.538i	−0.288847	−	0.500298i
$69$		0			0
$70$	−79.9544	+	138.485i	−0.136520	+	0.236459i
$71$	549.163			0.917939			0.458970	−	0.888452i	$-0.348219\pi$
$71$	549.163			0.917939			0.458970	+	0.888452i	$0.348219\pi$
$72$		0			0
$73$	−185.505			−0.297420			−0.148710	−	0.988881i	$-0.547512\pi$
$73$	−185.505			−0.297420			−0.148710	+	0.988881i	$0.547512\pi$
$74$	46.7878	−	81.0388i	0.0734996	−	0.127305i
$75$		0			0
$76$	−276.135	−	478.279i	−0.416774	−	0.721874i
$77$	−99.7458	−	172.765i	−0.147624	−	0.255693i
$78$		0			0
$79$	−40.2456	+	69.7075i	−0.0573163	+	0.0992748i	−0.893260	−	0.449541i	$-0.851588\pi$
$79$	−40.2456	+	69.7075i	−0.0573163	+	0.0992748i	0.835944	+	0.548815i	$0.184921\pi$
$80$	−124.245			−0.173637
$81$		0			0
$82$	−486.857			−0.655663
$83$	−288.377	+	499.483i	−0.381367	+	0.660547i	−0.991258	−	0.131939i	$-0.957880\pi$
$83$	−288.377	+	499.483i	−0.381367	+	0.660547i	0.609891	+	0.792485i	$0.291213\pi$
$84$		0			0
$85$	−129.792	−	224.806i	−0.165622	−	0.286866i
$86$	129.727	+	224.694i	0.162661	+	0.281736i
$87$		0			0
$88$	−78.1818	+	135.415i	−0.0947070	+	0.164037i
$89$	−224.516			−0.267401			−0.133700	−	0.991022i	$-0.542686\pi$
$89$	−224.516			−0.267401			−0.133700	+	0.991022i	$0.542686\pi$
$90$		0			0
$91$	−2099.83			−2.41893
$92$	403.216	−	698.391i	0.456937	−	0.791438i
$93$		0			0
$94$	−238.290	−	412.730i	−0.261465	−	0.452870i
$95$	−221.277	−	383.263i	−0.238974	−	0.413915i
$96$		0			0
$97$	277.508	−	480.658i	0.290481	−	0.503128i	−0.683442	−	0.730005i	$-0.739518\pi$
$97$	277.508	−	480.658i	0.290481	−	0.503128i	0.973924	+	0.226876i	$0.0728513\pi$
$98$	315.801			0.325518
$99$		0			0
$100$	−155.989			−0.155989
$101$	113.500	−	196.587i	0.111818	−	0.193675i	−0.804685	−	0.593702i	$-0.797666\pi$
$101$	113.500	−	196.587i	0.111818	−	0.193675i	0.916503	+	0.400027i	$0.130999\pi$
$102$		0			0
$103$	−191.762	−	332.142i	−0.183446	−	0.317737i	0.759606	−	0.650383i	$-0.225392\pi$
$103$	−191.762	−	332.142i	−0.183446	−	0.317737i	−0.943052	+	0.332646i	$0.892058\pi$
$104$	822.936	+	1425.37i	0.775918	+	1.34393i
$105$		0			0
$106$	19.5632	−	33.8844i	0.0179259	−	0.0310485i
$107$	1775.32			1.60399			0.801993	−	0.597334i	$-0.203773\pi$
$107$	1775.32			1.60399			0.801993	+	0.597334i	$0.203773\pi$
$108$		0			0
$109$	1530.50			1.34491			0.672454	−	0.740139i	$-0.265240\pi$
$109$	1530.50			1.34491			0.672454	+	0.740139i	$0.265240\pi$
$110$	−27.4523	+	47.5488i	−0.0237952	+	0.0412145i
$111$		0			0
$112$	299.483	+	518.720i	0.252665	+	0.437629i
$113$	−420.391	−	728.138i	−0.349974	−	0.606173i	0.636271	−	0.771466i	$-0.280476\pi$
$113$	−420.391	−	728.138i	−0.349974	−	0.606173i	−0.986244	+	0.165293i	$0.947143\pi$
$114$		0			0
$115$	323.112	−	559.646i	0.262003	−	0.453802i
$116$	−1691.60			−1.35398
$117$		0			0
$118$	1138.54			0.888230
$119$	−625.706	+	1083.75i	−0.482003	+	0.834854i
$120$		0			0
$121$	631.252	+	1093.36i	0.474269	+	0.821458i
$122$	369.391	+	639.804i	0.274124	+	0.474796i
$123$		0			0
$124$	700.539	−	1213.37i	0.507341	−	0.878740i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1038.45			0.725572			0.362786	−	0.931873i	$-0.381826\pi$
$127$	1038.45			0.725572			0.362786	+	0.931873i	$0.381826\pi$
$128$	706.281	−	1223.31i	0.487711	−	0.844740i
$129$		0			0
$130$	288.961	+	500.495i	0.194950	+	0.337664i
$131$	−401.705	−	695.774i	−0.267917	−	0.464046i	0.700407	−	0.713744i	$-0.253002\pi$
$131$	−401.705	−	695.774i	−0.267917	−	0.464046i	−0.968324	+	0.249698i	$0.919669\pi$
$132$		0			0
$133$	−1066.74	+	1847.65i	−0.695476	+	1.20460i
$134$	−55.5915			−0.0358386
$135$		0			0
$136$	980.871			0.618449
$137$	434.576	−	752.707i	0.271010	−	0.469402i	−0.698111	−	0.715990i	$-0.745976\pi$
$137$	434.576	−	752.707i	0.271010	−	0.469402i	0.969121	+	0.246587i	$0.0793091\pi$
$138$		0			0
$139$	98.1990	+	170.086i	0.0599218	+	0.103788i	0.894430	−	0.447208i	$-0.147582\pi$
$139$	98.1990	+	170.086i	0.0599218	+	0.103788i	−0.834508	+	0.550995i	$0.814248\pi$
$140$	376.001	+	651.253i	0.226985	+	0.393150i
$141$		0			0
$142$	−364.317	+	631.016i	−0.215302	+	0.372913i
$143$	−720.976			−0.421616
$144$		0			0
$145$	−1355.54			−0.776357
$146$	123.065	−	213.154i	0.0697596	−	0.120827i
$147$		0			0
$148$	−220.029	−	381.101i	−0.122204	−	0.211664i
$149$	377.142	+	653.228i	0.207360	+	0.359158i	0.950882	−	0.309553i	$-0.100180\pi$
$149$	377.142	+	653.228i	0.207360	+	0.359158i	−0.743522	+	0.668711i	$0.766846\pi$
$150$		0			0
$151$	−1028.82	+	1781.96i	−0.554464	+	0.960359i	0.443481	+	0.896284i	$0.353743\pi$
$151$	−1028.82	+	1781.96i	−0.554464	+	0.960359i	−0.997945	+	0.0640756i	$0.979590\pi$
$152$	1672.25			0.892351
$153$		0			0
$154$	264.687			0.138501
$155$	561.367	−	972.316i	0.290904	−	0.503860i
$156$		0			0
$157$	1679.29	+	2908.62i	0.853644	+	1.47855i	0.877897	+	0.478849i	$0.158946\pi$
$157$	1679.29	+	2908.62i	0.853644	+	1.47855i	−0.0242531	+	0.999706i	$0.507721\pi$
$158$	−53.3983	−	92.4885i	−0.0268869	−	0.0465696i
$159$		0			0
$160$	460.289	−	797.243i	0.227431	−	0.393922i
$161$	−3115.35			−1.52499
$162$		0			0
$163$	−710.376			−0.341356			−0.170678	−	0.985327i	$-0.554596\pi$
$163$	−710.376			−0.341356			−0.170678	+	0.985327i	$0.554596\pi$
$164$	−1144.77	+	1982.80i	−0.545070	+	0.944089i
$165$		0			0
$166$	−382.621	−	662.718i	−0.178898	−	0.309861i
$167$	−574.134	−	994.430i	−0.266035	−	0.460786i	0.701799	−	0.712375i	$-0.252380\pi$
$167$	−574.134	−	994.430i	−0.266035	−	0.460786i	−0.967834	+	0.251589i	$0.919047\pi$
$168$		0			0
$169$	−2695.97	+	4669.56i	−1.22711	+	2.12542i
$170$	344.417			0.155386
$171$		0			0
$172$	1220.13			0.540897
$173$	−1462.64	+	2533.37i	−0.642790	+	1.11335i	0.342017	+	0.939694i	$0.388890\pi$
$173$	−1462.64	+	2533.37i	−0.642790	+	1.11335i	−0.984807	+	0.173652i	$0.944443\pi$
$174$		0			0
$175$	301.303	+	521.873i	0.130151	+	0.225428i
$176$	102.827	+	178.102i	0.0440393	+	0.0762782i
$177$		0			0
$178$	148.945	−	257.980i	0.0627185	−	0.108632i
$179$	−3422.02			−1.42890			−0.714452	−	0.699685i	$-0.753324\pi$
$179$	−3422.02			−1.42890			−0.714452	+	0.699685i	$0.753324\pi$
$180$		0			0
$181$	−1151.58			−0.472907			−0.236454	−	0.971643i	$-0.575985\pi$
$181$	−1151.58			−0.472907			−0.236454	+	0.971643i	$0.575985\pi$
$182$	1393.04	−	2412.81i	0.567356	−	0.982690i
$183$		0			0
$184$	1220.92	+	2114.70i	0.489172	+	0.847271i
$185$	−176.317	−	305.390i	−0.0700707	−	0.121366i
$186$		0			0
$187$	−214.836	+	372.107i	−0.0840126	+	0.145514i
$188$	−2241.21			−0.869451
$189$		0			0
$190$	587.184			0.224204
$191$	466.060	−	807.240i	0.176560	−	0.305811i	−0.764140	−	0.645050i	$-0.776836\pi$
$191$	466.060	−	807.240i	0.176560	−	0.305811i	0.940700	+	0.339240i	$0.110170\pi$
$192$		0			0
$193$	2136.41	+	3700.36i	0.796797	+	1.38009i	0.921692	+	0.387923i	$0.126807\pi$
$193$	2136.41	+	3700.36i	0.796797	+	1.38009i	−0.124894	+	0.992170i	$0.539859\pi$
$194$	368.200	+	637.742i	0.136264	+	0.236016i
$195$		0			0
$196$	742.559	−	1286.15i	0.270612	−	0.468713i
$197$	1924.15			0.695888			0.347944	−	0.937515i	$-0.386880\pi$
$197$	1924.15			0.695888			0.347944	+	0.937515i	$0.386880\pi$
$198$		0			0
$199$	1738.84			0.619414			0.309707	−	0.950832i	$-0.399769\pi$
$199$	1738.84			0.619414			0.309707	+	0.950832i	$0.399769\pi$
$200$	236.165	−	409.050i	0.0834969	−	0.144621i
$201$		0			0
$202$	150.593	+	260.834i	0.0524537	+	0.0908525i
$203$	3267.44	+	5659.37i	1.12970	+	1.95670i
$204$		0			0
$205$	−917.345	+	1588.89i	−0.312537	+	0.541331i
$206$	508.864			0.172108
$207$		0			0
$208$	2164.71			0.721613
$209$	−366.266	+	634.391i	−0.121221	+	0.209960i
$210$		0			0
$211$	−1601.17	−	2773.32i	−0.522414	−	0.904848i	−0.999660	−	0.0260782i	$-0.991698\pi$
$211$	−1601.17	−	2773.32i	−0.522414	−	0.904848i	0.477246	−	0.878770i	$-0.341635\pi$
$212$	−91.9996	−	159.348i	−0.0298045	−	0.0516229i
$213$		0			0
$214$	−1177.75	+	2039.93i	−0.376213	+	0.651620i
$215$	977.736			0.310144
$216$		0			0
$217$	−5412.54			−1.69321
$218$	−1015.34	+	1758.62i	−0.315447	+	0.546370i
$219$		0			0
$220$	129.100	+	223.608i	0.0395632	+	0.0685255i
$221$	2261.35	+	3916.77i	0.688301	+	1.19217i
$222$		0			0
$223$	702.494	−	1216.76i	0.210953	−	0.365381i	−0.741060	−	0.671439i	$-0.765677\pi$
$223$	702.494	−	1216.76i	0.210953	−	0.365381i	0.952013	+	0.306058i	$0.0990100\pi$
$224$	−4437.97			−1.32377
$225$		0			0
$226$	1115.56			0.328344
$227$	3119.17	−	5402.57i	0.912012	−	1.57965i	0.100794	−	0.994907i	$-0.467862\pi$
$227$	3119.17	−	5402.57i	0.912012	−	1.57965i	0.811218	−	0.584744i	$-0.198805\pi$
$228$		0			0
$229$	−3315.15	−	5742.01i	−0.956644	−	1.65696i	−0.730560	−	0.682848i	$-0.760741\pi$
$229$	−3315.15	−	5742.01i	−0.956644	−	1.65696i	−0.226084	−	0.974108i	$-0.572592\pi$
$230$	428.708	+	742.543i	0.122905	+	0.212878i
$231$		0			0
$232$	2561.05	−	4435.88i	0.724747	−	1.25530i
$233$	−2453.48			−0.689840			−0.344920	−	0.938632i	$-0.612094\pi$
$233$	−2453.48			−0.689840			−0.344920	+	0.938632i	$0.612094\pi$
$234$		0			0
$235$	−1795.96			−0.498534
$236$	2677.10	−	4636.88i	0.738410	−	1.27896i
$237$		0			0
$238$	−830.192	−	1437.94i	−0.226107	−	0.391628i
$239$	−3474.87	−	6018.65i	−0.940463	−	1.62893i	−0.764590	−	0.644517i	$-0.777058\pi$
$239$	−3474.87	−	6018.65i	−0.940463	−	1.62893i	−0.175873	−	0.984413i	$-0.556275\pi$
$240$		0			0
$241$	−3167.60	+	5486.44i	−0.846651	+	1.46644i	0.0375294	+	0.999296i	$0.488051\pi$
$241$	−3167.60	+	5486.44i	−0.846651	+	1.46644i	−0.884180	+	0.467146i	$0.845282\pi$
$242$	−1675.10			−0.444957
$243$		0			0
$244$	3474.27			0.911546
$245$	595.039	−	1030.64i	0.155166	−	0.268755i
$246$		0			0
$247$	3855.28	+	6677.55i	0.993141	+	1.72017i
$248$	2121.20	+	3674.03i	0.543131	+	0.940731i
$249$		0			0
$250$	82.9255	−	143.631i	0.0209787	−	0.0363362i
$251$	4022.65			1.01158			0.505792	−	0.862656i	$-0.331200\pi$
$251$	4022.65			1.01158			0.505792	+	0.862656i	$0.331200\pi$
$252$		0			0
$253$	−1069.65			−0.265805
$254$	−688.913	+	1193.23i	−0.170182	+	0.294764i
$255$		0			0
$256$	1119.08	+	1938.30i	0.273212	+	0.473217i
$257$	−2498.87	−	4328.17i	−0.606519	−	1.05052i	−0.991809	−	0.127726i	$-0.959232\pi$
$257$	−2498.87	−	4328.17i	−0.606519	−	1.05052i	0.385291	−	0.922795i	$-0.374101\pi$
$258$		0			0
$259$	−849.998	+	1472.24i	−0.203924	+	0.353207i
$260$	2717.79			0.648270
$261$		0			0
$262$	1065.97			0.251359
$263$	−1996.24	+	3457.58i	−0.468035	+	0.810660i	−0.999333	−	0.0365251i	$-0.988371\pi$
$263$	−1996.24	+	3457.58i	−0.468035	+	0.810660i	0.531298	+	0.847185i	$0.321704\pi$
$264$		0			0
$265$	−73.7226	−	127.691i	−0.0170896	−	0.0296000i
$266$	−1415.36	−	2451.48i	−0.326246	−	0.565075i
$267$		0			0
$268$	−130.715	+	226.405i	−0.0297936	+	0.0516041i
$269$	2188.89			0.496131			0.248065	−	0.968743i	$-0.420205\pi$
$269$	2188.89			0.496131			0.248065	+	0.968743i	$0.420205\pi$
$270$		0			0
$271$	4280.26			0.959437			0.479718	−	0.877423i	$-0.340739\pi$
$271$	4280.26			0.959437			0.479718	+	0.877423i	$0.340739\pi$
$272$	645.038	−	1117.24i	0.143791	−	0.249053i
$273$		0			0
$274$	576.599	+	998.699i	0.127130	+	0.220196i
$275$	103.452	+	179.185i	0.0226851	+	0.0392918i
$276$		0			0
$277$	1939.98	−	3360.15i	0.420803	−	0.728852i	−0.575215	−	0.818002i	$-0.695082\pi$
$277$	1939.98	−	3360.15i	0.420803	−	0.728852i	0.996018	+	0.0891502i	$0.0284151\pi$
$278$	−260.583			−0.0562184
$279$		0			0
$280$	−2277.03			−0.485996
$281$	−3198.72	+	5540.34i	−0.679072	+	1.17619i	0.296188	+	0.955130i	$0.404284\pi$
$281$	−3198.72	+	5540.34i	−0.679072	+	1.17619i	−0.975261	+	0.221058i	$0.929049\pi$
$282$		0			0
$283$	171.435	+	296.933i	0.0360096	+	0.0623705i	0.883469	−	0.468490i	$-0.155202\pi$
$283$	171.435	+	296.933i	0.0360096	+	0.0623705i	−0.847459	+	0.530861i	$0.821869\pi$
$284$	1713.27	+	2967.48i	0.357972	+	0.620026i
$285$		0			0
$286$	478.299	−	828.438i	0.0988895	−	0.171282i
$287$	8844.78			1.81913
$288$		0			0
$289$	−2217.66			−0.451387
$290$	899.273	−	1557.59i	0.182094	−	0.315395i
$291$		0			0
$292$	−578.735	−	1002.40i	−0.115986	−	0.200894i
$293$	−3666.71	−	6350.94i	−0.731098	−	1.26630i	−0.956414	−	0.292013i	$-0.905675\pi$
$293$	−3666.71	−	6350.94i	−0.731098	−	1.26630i	0.225316	−	0.974286i	$-0.427659\pi$
$294$		0			0
$295$	2145.26	−	3715.70i	0.423396	−	0.733344i
$296$	1332.48			0.261651
$297$		0			0
$298$	−1000.79			−0.194544
$299$	−5629.55	+	9750.66i	−1.08885	+	1.88594i
$300$		0			0
$301$	−2356.76	−	4082.03i	−0.451300	−	0.781675i
$302$	−1365.04	−	2364.33i	−0.260098	−	0.450502i
$303$		0			0
$304$	1099.70	−	1904.74i	0.207474	−	0.359356i
$305$	2784.06			0.522671
$306$		0			0
$307$	−7965.33			−1.48080			−0.740399	−	0.672167i	$-0.765364\pi$
$307$	−7965.33			−1.48080			−0.740399	+	0.672167i	$0.765364\pi$
$308$	622.372	−	1077.98i	0.115139	−	0.199427i
$309$		0			0
$310$	744.827	+	1290.08i	0.136462	+	0.236360i
$311$	−1093.10	−	1893.30i	−0.199305	−	0.345206i	0.748998	−	0.662572i	$-0.230535\pi$
$311$	−1093.10	−	1893.30i	−0.199305	−	0.345206i	−0.948303	+	0.317366i	$0.897202\pi$
$312$		0			0
$313$	−19.3152	+	33.4549i	−0.00348805	+	0.00604147i	−0.867764	−	0.496976i	$-0.834444\pi$
$313$	−19.3152	+	33.4549i	−0.00348805	+	0.00604147i	0.864276	+	0.503018i	$0.167777\pi$
$314$	−4456.20			−0.800885
$315$		0			0
$316$	−502.232			−0.0894074
$317$	4767.91	−	8258.25i	0.844770	−	1.46319i	−0.0410499	−	0.999157i	$-0.513070\pi$
$317$	4767.91	−	8258.25i	0.844770	−	1.46319i	0.885820	−	0.464028i	$-0.153596\pi$
$318$		0			0
$319$	1121.87	+	1943.14i	0.196905	+	0.341050i
$320$	113.735	+	196.995i	0.0198687	+	0.0344137i
$321$		0			0
$322$	2066.74	−	3579.69i	0.357686	−	0.619530i
$323$	4595.18			0.791587
$324$		0			0
$325$	2177.86			0.371711
$326$	471.267	−	816.258i	0.0800645	−	0.138676i
$327$		0			0
$328$	−3466.32	−	6003.84i	−0.583522	−	1.01069i
$329$	4329.03	+	7498.10i	0.725432	+	1.25649i
$330$		0			0
$331$	1505.29	−	2607.24i	0.249964	−	0.432951i	−0.713551	−	0.700603i	$-0.752914\pi$
$331$	1505.29	−	2607.24i	0.249964	−	0.432951i	0.963516	+	0.267652i	$0.0862477\pi$
$332$	−3598.70			−0.594892
$333$		0			0
$334$	1523.53			0.249593
$335$	−104.747	+	181.427i	−0.0170833	+	0.0295892i
$336$		0			0
$337$	−1406.36	−	2435.89i	−0.227327	−	0.393742i	0.729688	−	0.683780i	$-0.239665\pi$
$337$	−1406.36	−	2435.89i	−0.227327	−	0.393742i	−0.957015	+	0.290038i	$0.906332\pi$
$338$	−3577.04	−	6195.61i	−0.575637	−	0.997032i
$339$		0			0
$340$	809.845	−	1402.69i	0.129176	−	0.223740i
$341$	−1858.39			−0.295125
$342$		0			0
$343$	2530.57			0.398361
$344$	−1847.26	+	3199.54i	−0.289527	+	0.501476i
$345$		0			0
$346$	−1940.65	−	3361.30i	−0.301531	−	0.522268i
$347$	−69.6769	−	120.684i	−0.0107794	−	0.0186705i	0.860585	−	0.509306i	$-0.170098\pi$
$347$	−69.6769	−	120.684i	−0.0107794	−	0.0186705i	−0.871365	+	0.490636i	$0.836765\pi$
$348$		0			0
$349$	2605.14	−	4512.24i	0.399571	−	0.692077i	−0.594102	−	0.804390i	$-0.702493\pi$
$349$	2605.14	−	4512.24i	0.399571	−	0.692077i	0.993673	+	0.112313i	$0.0358259\pi$
$350$	−799.544			−0.122107
$351$		0			0
$352$	−1523.77			−0.230731
$353$	−1705.23	+	2953.55i	−0.257112	+	0.445330i	−0.965467	−	0.260526i	$-0.916104\pi$
$353$	−1705.23	+	2953.55i	−0.257112	+	0.445330i	0.708355	+	0.705856i	$0.249437\pi$
$354$		0			0
$355$	1372.91	+	2377.95i	0.205257	+	0.355516i
$356$	−700.443	−	1213.20i	−0.104279	−	0.180617i
$357$		0			0
$358$	2270.18	−	3932.07i	0.335148	−	0.580493i
$359$	−8131.85			−1.19549			−0.597747	−	0.801685i	$-0.703937\pi$
$359$	−8131.85			−1.19549			−0.597747	+	0.801685i	$0.703937\pi$
$360$		0			0
$361$	975.143			0.142170
$362$	763.963	−	1323.22i	0.110920	−	0.192119i
$363$		0			0
$364$	−6551.03	−	11346.7i	−0.943317	−	1.63387i
$365$	−463.762	−	803.259i	−0.0665052	−	0.115190i
$366$		0			0
$367$	−66.0757	+	114.447i	−0.00939816	+	0.0162781i	−0.870686	−	0.491839i	$-0.836325\pi$
$367$	−66.0757	+	114.447i	−0.00939816	+	0.0162781i	0.861288	+	0.508117i	$0.169658\pi$
$368$	3211.60			0.454935
$369$		0			0
$370$	467.878			0.0657400
$371$	−355.406	+	615.581i	−0.0497352	+	0.0861438i
$372$		0			0
$373$	−4674.44	−	8096.37i	−0.648883	−	1.12390i	−0.983390	−	0.181505i	$-0.941903\pi$
$373$	−4674.44	−	8096.37i	−0.648883	−	1.12390i	0.334507	−	0.942393i	$-0.391430\pi$
$374$	−285.046	−	493.715i	−0.0394101	−	0.0682603i
$375$		0			0
$376$	3393.14	−	5877.10i	0.465394	−	0.806085i
$377$	23617.5			3.22643
$378$		0			0
$379$	6164.17			0.835441			0.417720	−	0.908576i	$-0.362829\pi$
$379$	6164.17			0.835441			0.417720	+	0.908576i	$0.362829\pi$
$380$	1380.67	−	2391.40i	0.186387	−	0.322832i
$381$		0			0
$382$	618.373	+	1071.05i	0.0828238	+	0.143455i
$383$	−1300.64	−	2252.77i	−0.173523	−	0.300551i	0.766126	−	0.642690i	$-0.222182\pi$
$383$	−1300.64	−	2252.77i	−0.173523	−	0.300551i	−0.939649	+	0.342139i	$0.888849\pi$
$384$		0			0
$385$	498.729	−	863.824i	0.0660197	−	0.114349i
$386$	−5669.20			−0.747552
$387$		0			0
$388$	3463.07			0.453120
$389$	−2125.29	+	3681.10i	−0.277008	+	0.479793i	−0.970640	−	0.240537i	$-0.922676\pi$
$389$	−2125.29	+	3681.10i	−0.277008	+	0.479793i	0.693631	+	0.720330i	$0.256010\pi$
$390$		0			0
$391$	3354.98	+	5810.99i	0.433935	+	0.751597i
$392$	2248.44	+	3894.41i	0.289702	+	0.501779i
$393$		0			0
$394$	−1276.49	+	2210.94i	−0.163220	+	0.282705i
$395$	−402.456			−0.0512653
$396$		0			0
$397$	−6088.35			−0.769687			−0.384843	−	0.922982i	$-0.625745\pi$
$397$	−6088.35			−0.769687			−0.384843	+	0.922982i	$0.625745\pi$
$398$	−1153.56	+	1998.02i	−0.145283	+	0.251637i
$399$		0			0
$400$	−310.612	−	537.996i	−0.0388265	−	0.0672495i
$401$	−3862.90	−	6690.73i	−0.481057	−	0.833215i	0.518707	−	0.854952i	$-0.326414\pi$
$401$	−3862.90	−	6690.73i	−0.481057	−	0.833215i	−0.999764	+	0.0217370i	$0.993080\pi$
$402$		0			0
$403$	−9780.64	+	16940.6i	−1.20895	+	2.09397i
$404$	1416.38			0.174425
$405$		0			0
$406$	−8670.53			−1.05988
$407$	−291.847	+	505.493i	−0.0355437	+	0.0615635i
$408$		0			0
$409$	−8160.57	−	14134.5i	−0.986588	−	1.70882i	−0.634657	−	0.772794i	$-0.718859\pi$
$409$	−8160.57	−	14134.5i	−0.986588	−	1.70882i	−0.351930	−	0.936026i	$-0.614475\pi$
$410$	−1217.14	−	2108.15i	−0.146611	−	0.253937i
$411$		0			0
$412$	1196.52	−	2072.43i	0.143078	−	0.247818i
$413$	−20684.0			−2.46439
$414$		0			0
$415$	−2883.77			−0.341105
$416$	−8019.56	+	13890.3i	−0.945172	+	1.63709i
$417$		0			0
$418$	−485.964	−	841.715i	−0.0568644	−	0.0984920i
$419$	−576.228	−	998.055i	−0.0671851	−	0.116368i	0.830476	−	0.557054i	$-0.188068\pi$
$419$	−576.228	−	998.055i	−0.0671851	−	0.116368i	−0.897661	+	0.440686i	$0.854735\pi$
$420$		0			0
$421$	1765.61	−	3058.13i	0.204396	−	0.354024i	−0.745544	−	0.666456i	$-0.767810\pi$
$421$	1765.61	−	3058.13i	0.204396	−	0.354024i	0.949940	+	0.312432i	$0.101144\pi$
$422$	4248.91			0.490127
$423$		0			0
$424$	557.142			0.0638142
$425$	648.958	−	1124.03i	0.0740684	−	0.128290i
$426$		0			0
$427$	−6710.76	−	11623.4i	−0.760554	−	1.31732i
$428$	5538.62	+	9593.17i	0.625512	+	1.08342i
$429$		0			0
$430$	−648.634	+	1123.47i	−0.0727440	+	0.125996i
$431$	10230.6			1.14337			0.571683	−	0.820475i	$-0.306291\pi$
$431$	10230.6			1.14337			0.571683	+	0.820475i	$0.306291\pi$
$432$		0			0
$433$	−7311.31			−0.811453			−0.405726	−	0.913995i	$-0.632981\pi$
$433$	−7311.31			−0.811453			−0.405726	+	0.913995i	$0.632981\pi$
$434$	3590.70	−	6219.27i	0.397141	−	0.687868i
$435$		0			0
$436$	4774.83	+	8270.24i	0.524479	+	0.908424i
$437$	5719.77	+	9906.94i	0.626119	+	1.08447i
$438$		0			0
$439$	−3526.59	+	6108.23i	−0.383405	+	0.664077i	−0.991547	−	0.129751i	$-0.958582\pi$
$439$	−3526.59	+	6108.23i	−0.383405	+	0.664077i	0.608141	+	0.793829i	$0.291915\pi$
$440$	−781.818			−0.0847085
$441$		0			0
$442$	−6000.75			−0.645761
$443$	83.8952	−	145.311i	0.00899770	−	0.0155845i	−0.861491	−	0.507772i	$-0.830469\pi$
$443$	83.8952	−	145.311i	0.00899770	−	0.0155845i	0.870489	+	0.492188i	$0.163803\pi$
$444$		0			0
$445$	−561.290	−	972.183i	−0.0597926	−	0.103564i
$446$	932.076	+	1614.40i	0.0989575	+	0.171399i
$447$		0			0
$448$	548.301	−	949.685i	0.0578232	−	0.100153i
$449$	3949.94			0.415165			0.207582	−	0.978218i	$-0.433440\pi$
$449$	3949.94			0.415165			0.207582	+	0.978218i	$0.433440\pi$
$450$		0			0
$451$	3036.85			0.317072
$452$	2623.06	−	4543.28i	0.272961	−	0.472783i
$453$		0			0
$454$	4138.55	+	7168.17i	0.427823	+	0.741011i
$455$	−5249.58	−	9092.54i	−0.540888	−	0.936846i
$456$		0			0
$457$	3723.80	−	6449.81i	0.381164	−	0.660195i	−0.610065	−	0.792351i	$-0.708857\pi$
$457$	3723.80	−	6449.81i	0.381164	−	0.660195i	0.991229	+	0.132156i	$0.0421900\pi$
$458$	8797.15			0.897519
$459$		0			0
$460$	4032.16			0.408697
$461$	−5443.82	+	9428.98i	−0.549987	+	0.952606i	0.448287	+	0.893889i	$0.352034\pi$
$461$	−5443.82	+	9428.98i	−0.549987	+	0.952606i	−0.998275	+	0.0587164i	$0.981299\pi$
$462$		0			0
$463$	1585.63	+	2746.40i	0.159159	+	0.275672i	0.934566	−	0.355791i	$-0.115788\pi$
$463$	1585.63	+	2746.40i	0.159159	+	0.275672i	−0.775407	+	0.631462i	$0.782455\pi$
$464$	−3368.39	−	5834.21i	−0.337012	−	0.583721i
$465$		0			0
$466$	1627.65	−	2819.17i	0.161801	−	0.280248i
$467$	−16348.5			−1.61996			−0.809978	−	0.586461i	$-0.800521\pi$
$467$	−16348.5			−1.61996			−0.809978	+	0.586461i	$0.800521\pi$
$468$		0			0
$469$	1009.94			0.0994340
$470$	1191.45	−	2063.65i	0.116931	−	0.202530i
$471$		0			0
$472$	8106.17	+	14040.3i	0.790501	+	1.36919i
$473$	−809.193	−	1401.56i	−0.0786612	−	0.136245i
$474$		0			0
$475$	1106.38	−	1916.31i	0.106872	−	0.185108i
$476$	−7808.29			−0.751874
$477$		0			0
$478$	9220.98			0.882338
$479$	−4706.84	+	8152.49i	−0.448980	+	0.777655i	−0.998320	−	0.0579433i	$-0.981546\pi$
$479$	−4706.84	+	8152.49i	−0.448980	+	0.777655i	0.549340	+	0.835599i	$0.314879\pi$
$480$		0			0
$481$	3071.95	+	5320.78i	0.291204	+	0.504380i
$482$	−4202.79	−	7279.45i	−0.397162	−	0.687905i
$483$		0			0
$484$	−3938.75	+	6822.11i	−0.369905	+	0.640694i
$485$	2775.08			0.259814
$486$		0			0
$487$	13482.3			1.25450			0.627250	−	0.778818i	$-0.284180\pi$
$487$	13482.3			1.25450			0.627250	+	0.778818i	$0.284180\pi$
$488$	−5259.97	+	9110.54i	−0.487926	+	0.845112i
$489$		0			0
$490$	789.503	+	1367.46i	0.0727880	+	0.126073i
$491$	4609.87	+	7984.53i	0.423708	+	0.733883i	0.996299	−	0.0859577i	$-0.0273950\pi$
$491$	4609.87	+	7984.53i	0.423708	+	0.733883i	−0.572591	+	0.819841i	$0.694062\pi$
$492$		0			0
$493$	7037.52	−	12189.3i	0.642909	−	1.11355i
$494$	−10230.4			−0.931760
$495$		0			0
$496$	5579.76			0.505118
$497$	6618.59	−	11463.7i	0.597353	−	1.03465i
$498$		0			0
$499$	−52.1731	−	90.3665i	−0.00468054	−	0.00810693i	0.863676	−	0.504048i	$-0.168157\pi$
$499$	−52.1731	−	90.3665i	−0.00468054	−	0.00810693i	−0.868356	+	0.495941i	$0.834823\pi$
$500$	−389.974	−	675.454i	−0.0348803	−	0.0604145i
$501$		0			0
$502$	−2668.64	+	4622.23i	−0.237266	+	0.410956i
$503$	490.652			0.0434933			0.0217466	−	0.999764i	$-0.493077\pi$
$503$	490.652			0.0434933			0.0217466	+	0.999764i	$0.493077\pi$
$504$		0			0
$505$	1135.00			0.100013
$506$	709.613	−	1229.09i	0.0623442	−	0.107983i
$507$		0			0
$508$	3239.75	+	5611.41i	0.282954	+	0.490090i
$509$	1706.40	+	2955.58i	0.148595	+	0.257375i	0.930709	−	0.365762i	$-0.119192\pi$
$509$	1706.40	+	2955.58i	0.148595	+	0.257375i	−0.782113	+	0.623136i	$0.785858\pi$
$510$		0			0
$511$	−2235.73	+	3872.39i	−0.193547	+	0.335234i
$512$	8330.89			0.719095
$513$		0			0
$514$	6631.05			0.569033
$515$	958.811	−	1660.71i	0.0820393	−	0.142096i
$516$		0			0
$517$	1486.37	+	2574.47i	0.126442	+	0.219004i
$518$	−1127.79	−	1953.38i	−0.0956603	−	0.165688i
$519$		0			0
$520$	−4114.68	+	7126.83i	−0.347001	+	0.601024i
$521$	−3486.31			−0.293163			−0.146582	−	0.989199i	$-0.546827\pi$
$521$	−3486.31			−0.293163			−0.146582	+	0.989199i	$0.546827\pi$
$522$		0			0
$523$	14465.6			1.20943			0.604717	−	0.796440i	$-0.293286\pi$
$523$	14465.6			1.20943			0.604717	+	0.796440i	$0.293286\pi$
$524$	2506.47	−	4341.34i	0.208961	−	0.361932i
$525$		0			0
$526$	−2648.62	−	4587.55i	−0.219554	−	0.380279i
$527$	5828.86	+	10095.9i	0.481801	+	0.834503i
$528$		0			0
$529$	−2268.60	+	3929.34i	−0.186456	+	0.322950i
$530$	195.632			0.0160334
$531$		0			0
$532$	−13312.1			−1.08487
$533$	15982.8	−	27683.1i	1.29886	−	2.24969i
$534$		0			0
$535$	4438.29	+	7687.35i	0.358662	+	0.621221i
$536$	−395.800	−	685.545i	−0.0318954	−	0.0552445i
$537$		0			0
$538$	−1452.12	+	2515.15i	−0.116367	+	0.201553i
$539$	−1969.86			−0.157417
$540$		0			0
$541$	6602.78			0.524724			0.262362	−	0.964970i	$-0.415499\pi$
$541$	6602.78			0.524724			0.262362	+	0.964970i	$0.415499\pi$
$542$	−2839.54	+	4918.23i	−0.225035	+	0.389772i
$543$		0			0
$544$	4779.33	+	8278.03i	0.376676	+	0.652422i
$545$	3826.24	+	6627.24i	0.300731	+	0.520881i
$546$		0			0
$547$	−3179.28	+	5506.67i	−0.248512	+	0.430436i	−0.963113	−	0.269096i	$-0.913275\pi$
$547$	−3179.28	+	5506.67i	−0.248512	+	0.430436i	0.714601	+	0.699532i	$0.246608\pi$
$548$	5423.14			0.422746
$549$		0			0
$550$	−274.523			−0.0212831
$551$	11998.0	−	20781.2i	0.927645	−	1.60673i
$552$		0			0
$553$	970.092	+	1680.25i	0.0745976	+	0.129207i
$554$	2573.99	+	4458.28i	0.197398	+	0.341903i
$555$		0			0
$556$	−612.720	+	1061.26i	−0.0467359	+	0.0809489i
$557$	−20782.3			−1.58092			−0.790461	−	0.612513i	$-0.790159\pi$
$557$	−20782.3			−1.58092			−0.790461	+	0.612513i	$0.790159\pi$
$558$		0			0
$559$	−17035.0			−1.28892
$560$	−1497.42	+	2593.60i	−0.112995	+	0.195714i
$561$		0			0
$562$	−4244.08	−	7350.97i	−0.318551	−	0.551747i
$563$	−8234.98	−	14263.4i	−0.616453	−	1.06773i	−0.990128	−	0.140168i	$-0.955236\pi$
$563$	−8234.98	−	14263.4i	−0.616453	−	1.06773i	0.373675	−	0.927560i	$-0.378098\pi$
$564$		0			0
$565$	2101.95	−	3640.69i	0.156513	−	0.271089i
$566$	−454.922			−0.0337841
$567$		0			0
$568$	−10375.4			−0.766451
$569$	−2712.69	+	4698.52i	−0.199863	+	0.346173i	−0.948484	−	0.316825i	$-0.897383\pi$
$569$	−2712.69	+	4698.52i	−0.199863	+	0.346173i	0.748621	+	0.662998i	$0.230716\pi$
$570$		0			0
$571$	7344.86	+	12721.7i	0.538306	+	0.932374i	0.998995	+	0.0448121i	$0.0142689\pi$
$571$	7344.86	+	12721.7i	0.538306	+	0.932374i	−0.460689	+	0.887561i	$0.652398\pi$
$572$	−2249.29	−	3895.89i	−0.164419	−	0.284782i
$573$		0			0
$574$	−5867.66	+	10163.1i	−0.426675	+	0.739023i
$575$	3231.12			0.234343
$576$		0			0
$577$	−17933.1			−1.29387			−0.646935	−	0.762545i	$-0.723949\pi$
$577$	−17933.1			−1.29387			−0.646935	+	0.762545i	$0.723949\pi$
$578$	1471.21	−	2548.21i	0.105872	−	0.183376i
$579$		0			0
$580$	−4229.01	−	7324.86i	−0.302759	−	0.524393i
$581$	6951.11	+	12039.7i	0.496352	+	0.859707i
$582$		0			0
$583$	−122.028	+	211.359i	−0.00866878	+	0.0150148i
$584$	3504.78			0.248337
$585$		0			0
$586$	9730.06			0.685913
$587$	−2400.58	+	4157.92i	−0.168795	+	0.292361i	−0.937996	−	0.346645i	$-0.887321\pi$
$587$	−2400.58	+	4157.92i	−0.168795	+	0.292361i	0.769202	+	0.639006i	$0.220654\pi$
$588$		0			0
$589$	9937.40	+	17212.1i	0.695184	+	1.20409i
$590$	2846.35	+	4930.02i	0.198614	+	0.344010i
$591$		0			0
$592$	876.259	−	1517.73i	0.0608345	−	0.105368i
$593$	−16803.8			−1.16366			−0.581830	−	0.813311i	$-0.697663\pi$
$593$	−16803.8			−1.16366			−0.581830	+	0.813311i	$0.697663\pi$
$594$		0			0
$595$	−6257.06			−0.431117
$596$	−2353.20	+	4075.87i	−0.161730	+	0.280124i
$597$		0			0
$598$	−7469.33	−	12937.3i	−0.510775	−	0.884689i
$599$	8610.08	+	14913.1i	0.587310	+	1.01725i	0.994583	+	0.103944i	$0.0331462\pi$
$599$	8610.08	+	14913.1i	0.587310	+	1.01725i	−0.407274	+	0.913306i	$0.633520\pi$
$600$		0			0
$601$	−10768.3	+	18651.3i	−0.730865	+	1.26590i	0.225649	+	0.974209i	$0.427550\pi$
$601$	−10768.3	+	18651.3i	−0.730865	+	1.26590i	−0.956514	+	0.291687i	$0.905784\pi$
$602$	6253.94			0.423408
$603$		0			0
$604$	−12838.8			−0.864905
$605$	−3156.26	+	5466.81i	−0.212100	+	0.367367i
$606$		0			0
$607$	12025.5	+	20828.7i	0.804117	+	1.39277i	0.916886	+	0.399150i	$0.130695\pi$
$607$	12025.5	+	20828.7i	0.804117	+	1.39277i	−0.112769	+	0.993621i	$0.535972\pi$
$608$	8148.09	+	14112.9i	0.543501	+	0.941372i
$609$		0			0
$610$	−1846.96	+	3199.02i	−0.122592	+	0.212335i
$611$	31290.8			2.07184
$612$		0			0
$613$	−3554.49			−0.234200			−0.117100	−	0.993120i	$-0.537360\pi$
$613$	−3554.49			−0.234200			−0.117100	+	0.993120i	$0.537360\pi$
$614$	5284.23	−	9152.56i	0.347320	−	0.601575i
$615$		0			0
$616$	1884.52	+	3264.08i	0.123262	+	0.213496i
$617$	−4873.71	−	8441.51i	−0.318003	−	0.550798i	0.662068	−	0.749444i	$-0.269679\pi$
$617$	−4873.71	−	8441.51i	−0.318003	−	0.550798i	−0.980071	+	0.198646i	$0.936346\pi$
$618$		0			0
$619$	10382.6	−	17983.2i	0.674171	−	1.16770i	−0.302540	−	0.953137i	$-0.597835\pi$
$619$	10382.6	−	17983.2i	0.674171	−	1.16770i	0.976711	−	0.214561i	$-0.0688321\pi$
$620$	7005.39			0.453779
$621$		0			0
$622$	2900.66			0.186987
$623$	−2705.90	+	4686.75i	−0.174012	+	0.301398i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	−25.6275	−	44.3882i	−0.00163623	−	0.00283404i
$627$		0			0
$628$	−10478.1	+	18148.6i	−0.665797	+	1.15320i
$629$	3661.51			0.232105
$630$		0			0
$631$	23740.6			1.49778			0.748890	−	0.662694i	$-0.230587\pi$
$631$	23740.6			1.49778			0.748890	+	0.662694i	$0.230587\pi$
$632$	760.369	−	1317.00i	0.0478574	−	0.0828914i
$633$		0			0
$634$	6326.10	+	10957.1i	0.396280	+	0.686377i
$635$	2596.13	+	4496.62i	0.162243	+	0.281013i
$636$		0			0
$637$	−10367.3	+	17956.7i	−0.644848	+	1.11691i
$638$	−2977.02			−0.184736
$639$		0			0
$640$	7062.81			0.436222
$641$	13054.4	−	22610.8i	0.804394	−	1.39325i	−0.112306	−	0.993674i	$-0.535824\pi$
$641$	13054.4	−	22610.8i	0.804394	−	1.39325i	0.916700	−	0.399577i	$-0.130843\pi$
$642$		0			0
$643$	12008.9	+	20800.1i	0.736526	+	1.27570i	0.954051	+	0.299646i	$0.0968685\pi$
$643$	12008.9	+	20800.1i	0.736526	+	1.27570i	−0.217525	+	0.976055i	$0.569798\pi$
$644$	−9719.24	−	16834.2i	−0.594707	−	1.03006i
$645$		0			0
$646$	−3048.46	+	5280.09i	−0.185666	+	0.321582i
$647$	−18588.3			−1.12949			−0.564747	−	0.825264i	$-0.691026\pi$
$647$	−18588.3			−1.12949			−0.564747	+	0.825264i	$0.691026\pi$
$648$		0			0
$649$	−7101.83			−0.429540
$650$	−1444.80	+	2502.47i	−0.0871844	+	0.151008i
$651$		0			0
$652$	−2216.22	−	3838.61i	−0.133120	−	0.230570i
$653$	6085.97	+	10541.2i	0.364721	+	0.631714i	0.988731	−	0.149701i	$-0.0478312\pi$
$653$	6085.97	+	10541.2i	0.364721	+	0.631714i	−0.624011	+	0.781416i	$0.714498\pi$
$654$		0			0
$655$	2008.53	−	3478.87i	0.119816	−	0.207528i
$656$	−9118.04			−0.542682
$657$		0			0
$658$	−11487.6			−0.680597
$659$	4294.82	−	7438.84i	0.253873	−	0.439721i	−0.710716	−	0.703479i	$-0.751629\pi$
$659$	4294.82	−	7438.84i	0.253873	−	0.439721i	0.964589	+	0.263758i	$0.0849621\pi$
$660$		0			0
$661$	−12497.7	−	21646.6i	−0.735405	−	1.27376i	−0.954545	−	0.298066i	$-0.903659\pi$
$661$	−12497.7	−	21646.6i	−0.735405	−	1.27376i	0.219140	−	0.975693i	$-0.429675\pi$
$662$	1997.23	+	3459.31i	0.117258	+	0.203096i
$663$		0			0
$664$	5448.36	−	9436.83i	0.318430	−	0.551536i
$665$	−10667.4			−0.622053
$666$		0			0
$667$	35039.4			2.03408
$668$	3582.36	−	6204.82i	0.207493	−	0.359389i
$669$		0			0
$670$	−138.979	−	240.718i	−0.00801376	−	0.0138802i
$671$	−2304.14	−	3990.88i	−0.132564	−	0.229607i
$672$		0			0
$673$	1270.08	−	2199.85i	0.0727461	−	0.126000i	−0.827358	−	0.561675i	$-0.810157\pi$
$673$	1270.08	−	2199.85i	0.0727461	−	0.126000i	0.900104	+	0.435675i	$0.143490\pi$
$674$	3731.94			0.213277
$675$		0			0
$676$	−33643.4			−1.91417
$677$	2098.89	−	3635.39i	0.119154	−	0.206380i	−0.800279	−	0.599628i	$-0.795315\pi$
$677$	2098.89	−	3635.39i	0.119154	−	0.206380i	0.919433	+	0.393248i	$0.128649\pi$
$678$		0			0
$679$	−6689.13	−	11585.9i	−0.378064	−	0.654826i
$680$	2452.18	+	4247.30i	0.138289	+	0.239524i
$681$		0			0
$682$	1232.87	−	2135.39i	0.0692212	−	0.119895i
$683$	−8523.02			−0.477488			−0.238744	−	0.971083i	$-0.576736\pi$
$683$	−8523.02			−0.477488			−0.238744	+	0.971083i	$0.576736\pi$
$684$		0			0
$685$	4345.76			0.242398
$686$	−1678.79	+	2907.75i	−0.0934352	+	0.161834i
$687$		0			0
$688$	2429.57	+	4208.15i	0.134632	+	0.233189i
$689$	1284.46	+	2224.75i	0.0710219	+	0.123014i
$690$		0			0
$691$	10146.0	−	17573.4i	0.558571	−	0.967473i	−0.439045	−	0.898465i	$-0.644683\pi$
$691$	10146.0	−	17573.4i	0.558571	−	0.967473i	0.997616	−	0.0690084i	$-0.0219835\pi$
$692$	−18252.6			−1.00269
$693$		0			0
$694$	184.896			0.0101132
$695$	−490.995	+	850.428i	−0.0267978	+	0.0464152i
$696$		0			0
$697$	−9525.10	−	16498.0i	−0.517631	−	0.896563i
$698$	3456.53	+	5986.88i	0.187438	+	0.324652i
$699$		0			0
$700$	−1880.01	+	3256.27i	−0.101511	+	0.175822i
$701$	−11223.6			−0.604721			−0.302361	−	0.953194i	$-0.597775\pi$
$701$	−11223.6			−0.604721			−0.302361	+	0.953194i	$0.597775\pi$
$702$		0			0
$703$	6242.38			0.334901
$704$	188.259	−	326.074i	0.0100785	−	0.0174565i
$705$		0			0
$706$	−2262.52	−	3918.79i	−0.120610	−	0.208903i
$707$	−2735.83	−	4738.60i	−0.145533	−	0.252070i
$708$		0			0
$709$	9965.49	−	17260.7i	0.527873	−	0.914302i	−0.471600	−	0.881813i	$-0.656323\pi$
$709$	9965.49	−	17260.7i	0.527873	−	0.914302i	0.999472	−	0.0324893i	$-0.0103435\pi$
$710$	−3643.17			−0.192572
$711$		0			0
$712$	4241.83			0.223271
$713$	−14510.7	+	25133.4i	−0.762177	+	1.32013i
$714$		0			0
$715$	−1802.44	−	3121.92i	−0.0942762	−	0.163291i
$716$	−10676.0	−	18491.3i	−0.557235	−	0.965159i
$717$		0			0
$718$	5394.70	−	9343.90i	0.280402	−	0.485670i
$719$	−3186.13			−0.165261			−0.0826305	−	0.996580i	$-0.526332\pi$
$719$	−3186.13			−0.165261			−0.0826305	+	0.996580i	$0.526332\pi$
$720$		0			0
$721$	−9244.58			−0.477512
$722$	−646.914	+	1120.49i	−0.0333458	+	0.0577566i
$723$		0			0
$724$	−3592.69	−	6222.72i	−0.184421	−	0.319427i
$725$	−3388.86	−	5869.67i	−0.173599	−	0.300682i
$726$		0			0
$727$	11544.1	−	19995.0i	0.588923	−	1.02004i	−0.405451	−	0.914117i	$-0.632886\pi$
$727$	11544.1	−	19995.0i	0.588923	−	1.02004i	0.994374	−	0.105927i	$-0.0337810\pi$
$728$	39672.5			2.01973
$729$		0			0
$730$	1230.65			0.0623949
$731$	−5076.08	+	8792.02i	−0.256834	+	0.444849i
$732$		0			0
$733$	14900.7	+	25808.8i	0.750846	+	1.30050i	0.947413	+	0.320013i	$0.103687\pi$
$733$	14900.7	+	25808.8i	0.750846	+	1.30050i	−0.196567	+	0.980490i	$0.562979\pi$
$734$	−87.6698	−	151.849i	−0.00440866	−	0.00763602i
$735$		0			0
$736$	−11898.0	+	20607.9i	−0.595877	+	1.03209i
$737$	346.761			0.0173312
$738$		0			0
$739$	39741.5			1.97823			0.989117	−	0.147134i	$-0.0470047\pi$
$739$	39741.5			1.97823			0.989117	+	0.147134i	$0.0470047\pi$
$740$	1100.14	−	1905.50i	0.0546515	−	0.0946591i
$741$		0			0
$742$	−471.555	−	816.758i	−0.0233307	−	0.0404099i
$743$	−4856.77	−	8412.17i	−0.239808	−	0.415360i	0.720851	−	0.693090i	$-0.243751\pi$
$743$	−4856.77	−	8412.17i	−0.239808	−	0.415360i	−0.960659	+	0.277730i	$0.910418\pi$
$744$		0			0
$745$	−1885.71	+	3266.14i	−0.0927342	+	0.160620i
$746$	12404.2			0.608779
$747$		0			0
$748$	−2680.97			−0.131051
$749$	21396.4	−	37059.6i	1.04380	−	1.80791i
$750$		0			0
$751$	1254.64	+	2173.10i	0.0609619	+	0.105589i	0.894896	−	0.446275i	$-0.147250\pi$
$751$	1254.64	+	2173.10i	0.0609619	+	0.105589i	−0.833934	+	0.551865i	$0.813917\pi$
$752$	−4462.78	−	7729.76i	−0.216411	−	0.374834i
$753$		0			0
$754$	−15667.9	+	27137.7i	−0.756755	+	1.31074i
$755$	−10288.2			−0.495927
$756$		0			0
$757$	9705.73			0.465998			0.232999	−	0.972477i	$-0.425146\pi$
$757$	9705.73			0.465998			0.232999	+	0.972477i	$0.425146\pi$
$758$	−4089.34	+	7082.94i	−0.195952	+	0.339398i
$759$		0			0
$760$	4180.62	+	7241.05i	0.199536	+	0.345606i
$761$	3889.22	+	6736.33i	0.185262	+	0.320883i	0.943665	−	0.330903i	$-0.107353\pi$
$761$	3889.22	+	6736.33i	0.185262	+	0.320883i	−0.758403	+	0.651786i	$0.774020\pi$
$762$		0			0
$763$	18445.7	−	31949.0i	0.875204	−	1.51590i
$764$	5816.04			0.275415
$765$		0			0
$766$	3451.39			0.162799
$767$	−37376.7	+	64738.3i	−1.75957	+	3.04767i
$768$		0			0
$769$	−693.951	−	1201.96i	−0.0325416	−	0.0563638i	0.849296	−	0.527917i	$-0.177027\pi$
$769$	−693.951	−	1201.96i	−0.0325416	−	0.0563638i	−0.881838	+	0.471553i	$0.843693\pi$
$770$	661.718	+	1146.13i	0.0309697	+	0.0536410i
$771$		0			0
$772$	−13330.3	+	23088.7i	−0.621460	+	1.07640i
$773$	20692.0			0.962796			0.481398	−	0.876502i	$-0.340129\pi$
$773$	20692.0			0.962796			0.481398	+	0.876502i	$0.340129\pi$
$774$		0			0
$775$	5613.67			0.260192
$776$	−5243.02	+	9081.17i	−0.242543	+	0.420097i
$777$		0			0
$778$	−2819.85	−	4884.12i	−0.129944	−	0.225070i
$779$	−16239.0	−	28126.7i	−0.746883	−	1.29364i
$780$		0			0
$781$	2272.49	−	3936.07i	0.104118	−	0.180338i
$782$	−8902.82			−0.407115
$783$		0			0
$784$	5914.45			0.269426
$785$	−8396.46	+	14543.1i	−0.381761	+	0.661230i
$786$		0

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.663404 + 1.14905i) q^{2} +(3.11979 + 5.40363i) q^{4} +(2.50000 + 4.33013i) q^{5} +(12.0521 - 20.8749i) q^{7} -18.8932 q^{8} +O(q^{10})\)	\(q+(-0.663404 + 1.14905i) q^{2} +(3.11979 + 5.40363i) q^{4} +(2.50000 + 4.33013i) q^{5} +(12.0521 - 20.8749i) q^{7} -18.8932 q^{8} -6.63404 q^{10} +(4.13810 - 7.16739i) q^{11} +(-43.5573 - 75.4434i) q^{13} +(15.9909 + 27.6970i) q^{14} +(-12.4245 + 21.5198i) q^{16} -51.9166 q^{17} -88.5107 q^{19} +(-15.5989 + 27.0182i) q^{20} +(5.49046 + 9.50976i) q^{22} +(-64.6224 - 111.929i) q^{23} +(-12.5000 + 21.6506i) q^{25} +115.584 q^{26} +150.400 q^{28} +(-135.554 + 234.787i) q^{29} +(-112.273 - 194.463i) q^{31} +(-92.0577 - 159.449i) q^{32} +(34.4417 - 59.6548i) q^{34} +120.521 q^{35} -70.5268 q^{37} +(58.7184 - 101.703i) q^{38} +(-47.2330 - 81.8099i) q^{40} +(183.469 + 317.778i) q^{41} +(97.7736 - 169.349i) q^{43} +51.6399 q^{44} +171.483 q^{46} +(-179.596 + 311.069i) q^{47} +(-119.008 - 206.128i) q^{49} +(-16.5851 - 28.7262i) q^{50} +(271.779 - 470.735i) q^{52} -29.4890 q^{53} +41.3810 q^{55} +(-227.703 + 394.394i) q^{56} +(-179.855 - 311.517i) q^{58} +(-429.052 - 743.140i) q^{59} +(278.406 - 482.213i) q^{61} +297.931 q^{62} +45.4941 q^{64} +(217.786 - 377.217i) q^{65} +(20.9493 + 36.2853i) q^{67} +(-161.969 - 280.538i) q^{68} +(-79.9544 + 138.485i) q^{70} +549.163 q^{71} -185.505 q^{73} +(46.7878 - 81.0388i) q^{74} +(-276.135 - 478.279i) q^{76} +(-99.7458 - 172.765i) q^{77} +(-40.2456 + 69.7075i) q^{79} -124.245 q^{80} -486.857 q^{82} +(-288.377 + 499.483i) q^{83} +(-129.792 - 224.806i) q^{85} +(129.727 + 224.694i) q^{86} +(-78.1818 + 135.415i) q^{88} -224.516 q^{89} -2099.83 q^{91} +(403.216 - 698.391i) q^{92} +(-238.290 - 412.730i) q^{94} +(-221.277 - 383.263i) q^{95} +(277.508 - 480.658i) q^{97} +315.801 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8}+O(q^{10}) \) 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8	\( 6 q - q^{2} - 5 q^{4} + 15 q^{5} + 25 q^{7} - 54 q^{8} - 10 q^{10} - 58 q^{11} + 47 q^{13} - 159 q^{14} + 127 q^{16} + 68 q^{17} - 10 q^{19} + 25 q^{20} - 260 q^{22} + 51 q^{23} - 75 q^{25} + 506 q^{26} + 166 q^{28} - 350 q^{29} - 638 q^{31} - 245 q^{32} + 154 q^{34} + 250 q^{35} - 828 q^{37} - 397 q^{38} - 135 q^{40} - 179 q^{41} + 836 q^{43} + 664 q^{44} + 522 q^{46} - 235 q^{47} - 892 q^{49} - 25 q^{50} + 1335 q^{52} + 1010 q^{53} - 580 q^{55} - 15 q^{56} - 1876 q^{58} - 535 q^{59} + 104 q^{61} + 696 q^{62} - 606 q^{64} - 235 q^{65} + 40 q^{67} - 830 q^{68} + 795 q^{70} + 904 q^{71} - 1420 q^{73} - 1394 q^{74} - 849 q^{76} - 2148 q^{77} + 634 q^{79} + 1270 q^{80} + 1226 q^{82} - 1734 q^{83} + 170 q^{85} - 460 q^{86} + 768 q^{88} - 1704 q^{89} - 2458 q^{91} + 1839 q^{92} - 1751 q^{94} - 25 q^{95} + 76 q^{98}+O(q^{100}) \) 6 * q - q^2 - 5 * q^4 + 15 * q^5 + 25 * q^7 - 54 * q^8 - 10 * q^10 - 58 * q^11 + 47 * q^13 - 159 * q^14 + 127 * q^16 + 68 * q^17 - 10 * q^19 + 25 * q^20 - 260 * q^22 + 51 * q^23 - 75 * q^25 + 506 * q^26 + 166 * q^28 - 350 * q^29 - 638 * q^31 - 245 * q^32 + 154 * q^34 + 250 * q^35 - 828 * q^37 - 397 * q^38 - 135 * q^40 - 179 * q^41 + 836 * q^43 + 664 * q^44 + 522 * q^46 - 235 * q^47 - 892 * q^49 - 25 * q^50 + 1335 * q^52 + 1010 * q^53 - 580 * q^55 - 15 * q^56 - 1876 * q^58 - 535 * q^59 + 104 * q^61 + 696 * q^62 - 606 * q^64 - 235 * q^65 + 40 * q^67 - 830 * q^68 + 795 * q^70 + 904 * q^71 - 1420 * q^73 - 1394 * q^74 - 849 * q^76 - 2148 * q^77 + 634 * q^79 + 1270 * q^80 + 1226 * q^82 - 1734 * q^83 + 170 * q^85 - 460 * q^86 + 768 * q^88 - 1704 * q^89 - 2458 * q^91 + 1839 * q^92 - 1751 * q^94 - 25 * q^95 + 76 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more