LMFDB - Embedded newform 45.4.f.a.8.2

Newspace parameters

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$ x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 $ x^12 - 16x^10 - 14x^8 - 512x^6 + 3889x^4 + 126224*x^2 + 506944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.2
Root			$-0.347140 + 2.27426i$ of defining polynomial
Character	$\chi$	$=$	45.8
Dual form			45.4.f.a.17.2

$q$-expansion

$f(q)$	$=$	$q+(-2.62140 - 2.62140i) q^{2} +5.74350i q^{4} +(-8.38994 - 7.38978i) q^{5} +(-10.8652 + 10.8652i) q^{7} +(-5.91519 + 5.91519i) q^{8} +O(q^{10})$	\(q+(-2.62140 - 2.62140i) q^{2} +5.74350i q^{4} +(-8.38994 - 7.38978i) q^{5} +(-10.8652 + 10.8652i) q^{7} +(-5.91519 + 5.91519i) q^{8} +(2.62183 + 41.3650i) q^{10} +37.8905i q^{11} +(-48.1085 - 48.1085i) q^{13} +56.9640 q^{14} +76.9602 q^{16} +(-60.3458 - 60.3458i) q^{17} -109.678i q^{19} +(42.4432 - 48.1877i) q^{20} +(99.3262 - 99.3262i) q^{22} +(39.7660 - 39.7660i) q^{23} +(15.7823 + 124.000i) q^{25} +252.224i q^{26} +(-62.4042 - 62.4042i) q^{28} +90.3553 q^{29} -233.678 q^{31} +(-154.422 - 154.422i) q^{32} +316.381i q^{34} +(171.449 - 10.8670i) q^{35} +(-19.0042 + 19.0042i) q^{37} +(-287.509 + 287.509i) q^{38} +(93.3400 - 5.91616i) q^{40} -260.783i q^{41} +(176.216 + 176.216i) q^{43} -217.624 q^{44} -208.485 q^{46} +(-145.788 - 145.788i) q^{47} +106.896i q^{49} +(283.681 - 366.425i) q^{50} +(276.311 - 276.311i) q^{52} +(-183.285 + 183.285i) q^{53} +(280.002 - 317.899i) q^{55} -128.539i q^{56} +(-236.858 - 236.858i) q^{58} +279.564 q^{59} -390.267 q^{61} +(612.563 + 612.563i) q^{62} +193.924i q^{64} +(48.1164 + 759.139i) q^{65} +(150.444 - 150.444i) q^{67} +(346.596 - 346.596i) q^{68} +(-477.925 - 420.951i) q^{70} +470.042i q^{71} +(480.765 + 480.765i) q^{73} +99.6355 q^{74} +629.934 q^{76} +(-411.687 - 411.687i) q^{77} -1322.44i q^{79} +(-645.692 - 568.719i) q^{80} +(-683.618 + 683.618i) q^{82} +(456.192 - 456.192i) q^{83} +(60.3558 + 952.240i) q^{85} -923.868i q^{86} +(-224.129 - 224.129i) q^{88} -1364.85 q^{89} +1045.41 q^{91} +(228.396 + 228.396i) q^{92} +764.340i q^{94} +(-810.494 + 920.190i) q^{95} +(-785.308 + 785.308i) q^{97} +(280.218 - 280.218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−2.62140	−	2.62140i	−0.926806	−	0.926806i	0.0706924	−	0.997498i	$-0.477479\pi$
$2$	−2.62140	−	2.62140i	−0.926806	−	0.926806i	−0.997498	+	0.0706924i	$0.977479\pi$
$3$		0			0
$3$		0			0
$4$			5.74350i			0.717938i
$5$	−8.38994	−	7.38978i	−0.750419	−	0.660962i
$5$	−8.38994	−	7.38978i	−0.750419	−	0.660962i
$6$		0			0
$7$	−10.8652	+	10.8652i	−0.586664	+	0.586664i	−0.936726	−	0.350062i	$-0.886160\pi$
$7$	−10.8652	+	10.8652i	−0.586664	+	0.586664i	0.350062	+	0.936726i	$0.386160\pi$
$8$	−5.91519	+	5.91519i	−0.261417	+	0.261417i
$9$		0			0
$10$	2.62183	+	41.3650i	0.0829097	+	1.30808i
$11$			37.8905i			1.03858i	0.854597	+	0.519291i	$0.173804\pi$
$11$			37.8905i			1.03858i	−0.854597	+	0.519291i	$0.826196\pi$
$12$		0			0
$13$	−48.1085	−	48.1085i	−1.02638	−	1.02638i	−0.999643	−	0.0267344i	$-0.991489\pi$
$13$	−48.1085	−	48.1085i	−1.02638	−	1.02638i	−0.0267344	−	0.999643i	$-0.508511\pi$
$14$	56.9640			1.08745
$15$		0			0
$16$	76.9602			1.20250
$17$	−60.3458	−	60.3458i	−0.860942	−	0.860942i	0.130506	−	0.991448i	$-0.458340\pi$
$17$	−60.3458	−	60.3458i	−0.860942	−	0.860942i	−0.991448	+	0.130506i	$0.958340\pi$
$18$		0			0
$19$		−	109.678i		−	1.32430i	−0.749369	−	0.662152i	$-0.769643\pi$
$19$		−	109.678i		−	1.32430i	0.749369	−	0.662152i	$-0.230357\pi$
$20$	42.4432	−	48.1877i	0.474530	−	0.538755i
$21$		0			0
$22$	99.3262	−	99.3262i	0.962564	−	0.962564i
$23$	39.7660	−	39.7660i	0.360512	−	0.360512i	−0.503489	−	0.864002i	$-0.667951\pi$
$23$	39.7660	−	39.7660i	0.360512	−	0.360512i	0.864002	+	0.503489i	$0.167951\pi$
$24$		0			0
$25$	15.7823	+	124.000i	0.126259	+	0.991997i
$26$			252.224i			1.90250i
$27$		0			0
$28$	−62.4042	−	62.4042i	−0.421188	−	0.421188i
$29$	90.3553			0.578571			0.289285	−	0.957243i	$-0.406582\pi$
$29$	90.3553			0.578571			0.289285	+	0.957243i	$0.406582\pi$
$30$		0			0
$31$	−233.678			−1.35386			−0.676932	−	0.736046i	$-0.736691\pi$
$31$	−233.678			−1.35386			−0.676932	+	0.736046i	$0.736691\pi$
$32$	−154.422	−	154.422i	−0.853070	−	0.853070i
$33$		0			0
$34$			316.381i			1.59585i
$35$	171.449	−	10.8670i	0.828007	−	0.0524815i
$36$		0			0
$37$	−19.0042	+	19.0042i	−0.0844399	+	0.0844399i	−0.748065	−	0.663625i	$-0.769017\pi$
$37$	−19.0042	+	19.0042i	−0.0844399	+	0.0844399i	0.663625	+	0.748065i	$0.269017\pi$
$38$	−287.509	+	287.509i	−1.22737	+	1.22737i
$39$		0			0
$40$	93.3400	−	5.91616i	0.368959	−	0.0233857i
$41$		−	260.783i		−	0.993354i	−0.867935	−	0.496677i	$-0.834553\pi$
$41$		−	260.783i		−	0.993354i	0.867935	−	0.496677i	$-0.165447\pi$
$42$		0			0
$43$	176.216	+	176.216i	0.624948	+	0.624948i	0.946792	−	0.321845i	$-0.104303\pi$
$43$	176.216	+	176.216i	0.624948	+	0.624948i	−0.321845	+	0.946792i	$0.604303\pi$
$44$	−217.624			−0.745638
$45$		0			0
$46$	−208.485			−0.668250
$47$	−145.788	−	145.788i	−0.452456	−	0.452456i	0.443713	−	0.896169i	$-0.353661\pi$
$47$	−145.788	−	145.788i	−0.452456	−	0.452456i	−0.896169	+	0.443713i	$0.853661\pi$
$48$		0			0
$49$			106.896i			0.311650i
$50$	283.681	−	366.425i	0.802372	−	1.03641i
$51$		0			0
$52$	276.311	−	276.311i	0.736875	−	0.736875i
$53$	−183.285	+	183.285i	−0.475022	+	0.475022i	−0.903535	−	0.428513i	$-0.859038\pi$
$53$	−183.285	+	183.285i	−0.475022	+	0.475022i	0.428513	+	0.903535i	$0.359038\pi$
$54$		0			0
$55$	280.002	−	317.899i	0.686464	−	0.779373i
$56$		−	128.539i		−	0.306728i
$57$		0			0
$58$	−236.858	−	236.858i	−0.536223	−	0.536223i
$59$	279.564			0.616883			0.308441	−	0.951243i	$-0.400193\pi$
$59$	279.564			0.616883			0.308441	+	0.951243i	$0.400193\pi$
$60$		0			0
$61$	−390.267			−0.819157			−0.409578	−	0.912275i	$-0.634324\pi$
$61$	−390.267			−0.819157			−0.409578	+	0.912275i	$0.634324\pi$
$62$	612.563	+	612.563i	1.25477	+	1.25477i
$63$		0			0
$64$			193.924i			0.378757i
$65$	48.1164	+	759.139i	0.0918170	+	1.44861i
$66$		0			0
$67$	150.444	−	150.444i	0.274324	−	0.274324i	−0.556514	−	0.830838i	$-0.687861\pi$
$67$	150.444	−	150.444i	0.274324	−	0.274324i	0.830838	+	0.556514i	$0.187861\pi$
$68$	346.596	−	346.596i	0.618103	−	0.618103i
$69$		0			0
$70$	−477.925	−	420.951i	−0.816042	−	0.718761i
$71$			470.042i			0.785686i	0.919605	+	0.392843i	$0.128508\pi$
$71$			470.042i			0.785686i	−0.919605	+	0.392843i	$0.871492\pi$
$72$		0			0
$73$	480.765	+	480.765i	0.770812	+	0.770812i	0.978249	−	0.207436i	$-0.0665119\pi$
$73$	480.765	+	480.765i	0.770812	+	0.770812i	−0.207436	+	0.978249i	$0.566512\pi$
$74$	99.6355			0.156519
$75$		0			0
$76$	629.934			0.950769
$77$	−411.687	−	411.687i	−0.609299	−	0.609299i
$78$		0			0
$79$		−	1322.44i		−	1.88337i	−0.336497	−	0.941685i	$-0.609242\pi$
$79$		−	1322.44i		−	1.88337i	0.336497	−	0.941685i	$-0.390758\pi$
$80$	−645.692	−	568.719i	−0.902382	−	0.794809i
$81$		0			0
$82$	−683.618	+	683.618i	−0.920646	+	0.920646i
$83$	456.192	−	456.192i	0.603296	−	0.603296i	−0.337890	−	0.941186i	$-0.609713\pi$
$83$	456.192	−	456.192i	0.603296	−	0.603296i	0.941186	+	0.337890i	$0.109713\pi$
$84$		0			0
$85$	60.3558	+	952.240i	0.0770177	+	1.21512i
$86$		−	923.868i		−	1.15841i
$87$		0			0
$88$	−224.129	−	224.129i	−0.271503	−	0.271503i
$89$	−1364.85			−1.62555			−0.812774	−	0.582580i	$-0.802043\pi$
$89$	−1364.85			−1.62555			−0.812774	+	0.582580i	$0.802043\pi$
$90$		0			0
$91$	1045.41			1.20428
$92$	228.396	+	228.396i	0.258825	+	0.258825i
$93$		0			0
$94$			764.340i			0.838677i
$95$	−810.494	+	920.190i	−0.875315	+	0.993784i
$96$		0			0
$97$	−785.308	+	785.308i	−0.822020	+	0.822020i	−0.986398	−	0.164377i	$-0.947439\pi$
$97$	−785.308	+	785.308i	−0.822020	+	0.822020i	0.164377	+	0.986398i	$0.447439\pi$
$98$	280.218	−	280.218i	0.288839	−	0.288839i
$99$		0			0
$100$	−712.193	+	90.6458i	−0.712193	+	0.0906458i
$101$			111.555i			0.109902i	0.998489	+	0.0549510i	$0.0175003\pi$
$101$			111.555i			0.109902i	−0.998489	+	0.0549510i	$0.982500\pi$
$102$		0			0
$103$	13.8820	+	13.8820i	0.0132800	+	0.0132800i	0.713716	−	0.700436i	$-0.247011\pi$
$103$	13.8820	+	13.8820i	0.0132800	+	0.0132800i	−0.700436	+	0.713716i	$0.747011\pi$
$104$	569.142			0.536624
$105$		0			0
$106$	960.929			0.880507
$107$	776.165	+	776.165i	0.701259	+	0.701259i	0.964681	−	0.263422i	$-0.0848510\pi$
$107$	776.165	+	776.165i	0.701259	+	0.701259i	−0.263422	+	0.964681i	$0.584851\pi$
$108$		0			0
$109$		−	410.579i		−	0.360792i	−0.983594	−	0.180396i	$-0.942262\pi$
$109$		−	410.579i		−	0.360792i	0.983594	−	0.180396i	$-0.0577379\pi$
$110$	−1567.34	+	99.3425i	−1.35855	+	0.0861085i
$111$		0			0
$112$	−836.186	+	836.186i	−0.705465	+	0.705465i
$113$	521.328	−	521.328i	0.434004	−	0.434004i	−0.455984	−	0.889988i	$-0.650713\pi$
$113$	521.328	−	521.328i	0.434004	−	0.434004i	0.889988	+	0.455984i	$0.150713\pi$
$114$		0			0
$115$	−627.496	+	39.7725i	−0.508820	+	0.0322505i
$116$			518.956i			0.415378i
$117$		0			0
$118$	−732.849	−	732.849i	−0.571730	−	0.571730i
$119$	1311.34			1.01017
$120$		0			0
$121$	−104.688			−0.0786538
$122$	1023.05	+	1023.05i	0.759199	+	0.759199i
$123$		0			0
$124$		−	1342.13i		−	0.971990i
$125$	783.917	−	1156.98i	0.560926	−	0.827866i
$126$		0			0
$127$	786.555	−	786.555i	0.549571	−	0.549571i	−0.376746	−	0.926317i	$-0.622957\pi$
$127$	786.555	−	786.555i	0.549571	−	0.549571i	0.926317	+	0.376746i	$0.122957\pi$
$128$	−727.025	+	727.025i	−0.502036	+	0.502036i
$129$		0			0
$130$	1863.88	−	2116.14i	1.25748	−	1.42768i
$131$		−	448.364i		−	0.299036i	−0.988759	−	0.149518i	$-0.952228\pi$
$131$		−	448.364i		−	0.299036i	0.988759	−	0.149518i	$-0.0477722\pi$
$132$		0			0
$133$	1191.67	+	1191.67i	0.776922	+	0.776922i
$134$	−788.751			−0.508490
$135$		0			0
$136$	713.914			0.450129
$137$	−950.120	−	950.120i	−0.592512	−	0.592512i	0.345797	−	0.938309i	$-0.387609\pi$
$137$	−950.120	−	950.120i	−0.592512	−	0.592512i	−0.938309	+	0.345797i	$0.887609\pi$
$138$		0			0
$139$		−	112.735i		−	0.0687918i	−0.999408	−	0.0343959i	$-0.989049\pi$
$139$		−	112.735i		−	0.0687918i	0.999408	−	0.0343959i	$-0.0109507\pi$
$140$	62.4144	+	984.720i	0.0376784	+	0.594458i
$141$		0			0
$142$	1232.17	−	1232.17i	0.728179	−	0.728179i
$143$	1822.85	−	1822.85i	1.06598	−	1.06598i
$144$		0			0
$145$	−758.076	−	667.706i	−0.434171	−	0.382413i
$146$		−	2520.56i		−	1.42879i
$147$		0			0
$148$	−109.151	−	109.151i	−0.0606226	−	0.0606226i
$149$	−1492.95			−0.820852			−0.410426	−	0.911894i	$-0.634620\pi$
$149$	−1492.95			−0.820852			−0.410426	+	0.911894i	$0.634620\pi$
$150$		0			0
$151$	−2939.54			−1.58422			−0.792108	−	0.610381i	$-0.791016\pi$
$151$	−2939.54			−1.58422			−0.792108	+	0.610381i	$0.791016\pi$
$152$	648.764	+	648.764i	0.346196	+	0.346196i
$153$		0			0
$154$			2158.39i			1.12940i
$155$	1960.54	+	1726.83i	1.01597	+	0.894852i
$156$		0			0
$157$	156.386	−	156.386i	0.0794967	−	0.0794967i	−0.666240	−	0.745737i	$-0.732098\pi$
$157$	156.386	−	156.386i	0.0794967	−	0.0794967i	0.745737	+	0.666240i	$0.232098\pi$
$158$	−3466.65	+	3466.65i	−1.74552	+	1.74552i
$159$		0			0
$160$	154.448	+	2436.74i	0.0763135	+	1.20401i
$161$			864.129i			0.422999i
$162$		0			0
$163$	−239.292	−	239.292i	−0.114986	−	0.114986i	0.647272	−	0.762259i	$-0.275910\pi$
$163$	−239.292	−	239.292i	−0.114986	−	0.114986i	−0.762259	+	0.647272i	$0.775910\pi$
$164$	1497.81			0.713167
$165$		0			0
$166$	−2391.72			−1.11828
$167$	−677.159	−	677.159i	−0.313773	−	0.313773i	0.532596	−	0.846369i	$-0.321216\pi$
$167$	−677.159	−	677.159i	−0.313773	−	0.313773i	−0.846369	+	0.532596i	$0.821216\pi$
$168$		0			0
$169$			2431.86i			1.10690i
$170$	2337.99	−	2654.42i	1.05480	−	1.19756i
$171$		0			0
$172$	−1012.10	+	1012.10i	−0.448674	+	0.448674i
$173$	−441.523	+	441.523i	−0.194037	+	0.194037i	−0.797438	−	0.603401i	$-0.793812\pi$
$173$	−441.523	+	441.523i	−0.194037	+	0.194037i	0.603401	+	0.797438i	$0.293812\pi$
$174$		0			0
$175$	−1518.76	−	1175.80i	−0.656041	−	0.507898i
$176$			2916.06i			1.24890i
$177$		0			0
$178$	3577.82	+	3577.82i	1.50657	+	1.50657i
$179$	−4724.27			−1.97267			−0.986337	−	0.164737i	$-0.947322\pi$
$179$	−4724.27			−1.97267			−0.986337	+	0.164737i	$0.947322\pi$
$180$		0			0
$181$	−1548.35			−0.635847			−0.317923	−	0.948116i	$-0.602985\pi$
$181$	−1548.35			−0.635847			−0.317923	+	0.948116i	$0.602985\pi$
$182$	−2740.45	−	2740.45i	−1.11613	−	1.11613i
$183$		0			0
$184$			470.447i			0.188488i
$185$	299.882	−	19.0074i	0.119177	−	0.00755378i
$186$		0			0
$187$	2286.53	−	2286.53i	0.894159	−	0.894159i
$188$	837.336	−	837.336i	0.324835	−	0.324835i
$189$		0			0
$190$	4536.82	−	287.557i	1.73229	−	0.109798i
$191$		−	1090.71i		−	0.413200i	−0.978426	−	0.206600i	$-0.933760\pi$
$191$		−	1090.71i		−	0.413200i	0.978426	−	0.206600i	$-0.0662398\pi$
$192$		0			0
$193$	−2825.69	−	2825.69i	−1.05387	−	1.05387i	−0.998464	−	0.0554112i	$-0.982353\pi$
$193$	−2825.69	−	2825.69i	−1.05387	−	1.05387i	−0.0554112	−	0.998464i	$-0.517647\pi$
$194$	4117.22			1.52371
$195$		0			0
$196$	−613.958			−0.223746
$197$	−1163.99	−	1163.99i	−0.420969	−	0.420969i	0.464569	−	0.885537i	$-0.346209\pi$
$197$	−1163.99	−	1163.99i	−0.420969	−	0.420969i	−0.885537	+	0.464569i	$0.846209\pi$
$198$		0			0
$199$			520.257i			0.185327i	0.995697	+	0.0926633i	$0.0295380\pi$
$199$			520.257i			0.185327i	−0.995697	+	0.0926633i	$0.970462\pi$
$200$	−826.837	−	640.126i	−0.292331	−	0.226319i
$201$		0			0
$202$	292.430	−	292.430i	0.101858	−	0.101858i
$203$	−981.726	+	981.726i	−0.339427	+	0.339427i
$204$		0			0
$205$	−1927.13	+	2187.96i	−0.656569	+	0.745432i
$206$		−	72.7808i		−	0.0246159i
$207$		0			0
$208$	−3702.44	−	3702.44i	−1.23422	−	1.23422i
$209$	4155.74			1.37540
$210$		0			0
$211$	5524.38			1.80244			0.901219	−	0.433365i	$-0.142674\pi$
$211$	5524.38			1.80244			0.901219	+	0.433365i	$0.142674\pi$
$212$	−1052.70	−	1052.70i	−0.341036	−	0.341036i
$213$		0			0
$214$		−	4069.28i		−	1.29986i
$215$	−176.245	−	2780.65i	−0.0559062	−	0.882039i
$216$		0			0
$217$	2538.95	−	2538.95i	0.794263	−	0.794263i
$218$	−1076.29	+	1076.29i	−0.334384	+	0.334384i
$219$		0			0
$220$	1825.85	+	1608.19i	0.559541	+	0.492838i
$221$			5806.30i			1.76730i
$222$		0			0
$223$	323.198	+	323.198i	0.0970537	+	0.0970537i	0.753967	−	0.656913i	$-0.228138\pi$
$223$	323.198	+	323.198i	0.0970537	+	0.0970537i	−0.656913	+	0.753967i	$0.728138\pi$
$224$	3355.65			1.00093
$225$		0			0
$226$	−2733.22			−0.804474
$227$	2739.76	+	2739.76i	0.801077	+	0.801077i	0.983264	−	0.182187i	$-0.0583177\pi$
$227$	2739.76	+	2739.76i	0.801077	+	0.801077i	−0.182187	+	0.983264i	$0.558318\pi$
$228$		0			0
$229$		−	230.334i		−	0.0664667i	−0.999448	−	0.0332333i	$-0.989420\pi$
$229$		−	230.334i		−	0.0664667i	0.999448	−	0.0332333i	$-0.0105804\pi$
$230$	1749.18	+	1540.66i	0.501468	+	0.441688i
$231$		0			0
$232$	−534.468	+	534.468i	−0.151248	+	0.151248i
$233$	350.156	−	350.156i	0.0984526	−	0.0984526i	−0.656165	−	0.754618i	$-0.727822\pi$
$233$	350.156	−	350.156i	0.0984526	−	0.0984526i	0.754618	+	0.656165i	$0.227822\pi$
$234$		0			0
$235$	145.812	+	2300.50i	0.0404755	+	0.638587i
$236$			1605.67i			0.442883i
$237$		0			0
$238$	−3437.54	−	3437.54i	−0.936229	−	0.936229i
$239$	−1013.20			−0.274218			−0.137109	−	0.990556i	$-0.543781\pi$
$239$	−1013.20			−0.274218			−0.137109	+	0.990556i	$0.543781\pi$
$240$		0			0
$241$	2584.30			0.690744			0.345372	−	0.938466i	$-0.387753\pi$
$241$	2584.30			0.690744			0.345372	+	0.938466i	$0.387753\pi$
$242$	274.430	+	274.430i	0.0728968	+	0.0728968i
$243$		0			0
$244$		−	2241.50i		−	0.588104i
$245$	789.938	−	896.852i	0.205989	−	0.233868i
$246$		0			0
$247$	−5276.43	+	5276.43i	−1.35924	+	1.35924i
$248$	1382.25	−	1382.25i	0.353923	−	0.353923i
$249$		0			0
$250$	−5087.87	+	977.942i	−1.28714	+	0.247402i
$251$			1708.34i			0.429599i	0.976658	+	0.214800i	$0.0689099\pi$
$251$			1708.34i			0.429599i	−0.976658	+	0.214800i	$0.931090\pi$
$252$		0			0
$253$	1506.75	+	1506.75i	0.374422	+	0.374422i
$254$	−4123.76			−1.01869
$255$		0			0
$256$	5363.04			1.30934
$257$	−2659.48	−	2659.48i	−0.645500	−	0.645500i	0.306402	−	0.951902i	$-0.400875\pi$
$257$	−2659.48	−	2659.48i	−0.645500	−	0.645500i	−0.951902	+	0.306402i	$0.900875\pi$
$258$		0			0
$259$		−	412.969i		−	0.0990757i
$260$	−4360.12	+	276.357i	−1.04001	+	0.0659189i
$261$		0			0
$262$	−1175.34	+	1175.34i	−0.277148	+	0.277148i
$263$	2307.03	−	2307.03i	0.540903	−	0.540903i	−0.382891	−	0.923794i	$-0.625071\pi$
$263$	2307.03	−	2307.03i	0.540903	−	0.540903i	0.923794	+	0.382891i	$0.125071\pi$
$264$		0			0
$265$	2892.19	−	183.315i	0.670437	−	0.0424943i
$266$		−	6247.68i		−	1.44011i
$267$		0			0
$268$	864.078	+	864.078i	0.196948	+	0.196948i
$269$	447.584			0.101449			0.0507243	−	0.998713i	$-0.483847\pi$
$269$	447.584			0.101449			0.0507243	+	0.998713i	$0.483847\pi$
$270$		0			0
$271$	1573.15			0.352627			0.176314	−	0.984334i	$-0.443583\pi$
$271$	1573.15			0.352627			0.176314	+	0.984334i	$0.443583\pi$
$272$	−4644.23	−	4644.23i	−1.03529	−	1.03529i
$273$		0			0
$274$			4981.29i			1.09829i
$275$	−4698.41	+	598.000i	−1.03027	+	0.131130i
$276$		0			0
$277$	5816.91	−	5816.91i	1.26175	−	1.26175i	0.311504	−	0.950245i	$-0.399167\pi$
$277$	5816.91	−	5816.91i	1.26175	−	1.26175i	0.950245	−	0.311504i	$-0.100833\pi$
$278$	−295.524	+	295.524i	−0.0637566	+	0.0637566i
$279$		0			0
$280$	−949.875	+	1078.44i	−0.202735	+	0.230174i
$281$		−	140.032i		−	0.0297281i	−0.999890	−	0.0148640i	$-0.995268\pi$
$281$		−	140.032i		−	0.0297281i	0.999890	−	0.0148640i	$-0.00473155\pi$
$282$		0			0
$283$	4431.85	+	4431.85i	0.930905	+	0.930905i	0.997763	−	0.0668572i	$-0.0212972\pi$
$283$	4431.85	+	4431.85i	0.930905	+	0.930905i	−0.0668572	+	0.997763i	$0.521297\pi$
$284$	−2699.69			−0.564074
$285$		0			0
$286$	−9556.87			−1.97591
$287$	2833.46	+	2833.46i	0.582765	+	0.582765i
$288$		0			0
$289$			2370.24i			0.482442i
$290$	236.897	+	3737.55i	0.0479691	+	0.756815i
$291$		0			0
$292$	−2761.28	+	2761.28i	−0.553395	+	0.553395i
$293$	1244.45	−	1244.45i	0.248127	−	0.248127i	−0.572074	−	0.820202i	$-0.693861\pi$
$293$	1244.45	−	1244.45i	0.248127	−	0.248127i	0.820202	+	0.572074i	$0.193861\pi$
$294$		0			0
$295$	−2345.52	−	2065.91i	−0.462921	−	0.407736i
$296$		−	224.827i		−	0.0441480i
$297$		0			0
$298$	3913.61	+	3913.61i	0.760771	+	0.760771i
$299$	−3826.17			−0.740043
$300$		0			0
$301$	−3829.24			−0.733269
$302$	7705.73	+	7705.73i	1.46826	+	1.46826i
$303$		0			0
$304$		−	8440.82i		−	1.59248i
$305$	3274.32	+	2883.99i	0.614711	+	0.541432i
$306$		0			0
$307$	−7327.60	+	7327.60i	−1.36224	+	1.36224i	−0.491189	+	0.871053i	$0.663437\pi$
$307$	−7327.60	+	7327.60i	−1.36224	+	1.36224i	−0.871053	+	0.491189i	$0.836563\pi$
$308$	2364.52	−	2364.52i	0.437439	−	0.437439i
$309$		0			0
$310$	−612.664	−	9666.08i	−0.112248	−	1.77096i
$311$		−	7489.38i		−	1.36554i	−0.730632	−	0.682771i	$-0.760775\pi$
$311$		−	7489.38i		−	1.36554i	0.730632	−	0.682771i	$-0.239225\pi$
$312$		0			0
$313$	−2456.04	−	2456.04i	−0.443525	−	0.443525i	0.449670	−	0.893195i	$-0.351542\pi$
$313$	−2456.04	−	2456.04i	−0.443525	−	0.443525i	−0.893195	+	0.449670i	$0.851542\pi$
$314$	−819.903			−0.147356
$315$		0			0
$316$	7595.44			1.35214
$317$	6724.30	+	6724.30i	1.19140	+	1.19140i	0.976674	+	0.214728i	$0.0688866\pi$
$317$	6724.30	+	6724.30i	1.19140	+	1.19140i	0.214728	+	0.976674i	$0.431113\pi$
$318$		0			0
$319$			3423.60i			0.600894i
$320$	1433.05	−	1627.01i	0.250344	−	0.284227i
$321$		0			0
$322$	2265.23	−	2265.23i	0.392038	−	0.392038i
$323$	−6618.59	+	6618.59i	−1.14015	+	1.14015i
$324$		0			0
$325$	5206.18	−	6724.70i	0.888574	−	1.14775i
$326$			1254.56i			0.213140i
$327$		0			0
$328$	1542.58	+	1542.58i	0.259679	+	0.259679i
$329$	3168.03			0.530879
$330$		0			0
$331$	−199.080			−0.0330586			−0.0165293	−	0.999863i	$-0.505262\pi$
$331$	−199.080			−0.0330586			−0.0165293	+	0.999863i	$0.505262\pi$
$332$	2620.14	+	2620.14i	0.433129	+	0.433129i
$333$		0			0
$334$			3550.21i			0.581614i
$335$	−2373.97	+	150.469i	−0.387176	+	0.0245403i
$336$		0			0
$337$	3308.77	−	3308.77i	0.534838	−	0.534838i	−0.387170	−	0.922008i	$-0.626547\pi$
$337$	3308.77	−	3308.77i	0.534838	−	0.534838i	0.922008	+	0.387170i	$0.126547\pi$
$338$	6374.88	−	6374.88i	1.02588	−	1.02588i
$339$		0			0
$340$	−5469.20	+	346.653i	−0.872379	+	0.0552939i
$341$		−	8854.16i		−	1.40610i
$342$		0			0
$343$	−4888.20	−	4888.20i	−0.769498	−	0.769498i
$344$	−2084.71			−0.326744
$345$		0			0
$346$	2314.82			0.359669
$347$	−2371.68	−	2371.68i	−0.366912	−	0.366912i	0.499438	−	0.866350i	$-0.333540\pi$
$347$	−2371.68	−	2371.68i	−0.366912	−	0.366912i	−0.866350	+	0.499438i	$0.833540\pi$
$348$		0			0
$349$		−	8208.92i		−	1.25906i	−0.776975	−	0.629532i	$-0.783247\pi$
$349$		−	8208.92i		−	1.25906i	0.776975	−	0.629532i	$-0.216753\pi$
$350$	899.024	+	7063.52i	0.137300	+	1.07875i
$351$		0			0
$352$	5851.13	−	5851.13i	0.885984	−	0.885984i
$353$	−6800.98	+	6800.98i	−1.02544	+	1.02544i	−0.0257707	+	0.999668i	$0.508204\pi$
$353$	−6800.98	+	6800.98i	−1.02544	+	1.02544i	−0.999668	+	0.0257707i	$0.991796\pi$
$354$		0			0
$355$	3473.51	−	3943.63i	0.519309	−	0.589594i
$356$		−	7839.01i		−	1.16704i
$357$		0			0
$358$	12384.2	+	12384.2i	1.82829	+	1.82829i
$359$	1778.14			0.261411			0.130705	−	0.991421i	$-0.458276\pi$
$359$	1778.14			0.261411			0.130705	+	0.991421i	$0.458276\pi$
$360$		0			0
$361$	−5170.20			−0.753783
$362$	4058.86	+	4058.86i	0.589306	+	0.589306i
$363$		0			0
$364$			6004.34i			0.864596i
$365$	−480.844	−	7586.34i	−0.0689549	−	1.08791i
$366$		0			0
$367$	−4673.56	+	4673.56i	−0.664735	+	0.664735i	−0.956492	−	0.291757i	$-0.905760\pi$
$367$	−4673.56	+	4673.56i	−0.664735	+	0.664735i	0.291757	+	0.956492i	$0.405760\pi$
$368$	3060.40	−	3060.40i	0.433517	−	0.433517i
$369$		0			0
$370$	−835.936	−	736.284i	−0.117455	−	0.103453i
$371$		−	3982.85i		−	0.557357i
$372$		0			0
$373$	−5729.37	−	5729.37i	−0.795323	−	0.795323i	0.187031	−	0.982354i	$-0.440114\pi$
$373$	−5729.37	−	5729.37i	−0.795323	−	0.795323i	−0.982354	+	0.187031i	$0.940114\pi$
$374$	−11987.8			−1.65742
$375$		0			0
$376$	1724.73			0.236559
$377$	−4346.86	−	4346.86i	−0.593832	−	0.593832i
$378$		0			0
$379$		−	153.883i		−	0.0208561i	−0.999946	−	0.0104280i	$-0.996681\pi$
$379$		−	153.883i		−	0.0208561i	0.999946	−	0.0104280i	$-0.00331941\pi$
$380$	−5285.11	−	4655.08i	−0.713475	−	0.628422i
$381$		0			0
$382$	−2859.19	+	2859.19i	−0.382956	+	0.382956i
$383$	4297.76	−	4297.76i	0.573382	−	0.573382i	−0.359690	−	0.933072i	$-0.617118\pi$
$383$	4297.76	−	4297.76i	0.573382	−	0.573382i	0.933072	+	0.359690i	$0.117118\pi$
$384$		0			0
$385$	411.754	+	6496.30i	0.0545063	+	0.859954i
$386$			14814.6i			1.95347i
$387$		0			0
$388$	−4510.42	−	4510.42i	−0.590160	−	0.590160i
$389$	7833.00			1.02095			0.510474	−	0.859893i	$-0.329470\pi$
$389$	7833.00			1.02095			0.510474	+	0.859893i	$0.329470\pi$
$390$		0			0
$391$	−4799.42			−0.620760
$392$	−632.310	−	632.310i	−0.0814706	−	0.0814706i
$393$		0			0
$394$			6102.57i			0.780312i
$395$	−9772.54	+	11095.2i	−1.24484	+	1.41332i
$396$		0			0
$397$	−10195.4	+	10195.4i	−1.28889	+	1.28889i	−0.353435	+	0.935459i	$0.614987\pi$
$397$	−10195.4	+	10195.4i	−1.28889	+	1.28889i	−0.935459	+	0.353435i	$0.885013\pi$
$398$	1363.80	−	1363.80i	0.171762	−	0.171762i
$399$		0			0
$400$	1214.61	+	9543.04i	0.151826	+	1.19288i
$401$			10778.3i			1.34226i	0.741341	+	0.671128i	$0.234190\pi$
$401$			10778.3i			1.34226i	−0.741341	+	0.671128i	$0.765810\pi$
$402$		0			0
$403$	11241.9	+	11241.9i	1.38957	+	1.38957i
$404$	−640.714			−0.0789028
$405$		0			0
$406$	5147.00			0.629165
$407$	−720.079	−	720.079i	−0.0876978	−	0.0876978i
$408$		0			0
$409$		−	4629.92i		−	0.559743i	−0.960037	−	0.279872i	$-0.909708\pi$
$409$		−	4629.92i		−	0.559743i	0.960037	−	0.279872i	$-0.0902919\pi$
$410$	10787.3	−	683.731i	1.29938	−	0.0823587i
$411$		0			0
$412$	−79.7315	+	79.7315i	−0.00953420	+	0.00953420i
$413$	−3037.51	+	3037.51i	−0.361903	+	0.361903i
$414$		0			0
$415$	−7198.58	+	456.267i	−0.851480	+	0.0539693i
$416$			14858.0i			1.75114i
$417$		0			0
$418$	−10893.9	−	10893.9i	−1.27473	−	1.27473i
$419$	11276.5			1.31478			0.657389	−	0.753551i	$-0.271661\pi$
$419$	11276.5			1.31478			0.657389	+	0.753551i	$0.271661\pi$
$420$		0			0
$421$	3387.58			0.392163			0.196082	−	0.980588i	$-0.437178\pi$
$421$	3387.58			0.392163			0.196082	+	0.980588i	$0.437178\pi$
$422$	−14481.6	−	14481.6i	−1.67051	−	1.67051i
$423$		0			0
$424$		−	2168.33i		−	0.248358i
$425$	6530.47	−	8435.26i	0.745351	−	0.962753i
$426$		0			0
$427$	4240.32	−	4240.32i	0.480570	−	0.480570i
$428$	−4457.91	+	4457.91i	−0.503461	+	0.503461i
$429$		0			0
$430$	−6827.18	+	7751.20i	−0.765665	+	0.869293i
$431$		−	16161.7i		−	1.80623i	−0.429402	−	0.903113i	$-0.641276\pi$
$431$		−	16161.7i		−	1.80623i	0.429402	−	0.903113i	$-0.358724\pi$
$432$		0			0
$433$	−3808.79	−	3808.79i	−0.422722	−	0.422722i	0.463418	−	0.886140i	$-0.346623\pi$
$433$	−3808.79	−	3808.79i	−0.422722	−	0.422722i	−0.886140	+	0.463418i	$0.846623\pi$
$434$	−13311.2			−1.47226
$435$		0			0
$436$	2358.16			0.259026
$437$	−4361.44	−	4361.44i	−0.477428	−	0.477428i
$438$		0			0
$439$		−	1607.30i		−	0.174744i	−0.996176	−	0.0873718i	$-0.972153\pi$
$439$		−	1607.30i		−	0.174744i	0.996176	−	0.0873718i	$-0.0278468\pi$
$440$	224.166	+	3536.70i	0.0242880	+	0.383194i
$441$		0			0
$442$	15220.6	−	15220.6i	1.63795	−	1.63795i
$443$	8296.60	−	8296.60i	0.889805	−	0.889805i	−0.104699	−	0.994504i	$-0.533388\pi$
$443$	8296.60	−	8296.60i	0.889805	−	0.889805i	0.994504	+	0.104699i	$0.0333880\pi$
$444$		0			0
$445$	11451.0	+	10085.9i	1.21984	+	1.07442i
$446$		−	1694.47i		−	0.179900i
$447$		0			0
$448$	−2107.02	−	2107.02i	−0.222203	−	0.222203i
$449$	15061.8			1.58309			0.791547	−	0.611108i	$-0.209276\pi$
$449$	15061.8			1.58309			0.791547	+	0.611108i	$0.209276\pi$
$450$		0			0
$451$	9881.20			1.03168
$452$	2994.25	+	2994.25i	0.311588	+	0.311588i
$453$		0			0
$454$		−	14364.0i		−	1.48489i
$455$	−8770.97	−	7725.38i	−0.903713	−	0.795981i
$456$		0			0
$457$	2671.27	−	2671.27i	0.273428	−	0.273428i	−0.557050	−	0.830479i	$-0.688067\pi$
$457$	2671.27	−	2671.27i	0.273428	−	0.273428i	0.830479	+	0.557050i	$0.188067\pi$
$458$	−603.797	+	603.797i	−0.0616017	+	0.0616017i
$459$		0			0
$460$	−228.434	−	3604.03i	−0.0231539	−	0.365301i
$461$			12809.1i			1.29410i	0.762449	+	0.647049i	$0.223997\pi$
$461$			12809.1i			1.29410i	−0.762449	+	0.647049i	$0.776003\pi$
$462$		0			0
$463$	−11154.1	−	11154.1i	−1.11960	−	1.11960i	−0.991799	−	0.127804i	$-0.959207\pi$
$463$	−11154.1	−	11154.1i	−1.11960	−	1.11960i	−0.127804	−	0.991799i	$-0.540793\pi$
$464$	6953.76			0.695733
$465$		0			0
$466$	−1835.80			−0.182493
$467$	7905.44	+	7905.44i	0.783341	+	0.783341i	0.980393	−	0.197052i	$-0.0631368\pi$
$467$	7905.44	+	7905.44i	0.783341	+	0.783341i	−0.197052	+	0.980393i	$0.563137\pi$
$468$		0			0
$469$			3269.21i			0.321872i
$470$	5648.30	−	6412.77i	0.554334	−	0.629360i
$471$		0			0
$472$	−1653.67	+	1653.67i	−0.161263	+	0.161263i
$473$	−6676.92	+	6676.92i	−0.649060	+	0.649060i
$474$		0			0
$475$	13600.0	−	1730.97i	1.31371	−	0.167205i
$476$			7531.66i			0.725238i
$477$		0			0
$478$	2655.99	+	2655.99i	0.254147	+	0.254147i
$479$	−10715.8			−1.02217			−0.511085	−	0.859530i	$-0.670756\pi$
$479$	−10715.8			−1.02217			−0.511085	+	0.859530i	$0.670756\pi$
$480$		0			0
$481$	1828.53			0.173334
$482$	−6774.48	−	6774.48i	−0.640185	−	0.640185i
$483$		0			0
$484$		−	601.277i		−	0.0564685i
$485$	12391.9	−	785.437i	1.16018	−	0.0735358i
$486$		0			0
$487$	−4489.84	+	4489.84i	−0.417770	+	0.417770i	−0.884435	−	0.466664i	$-0.845456\pi$
$487$	−4489.84	+	4489.84i	−0.417770	+	0.417770i	0.466664	+	0.884435i	$0.345456\pi$
$488$	2308.50	−	2308.50i	0.214141	−	0.214141i
$489$		0			0
$490$	−4421.76	+	280.264i	−0.407662	+	0.0258388i
$491$			15174.1i			1.39470i	0.716729	+	0.697352i	$0.245638\pi$
$491$			15174.1i			1.39470i	−0.716729	+	0.697352i	$0.754362\pi$
$492$		0			0
$493$	−5452.56	−	5452.56i	−0.498116	−	0.498116i
$494$	27663.3			2.51950
$495$		0			0
$496$	−17983.9			−1.62802
$497$	−5107.09	−	5107.09i	−0.460934	−	0.460934i
$498$		0			0
$499$			8887.20i			0.797286i	0.917106	+	0.398643i	$0.130519\pi$
$499$			8887.20i			0.797286i	−0.917106	+	0.398643i	$0.869481\pi$
$500$	6645.11	+	4502.43i	0.594357	+	0.402710i
$501$		0			0
$502$	4478.25	−	4478.25i	0.398155	−	0.398155i
$503$	−12740.5	+	12740.5i	−1.12936	+	1.12936i	−0.139084	+	0.990281i	$0.544416\pi$
$503$	−12740.5	+	12740.5i	−1.12936	+	1.12936i	−0.990281	+	0.139084i	$0.955584\pi$
$504$		0			0
$505$	824.364	−	935.937i	0.0726410	−	0.0824726i
$506$		−	7899.61i		−	0.694033i
$507$		0			0
$508$	4517.58	+	4517.58i	0.394558	+	0.394558i
$509$	−9257.00			−0.806108			−0.403054	−	0.915176i	$-0.632051\pi$
$509$	−9257.00			−0.806108			−0.403054	+	0.915176i	$0.632051\pi$
$510$		0			0
$511$	−10447.2			−0.904416
$512$	−8242.49	−	8242.49i	−0.711465	−	0.711465i
$513$		0			0
$514$			13943.1i			1.19651i
$515$	−13.8843	−	219.055i	−0.00118799	−	0.0187431i
$516$		0			0
$517$	5523.99	−	5523.99i	0.469913	−	0.469913i
$518$	−1082.56	+	1082.56i	−0.0918240	+	0.0918240i
$519$		0			0
$520$	−4775.07	−	4205.83i	−0.402693	−	0.354688i
$521$		−	13563.3i		−	1.14054i	−0.821458	−	0.570268i	$-0.806839\pi$
$521$		−	13563.3i		−	1.14054i	0.821458	−	0.570268i	$-0.193161\pi$
$522$		0			0
$523$	11193.8	+	11193.8i	0.935894	+	0.935894i	0.998065	−	0.0621714i	$-0.0198026\pi$
$523$	11193.8	+	11193.8i	0.935894	+	0.935894i	−0.0621714	+	0.998065i	$0.519803\pi$
$524$	2575.18			0.214689
$525$		0			0
$526$	−12095.3			−1.00262
$527$	14101.5	+	14101.5i	1.16560	+	1.16560i
$528$		0			0
$529$			9004.33i			0.740062i
$530$	−8062.14	−	7101.05i	−0.660749	−	0.581981i
$531$		0			0
$532$	−6844.34	+	6844.34i	−0.557782	+	0.557782i
$533$	−12545.9	+	12545.9i	−1.01956	+	1.01956i
$534$		0			0
$535$	−776.293	−	12247.7i	−0.0627329	−	0.989744i
$536$			1779.81i			0.143426i
$537$		0			0
$538$	−1173.30	−	1173.30i	−0.0940231	−	0.0940231i
$539$	−4050.34			−0.323675
$540$		0			0
$541$	1383.33			0.109934			0.0549668	−	0.998488i	$-0.482495\pi$
$541$	1383.33			0.109934			0.0549668	+	0.998488i	$0.482495\pi$
$542$	−4123.86	−	4123.86i	−0.326817	−	0.326817i
$543$		0			0
$544$			18637.5i			1.46889i
$545$	−3034.09	+	3444.73i	−0.238470	+	0.270745i
$546$		0			0
$547$	2181.76	−	2181.76i	0.170540	−	0.170540i	−0.616677	−	0.787217i	$-0.711521\pi$
$547$	2181.76	−	2181.76i	0.170540	−	0.170540i	0.787217	+	0.616677i	$0.211521\pi$
$548$	5457.01	−	5457.01i	0.425387	−	0.425387i
$549$		0			0
$550$	13884.0	+	10748.8i	1.07639	+	0.833329i
$551$		−	9909.96i		−	0.766204i
$552$		0			0
$553$	14368.5	+	14368.5i	1.10491	+	1.10491i
$554$	−30496.9			−2.33879
$555$		0			0
$556$	647.494			0.0493882
$557$	−15674.9	−	15674.9i	−1.19240	−	1.19240i	−0.976392	−	0.216007i	$-0.930697\pi$
$557$	−15674.9	−	15674.9i	−1.19240	−	1.19240i	−0.216007	−	0.976392i	$-0.569303\pi$
$558$		0			0
$559$		−	16955.0i		−	1.28286i
$560$	13194.8	−	836.323i	0.995681	−	0.0631091i
$561$		0			0
$562$	−367.080	+	367.080i	−0.0275522	+	0.0275522i
$563$	8213.60	−	8213.60i	0.614852	−	0.614852i	−0.329354	−	0.944206i	$-0.606831\pi$
$563$	8213.60	−	8213.60i	0.614852	−	0.614852i	0.944206	+	0.329354i	$0.106831\pi$
$564$		0			0
$565$	−8226.41	+	521.414i	−0.612545	+	0.0388248i
$566$		−	23235.3i		−	1.72554i
$567$		0			0
$568$	−2780.39	−	2780.39i	−0.205392	−	0.205392i
$569$	−16902.1			−1.24530			−0.622648	−	0.782502i	$-0.713943\pi$
$569$	−16902.1			−1.24530			−0.622648	+	0.782502i	$0.713943\pi$
$570$		0			0
$571$	6507.12			0.476908			0.238454	−	0.971154i	$-0.423359\pi$
$571$	6507.12			0.476908			0.238454	+	0.971154i	$0.423359\pi$
$572$	10469.6	+	10469.6i	0.765305	+	0.765305i
$573$		0			0
$574$		−	14855.3i		−	1.08022i
$575$	5558.57	+	4303.37i	0.403145	+	0.312109i
$576$		0			0
$577$	7884.54	−	7884.54i	0.568870	−	0.568870i	−0.362942	−	0.931812i	$-0.618228\pi$
$577$	7884.54	−	7884.54i	0.568870	−	0.568870i	0.931812	+	0.362942i	$0.118228\pi$
$578$	6213.35	−	6213.35i	0.447130	−	0.447130i
$579$		0			0
$580$	3834.97	−	4354.01i	0.274549	−	0.311708i
$581$			9913.20i			0.707864i
$582$		0			0
$583$	−6944.77	−	6944.77i	−0.493350	−	0.493350i
$584$	−5687.63			−0.403007
$585$		0			0
$586$	−6524.39			−0.459932
$587$	9548.44	+	9548.44i	0.671390	+	0.671390i	0.958037	−	0.286646i	$-0.0925404\pi$
$587$	9548.44	+	9548.44i	0.671390	+	0.671390i	−0.286646	+	0.958037i	$0.592540\pi$
$588$		0			0
$589$			25629.2i			1.79293i
$590$	732.969	+	11564.1i	0.0511455	+	0.806930i
$591$		0			0
$592$	−1462.57	+	1462.57i	−0.101539	+	0.101539i
$593$	3491.97	−	3491.97i	0.241818	−	0.241818i	−0.575784	−	0.817602i	$-0.695303\pi$
$593$	3491.97	−	3491.97i	0.241818	−	0.241818i	0.817602	+	0.575784i	$0.195303\pi$
$594$		0			0
$595$	−11002.0	−	9690.48i	−0.758049	−	0.667682i
$596$		−	8574.75i		−	0.589321i
$597$		0			0
$598$	10029.9	+	10029.9i	0.685876	+	0.685876i
$599$	11203.0			0.764175			0.382087	−	0.924126i	$-0.375205\pi$
$599$	11203.0			0.764175			0.382087	+	0.924126i	$0.375205\pi$
$600$		0			0
$601$	−13953.5			−0.947048			−0.473524	−	0.880781i	$-0.657018\pi$
$601$	−13953.5			−0.947048			−0.473524	+	0.880781i	$0.657018\pi$
$602$	10038.0	+	10038.0i	0.679598	+	0.679598i
$603$		0			0
$604$		−	16883.3i		−	1.13737i
$605$	878.328	+	773.622i	0.0590233	+	0.0519871i
$606$		0			0
$607$	−1292.54	+	1292.54i	−0.0864293	+	0.0864293i	−0.749000	−	0.662570i	$-0.769466\pi$
$607$	−1292.54	+	1292.54i	−0.0864293	+	0.0864293i	0.662570	+	0.749000i	$0.269466\pi$
$608$	−16936.7	+	16936.7i	−1.12972	+	1.12972i
$609$		0			0
$610$	−1023.22	−	16143.4i	−0.0679160	−	1.07152i
$611$			14027.3i			0.928780i
$612$		0			0
$613$	225.964	+	225.964i	0.0148884	+	0.0148884i	0.714512	−	0.699623i	$-0.246649\pi$
$613$	225.964	+	225.964i	0.0148884	+	0.0148884i	−0.699623	+	0.714512i	$0.746649\pi$
$614$	38417.2			2.52507
$615$		0			0
$616$	4870.41			0.318562
$617$	2717.41	+	2717.41i	0.177308	+	0.177308i	0.790181	−	0.612873i	$-0.209986\pi$
$617$	2717.41	+	2717.41i	0.177308	+	0.177308i	−0.612873	+	0.790181i	$0.709986\pi$
$618$		0			0
$619$		−	8870.55i		−	0.575989i	−0.957632	−	0.287995i	$-0.907011\pi$
$619$		−	8870.55i		−	0.575989i	0.957632	−	0.287995i	$-0.0929886\pi$
$620$	−9918.04	+	11260.4i	−0.642448	+	0.729400i
$621$		0			0
$622$	−19632.7	+	19632.7i	−1.26559	+	1.26559i
$623$	14829.3	−	14829.3i	0.953650	−	0.953650i
$624$		0			0
$625$	−15126.8	+	3914.00i	−0.968118	+	0.250496i
$626$			12876.5i			0.822123i
$627$		0			0
$628$	898.205	+	898.205i	0.0570737	+	0.0570737i
$629$	2293.65			0.145396
$630$		0			0
$631$	29762.7			1.87771			0.938853	−	0.344318i	$-0.111890\pi$
$631$	29762.7			1.87771			0.938853	+	0.344318i	$0.111890\pi$
$632$	7822.48	+	7822.48i	0.492344	+	0.492344i
$633$		0			0
$634$		−	35254.2i		−	2.20840i
$635$	−12411.6	+	786.685i	−0.775654	+	0.0491632i
$636$		0			0
$637$	5142.61	−	5142.61i	0.319871	−	0.319871i
$638$	8974.65	−	8974.65i	0.556912	−	0.556912i
$639$		0			0
$640$	11472.3	−	727.145i	0.708564	−	0.0449108i
$641$		−	5168.75i		−	0.318492i	−0.987239	−	0.159246i	$-0.949094\pi$
$641$		−	5168.75i		−	0.318492i	0.987239	−	0.159246i	$-0.0509063\pi$
$642$		0			0
$643$	−2078.46	−	2078.46i	−0.127475	−	0.127475i	0.640491	−	0.767966i	$-0.278731\pi$
$643$	−2078.46	−	2078.46i	−0.127475	−	0.127475i	−0.767966	+	0.640491i	$0.778731\pi$
$644$	−4963.13			−0.303687
$645$		0			0
$646$	34700.0			2.11339
$647$	10765.7	+	10765.7i	0.654164	+	0.654164i	0.953993	−	0.299829i	$-0.0969297\pi$
$647$	10765.7	+	10765.7i	0.654164	+	0.654164i	−0.299829	+	0.953993i	$0.596930\pi$
$648$		0			0
$649$			10592.8i			0.640684i
$650$	−31275.6	+	3980.67i	−1.88728	+	0.240207i
$651$		0			0
$652$	1374.37	−	1374.37i	0.0825530	−	0.0825530i
$653$	12815.1	−	12815.1i	0.767984	−	0.767984i	−0.209767	−	0.977751i	$-0.567271\pi$
$653$	12815.1	−	12815.1i	0.767984	−	0.767984i	0.977751	+	0.209767i	$0.0672707\pi$
$654$		0			0
$655$	−3313.31	+	3761.75i	−0.197651	+	0.224402i
$656$		−	20069.9i		−	1.19451i
$657$		0			0
$658$	−8304.68	−	8304.68i	−0.492022	−	0.492022i
$659$	−16634.8			−0.983307			−0.491654	−	0.870791i	$-0.663607\pi$
$659$	−16634.8			−0.983307			−0.491654	+	0.870791i	$0.663607\pi$
$660$		0			0
$661$	−10835.6			−0.637601			−0.318800	−	0.947822i	$-0.603280\pi$
$661$	−10835.6			−0.637601			−0.318800	+	0.947822i	$0.603280\pi$
$662$	521.868	+	521.868i	0.0306389	+	0.0306389i
$663$		0			0
$664$			5396.92i			0.315423i
$665$	−1191.86	−	18804.2i	−0.0695015	−	1.09653i
$666$		0			0
$667$	3593.07	−	3593.07i	0.208582	−	0.208582i
$668$	3889.27	−	3889.27i	0.225270	−	0.225270i
$669$		0			0
$670$	6617.58	+	5828.70i	0.381581	+	0.336093i
$671$		−	14787.4i		−	0.850762i
$672$		0			0
$673$	−16220.3	−	16220.3i	−0.929045	−	0.929045i	0.0685992	−	0.997644i	$-0.478147\pi$
$673$	−16220.3	−	16220.3i	−0.929045	−	0.929045i	−0.997644	+	0.0685992i	$0.978147\pi$
$674$	−17347.3			−0.991382
$675$		0			0
$676$	−13967.4			−0.794685
$677$	−11196.6	−	11196.6i	−0.635626	−	0.635626i	0.313847	−	0.949473i	$-0.398382\pi$
$677$	−11196.6	−	11196.6i	−0.635626	−	0.635626i	−0.949473	+	0.313847i	$0.898382\pi$
$678$		0			0
$679$		−	17065.0i		−	0.964500i
$680$	−5989.70	−	5275.66i	−0.337786	−	0.297518i
$681$		0			0
$682$	−23210.3	+	23210.3i	−1.30318	+	1.30318i
$683$	15428.1	−	15428.1i	0.864331	−	0.864331i	−0.127507	−	0.991838i	$-0.540697\pi$
$683$	15428.1	−	15428.1i	0.864331	−	0.864331i	0.991838	+	0.127507i	$0.0406975\pi$
$684$		0			0
$685$	950.276	+	14992.6i	0.0530046	+	0.836261i
$686$			25627.9i			1.42635i
$687$		0			0
$688$	13561.6	+	13561.6i	0.751501	+	0.751501i
$689$	17635.2			0.975104
$690$		0			0
$691$	21858.6			1.20339			0.601693	−	0.798728i	$-0.294493\pi$
$691$	21858.6			1.20339			0.601693	+	0.798728i	$0.294493\pi$
$692$	−2535.89	−	2535.89i	−0.139307	−	0.139307i
$693$		0			0
$694$			12434.2i			0.680112i
$695$	−833.087	+	945.841i	−0.0454688	+	0.0516227i
$696$		0			0
$697$	−15737.2	+	15737.2i	−0.855220	+	0.855220i
$698$	−21518.9	+	21518.9i	−1.16691	+	1.16691i
$699$		0			0
$700$	6753.21	−	8722.98i	0.364639	−	0.470996i
$701$			8846.82i			0.476662i	0.971184	+	0.238331i	$0.0766002\pi$
$701$			8846.82i			0.476662i	−0.971184	+	0.238331i	$0.923400\pi$
$702$		0			0
$703$	2084.34	+	2084.34i	0.111824	+	0.111824i
$704$	−7347.86			−0.393371
$705$		0			0
$706$	35656.2			1.90076
$707$	−1212.06	−	1212.06i	−0.0644755	−	0.0644755i
$708$		0			0
$709$			23189.8i			1.22837i	0.789163	+	0.614184i	$0.210515\pi$
$709$			23189.8i			1.22837i	−0.789163	+	0.614184i	$0.789485\pi$
$710$	−19443.3	+	1232.37i	−1.02774	+	0.0651410i
$711$		0			0
$712$	8073.34	−	8073.34i	0.424945	−	0.424945i
$713$	−9292.43	+	9292.43i	−0.488084	+	0.488084i
$714$		0			0
$715$	−28764.1	+	1823.15i	−1.50450	+	0.0953596i
$716$		−	27133.9i		−	1.41626i
$717$		0			0
$718$	−4661.21	−	4661.21i	−0.242277	−	0.242277i
$719$	−7005.29			−0.363356			−0.181678	−	0.983358i	$-0.558153\pi$
$719$	−7005.29			−0.363356			−0.181678	+	0.983358i	$0.558153\pi$
$720$		0			0
$721$	−301.661			−0.0155818
$722$	13553.2	+	13553.2i	0.698611	+	0.698611i
$723$		0			0
$724$		−	8892.98i		−	0.456498i
$725$	1426.02	+	11204.0i	0.0730495	+	0.573941i
$726$		0			0
$727$	−16025.7	+	16025.7i	−0.817553	+	0.817553i	−0.985753	−	0.168200i	$-0.946205\pi$
$727$	−16025.7	+	16025.7i	−0.817553	+	0.817553i	0.168200	+	0.985753i	$0.446205\pi$
$728$	−6183.82	+	6183.82i	−0.314818	+	0.314818i
$729$		0			0
$730$	−18626.4	+	21147.3i	−0.944374	+	1.07219i
$731$		−	21267.8i		−	1.07609i
$732$		0			0
$733$	−3319.43	−	3319.43i	−0.167266	−	0.167266i	0.618511	−	0.785777i	$-0.287736\pi$
$733$	−3319.43	−	3319.43i	−0.167266	−	0.167266i	−0.785777	+	0.618511i	$0.787736\pi$
$734$	24502.5			1.23216
$735$		0			0
$736$	−12281.5			−0.615084
$737$	5700.41	+	5700.41i	0.284908	+	0.284908i
$738$		0			0
$739$		−	19979.4i		−	0.994526i	−0.867600	−	0.497263i	$-0.834338\pi$
$739$		−	19979.4i		−	0.994526i	0.867600	−	0.497263i	$-0.165662\pi$
$740$	109.169	+	1722.37i	0.00542314	+	0.0855616i
$741$		0			0
$742$	−10440.7	+	10440.7i	−0.516562	+	0.516562i
$743$	14574.6	−	14574.6i	0.719637	−	0.719637i	−0.248894	−	0.968531i	$-0.580067\pi$
$743$	14574.6	−	14574.6i	0.719637	−	0.719637i	0.968531	+	0.248894i	$0.0800670\pi$
$744$		0			0
$745$	12525.7	+	11032.5i	0.615983	+	0.542552i
$746$			30038.0i			1.47422i
$747$		0			0
$748$	13132.7	+	13132.7i	0.641951	+	0.641951i
$749$	−16866.3			−0.822807
$750$		0			0
$751$	−11620.1			−0.564612			−0.282306	−	0.959324i	$-0.591099\pi$
$751$	−11620.1			−0.564612			−0.282306	+	0.959324i	$0.591099\pi$
$752$	−11219.9	−	11219.9i	−0.544079	−	0.544079i
$753$		0			0
$754$			22789.7i			1.10073i
$755$	24662.6	+	21722.6i	1.18883	+	1.04711i
$756$		0			0
$757$	10502.0	−	10502.0i	0.504229	−	0.504229i	−0.408520	−	0.912749i	$-0.633955\pi$
$757$	10502.0	−	10502.0i	0.504229	−	0.504229i	0.912749	+	0.408520i	$0.133955\pi$
$758$	−403.390	+	403.390i	−0.0193296	+	0.0193296i
$759$		0			0
$760$	−648.871	−	10237.3i	−0.0309698	−	0.488614i
$761$			36489.6i			1.73817i	0.494664	+	0.869085i	$0.335291\pi$
$761$			36489.6i			1.73817i	−0.494664	+	0.869085i	$0.664709\pi$
$762$		0			0
$763$	4461.01	+	4461.01i	0.211664	+	0.211664i
$764$	6264.50			0.296652
$765$		0			0
$766$	−22532.3			−1.06283
$767$	−13449.4	−	13449.4i	−0.633154	−	0.633154i
$768$		0			0
$769$		−	30323.9i		−	1.42199i	−0.703198	−	0.710994i	$-0.748245\pi$
$769$		−	30323.9i		−	1.42199i	0.703198	−	0.710994i	$-0.251755\pi$
$770$	15950.0	−	18108.8i	0.746493	−	0.847527i
$771$		0			0
$772$	16229.4	−	16229.4i	0.756617	−	0.756617i
$773$	10327.2	−	10327.2i	0.480522	−	0.480522i	−0.424777	−	0.905298i	$-0.639647\pi$
$773$	10327.2	−	10327.2i	0.480522	−	0.480522i	0.905298	+	0.424777i	$0.139647\pi$
$774$		0			0
$775$	−3687.98	−	28976.0i	−0.170937	−	1.34303i
$776$		−	9290.49i		−	0.429780i
$777$		0			0
$778$	−20533.4	−	20533.4i	−0.946220	−	0.946220i
$779$	−28602.1			−1.31550
$780$		0			0
$781$	−17810.1			−0.816000
$782$	12581.2	+	12581.2i	0.575324	+	0.575324i
$783$		0			0
$784$			8226.74i			0.374760i
$785$	−2467.73	+	156.412i	−0.112200	+	0.00711157i
$786$

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.62140 - 2.62140i) q^{2} +5.74350i q^{4} +(-8.38994 - 7.38978i) q^{5} +(-10.8652 + 10.8652i) q^{7} +(-5.91519 + 5.91519i) q^{8} +O(q^{10})\)	\(q+(-2.62140 - 2.62140i) q^{2} +5.74350i q^{4} +(-8.38994 - 7.38978i) q^{5} +(-10.8652 + 10.8652i) q^{7} +(-5.91519 + 5.91519i) q^{8} +(2.62183 + 41.3650i) q^{10} +37.8905i q^{11} +(-48.1085 - 48.1085i) q^{13} +56.9640 q^{14} +76.9602 q^{16} +(-60.3458 - 60.3458i) q^{17} -109.678i q^{19} +(42.4432 - 48.1877i) q^{20} +(99.3262 - 99.3262i) q^{22} +(39.7660 - 39.7660i) q^{23} +(15.7823 + 124.000i) q^{25} +252.224i q^{26} +(-62.4042 - 62.4042i) q^{28} +90.3553 q^{29} -233.678 q^{31} +(-154.422 - 154.422i) q^{32} +316.381i q^{34} +(171.449 - 10.8670i) q^{35} +(-19.0042 + 19.0042i) q^{37} +(-287.509 + 287.509i) q^{38} +(93.3400 - 5.91616i) q^{40} -260.783i q^{41} +(176.216 + 176.216i) q^{43} -217.624 q^{44} -208.485 q^{46} +(-145.788 - 145.788i) q^{47} +106.896i q^{49} +(283.681 - 366.425i) q^{50} +(276.311 - 276.311i) q^{52} +(-183.285 + 183.285i) q^{53} +(280.002 - 317.899i) q^{55} -128.539i q^{56} +(-236.858 - 236.858i) q^{58} +279.564 q^{59} -390.267 q^{61} +(612.563 + 612.563i) q^{62} +193.924i q^{64} +(48.1164 + 759.139i) q^{65} +(150.444 - 150.444i) q^{67} +(346.596 - 346.596i) q^{68} +(-477.925 - 420.951i) q^{70} +470.042i q^{71} +(480.765 + 480.765i) q^{73} +99.6355 q^{74} +629.934 q^{76} +(-411.687 - 411.687i) q^{77} -1322.44i q^{79} +(-645.692 - 568.719i) q^{80} +(-683.618 + 683.618i) q^{82} +(456.192 - 456.192i) q^{83} +(60.3558 + 952.240i) q^{85} -923.868i q^{86} +(-224.129 - 224.129i) q^{88} -1364.85 q^{89} +1045.41 q^{91} +(228.396 + 228.396i) q^{92} +764.340i q^{94} +(-810.494 + 920.190i) q^{95} +(-785.308 + 785.308i) q^{97} +(280.218 - 280.218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Introduction

L-functions

Modular forms

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Database

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