LMFDB - Embedded newform 936.2.m.h.181.1

Newspace parameters

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.m (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.47399762919$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
Defining polynomial:	$ x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 $ x^16 + 2x^14 - 16x^12 - 72x^10 + 26x^8 + 360x^6 + 725x^4 + 1000*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$-0.556839 - 1.81878i$ of defining polynomial
Character	$\chi$	$=$	936.181
Dual form			936.2.m.h.181.4

$q$-expansion

$f(q)$	$=$	$q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} -1.66251 q^{5} -3.57266i q^{7} +(-1.66251 - 2.28825i) q^{8} +O(q^{10})$	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} -1.66251 q^{5} -3.57266i q^{7} +(-1.66251 - 2.28825i) q^{8} +(2.23607 + 0.726543i) q^{10} -1.02749 q^{11} +(1.87826 - 3.07768i) q^{13} +(-1.56131 + 4.80522i) q^{14} +(1.23607 + 3.80423i) q^{16} +5.05251 q^{17} +1.16083 q^{19} +(-2.68999 - 1.95440i) q^{20} +(1.38197 + 0.449028i) q^{22} -8.17513 q^{23} -2.23607 q^{25} +(-3.87125 + 3.31865i) q^{26} +(4.19992 - 5.78069i) q^{28} +4.29792i q^{29} +7.98872i q^{31} -5.65685i q^{32} +(-6.79561 - 2.20803i) q^{34} +5.93958i q^{35} -9.83470 q^{37} +(-1.56131 - 0.507301i) q^{38} +(2.76393 + 3.80423i) q^{40} -1.62054i q^{41} -2.35114i q^{43} +(-1.66251 - 1.20788i) q^{44} +(10.9955 + 3.57266i) q^{46} -7.73877i q^{47} -5.76393 q^{49} +(3.00750 + 0.977198i) q^{50} +(6.65712 - 2.77177i) q^{52} -11.2521i q^{53} +1.70820 q^{55} +(-8.17513 + 5.93958i) q^{56} +(1.87826 - 5.78069i) q^{58} -9.73249 q^{59} -11.4127i q^{61} +(3.49120 - 10.7448i) q^{62} +(-2.47214 + 7.60845i) q^{64} +(-3.12262 + 5.11667i) q^{65} -7.23901 q^{67} +(8.17513 + 5.93958i) q^{68} +(2.59569 - 7.98872i) q^{70} -10.5672i q^{71} +4.41606i q^{73} +(13.2276 + 4.29792i) q^{74} +(1.87826 + 1.36464i) q^{76} +3.67086i q^{77} -6.94427 q^{79} +(-2.05497 - 6.32456i) q^{80} +(-0.708204 + 2.17963i) q^{82} -2.29753 q^{83} -8.39984 q^{85} +(-1.02749 + 3.16228i) q^{86} +(1.70820 + 2.35114i) q^{88} +9.02546i q^{89} +(-10.9955 - 6.71040i) q^{91} +(-13.2276 - 9.61045i) q^{92} +(-3.38197 + 10.4086i) q^{94} -1.92989 q^{95} +11.5614i q^{97} +(7.75247 + 2.51893i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94}+O(q^{100}) $ 16 * q + 8 * q^4 - 16 * q^16 + 40 * q^22 + 80 * q^40 - 128 * q^49 + 40 * q^52 - 80 * q^55 + 32 * q^64 + 32 * q^79 + 96 * q^82 - 80 * q^88 - 72 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$2$	−1.34500	−	0.437016i	−0.951057	−	0.309017i
$3$		0			0
$3$		0			0
$4$	1.61803	+	1.17557i	0.809017	+	0.587785i
$5$	−1.66251			−0.743496			−0.371748	−	0.928334i	$-0.621241\pi$
$5$	−1.66251			−0.743496			−0.371748	+	0.928334i	$0.621241\pi$
$6$		0			0
$7$		−	3.57266i		−	1.35034i	−0.737662	−	0.675170i	$-0.764070\pi$
$7$		−	3.57266i		−	1.35034i	0.737662	−	0.675170i	$-0.235930\pi$
$8$	−1.66251	−	2.28825i	−0.587785	−	0.809017i
$9$		0			0
$10$	2.23607	+	0.726543i	0.707107	+	0.229753i
$11$	−1.02749			−0.309799			−0.154899	−	0.987930i	$-0.549505\pi$
$11$	−1.02749			−0.309799			−0.154899	+	0.987930i	$0.549505\pi$
$12$		0			0
$13$	1.87826	−	3.07768i	0.520936	−	0.853596i
$13$	1.87826	−	3.07768i	0.520936	−	0.853596i
$14$	−1.56131	+	4.80522i	−0.417278	+	1.28425i
$15$		0			0
$16$	1.23607	+	3.80423i	0.309017	+	0.951057i
$17$	5.05251			1.22541			0.612707	−	0.790310i	$-0.290081\pi$
$17$	5.05251			1.22541			0.612707	+	0.790310i	$0.290081\pi$
$18$		0			0
$19$	1.16083			0.266312			0.133156	−	0.991095i	$-0.457489\pi$
$19$	1.16083			0.266312			0.133156	+	0.991095i	$0.457489\pi$
$20$	−2.68999	−	1.95440i	−0.601501	−	0.437016i
$21$		0			0
$22$	1.38197	+	0.449028i	0.294636	+	0.0957331i
$23$	−8.17513			−1.70463			−0.852317	−	0.523026i	$-0.824803\pi$
$23$	−8.17513			−1.70463			−0.852317	+	0.523026i	$0.824803\pi$
$24$		0			0
$25$	−2.23607			−0.447214
$26$	−3.87125	+	3.31865i	−0.759215	+	0.650840i
$27$		0			0
$28$	4.19992	−	5.78069i	0.793710	−	1.09245i
$29$			4.29792i			0.798104i	0.916928	+	0.399052i	$0.130661\pi$
$29$			4.29792i			0.798104i	−0.916928	+	0.399052i	$0.869339\pi$
$30$		0			0
$31$			7.98872i			1.43482i	0.696653	+	0.717408i	$0.254672\pi$
$31$			7.98872i			1.43482i	−0.696653	+	0.717408i	$0.745328\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	−6.79561	−	2.20803i	−1.16544	−	0.378674i
$35$			5.93958i			1.00397i
$36$		0			0
$37$	−9.83470			−1.61682			−0.808408	−	0.588623i	$-0.799670\pi$
$37$	−9.83470			−1.61682			−0.808408	+	0.588623i	$0.799670\pi$
$38$	−1.56131	−	0.507301i	−0.253278	−	0.0822951i
$39$		0			0
$40$	2.76393	+	3.80423i	0.437016	+	0.601501i
$41$		−	1.62054i		−	0.253087i	−0.991961	−	0.126543i	$-0.959612\pi$
$41$		−	1.62054i		−	0.253087i	0.991961	−	0.126543i	$-0.0403883\pi$
$42$		0			0
$43$		−	2.35114i		−	0.358546i	−0.983799	−	0.179273i	$-0.942626\pi$
$43$		−	2.35114i		−	0.358546i	0.983799	−	0.179273i	$-0.0573745\pi$
$44$	−1.66251	−	1.20788i	−0.250632	−	0.182095i
$45$		0			0
$46$	10.9955	+	3.57266i	1.62120	+	0.526761i
$47$		−	7.73877i		−	1.12882i	−0.825496	−	0.564408i	$-0.809105\pi$
$47$		−	7.73877i		−	1.12882i	0.825496	−	0.564408i	$-0.190895\pi$
$48$		0			0
$49$	−5.76393			−0.823419
$50$	3.00750	+	0.977198i	0.425325	+	0.138197i
$51$		0			0
$52$	6.65712	−	2.77177i	0.923177	−	0.384375i
$53$		−	11.2521i		−	1.54560i	−0.634652	−	0.772798i	$-0.718857\pi$
$53$		−	11.2521i		−	1.54560i	0.634652	−	0.772798i	$-0.281143\pi$
$54$		0			0
$55$	1.70820			0.230334
$56$	−8.17513	+	5.93958i	−1.09245	+	0.793710i
$57$		0			0
$58$	1.87826	−	5.78069i	0.246628	−	0.759042i
$59$	−9.73249			−1.26706			−0.633531	−	0.773717i	$-0.718395\pi$
$59$	−9.73249			−1.26706			−0.633531	+	0.773717i	$0.718395\pi$
$60$		0			0
$61$		−	11.4127i		−	1.46124i	−0.682782	−	0.730622i	$-0.739230\pi$
$61$		−	11.4127i		−	1.46124i	0.682782	−	0.730622i	$-0.260770\pi$
$62$	3.49120	−	10.7448i	0.443383	−	1.36459i
$63$		0			0
$64$	−2.47214	+	7.60845i	−0.309017	+	0.951057i
$65$	−3.12262	+	5.11667i	−0.387314	+	0.634645i
$66$		0			0
$67$	−7.23901			−0.884386			−0.442193	−	0.896920i	$-0.645799\pi$
$67$	−7.23901			−0.884386			−0.442193	+	0.896920i	$0.645799\pi$
$68$	8.17513	+	5.93958i	0.991381	+	0.720280i
$69$		0			0
$70$	2.59569	−	7.98872i	0.310245	−	0.954835i
$71$		−	10.5672i		−	1.25410i	−0.778981	−	0.627048i	$-0.784263\pi$
$71$		−	10.5672i		−	1.25410i	0.778981	−	0.627048i	$-0.215737\pi$
$72$		0			0
$73$			4.41606i			0.516860i	0.966030	+	0.258430i	$0.0832052\pi$
$73$			4.41606i			0.516860i	−0.966030	+	0.258430i	$0.916795\pi$
$74$	13.2276	+	4.29792i	1.53768	+	0.499623i
$75$		0			0
$76$	1.87826	+	1.36464i	0.215451	+	0.156535i
$77$			3.67086i			0.418334i
$78$		0			0
$79$	−6.94427			−0.781292			−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427			−0.781292			−0.390646	+	0.920541i	$0.627748\pi$
$80$	−2.05497	−	6.32456i	−0.229753	−	0.707107i
$81$		0			0
$82$	−0.708204	+	2.17963i	−0.0782080	+	0.240700i
$83$	−2.29753			−0.252187			−0.126093	−	0.992018i	$-0.540244\pi$
$83$	−2.29753			−0.252187			−0.126093	+	0.992018i	$0.540244\pi$
$84$		0			0
$85$	−8.39984			−0.911090
$86$	−1.02749	+	3.16228i	−0.110797	+	0.340997i
$87$		0			0
$88$	1.70820	+	2.35114i	0.182095	+	0.250632i
$89$			9.02546i			0.956697i	0.878170	+	0.478349i	$0.158764\pi$
$89$			9.02546i			0.956697i	−0.878170	+	0.478349i	$0.841236\pi$
$90$		0			0
$91$	−10.9955	−	6.71040i	−1.15264	−	0.703441i
$92$	−13.2276	−	9.61045i	−1.37908	−	1.00196i
$93$		0			0
$94$	−3.38197	+	10.4086i	−0.348823	+	1.07357i
$95$	−1.92989			−0.198002
$96$		0			0
$97$			11.5614i			1.17388i	0.809630	+	0.586940i	$0.199668\pi$
$97$			11.5614i			1.17388i	−0.809630	+	0.586940i	$0.800332\pi$
$98$	7.75247	+	2.51893i	0.783118	+	0.254450i
$99$		0			0
$100$	−3.61803	−	2.62866i	−0.361803	−	0.262866i
$101$			7.96879i			0.792924i	0.918051	+	0.396462i	$0.129762\pi$
$101$			7.96879i			0.792924i	−0.918051	+	0.396462i	$0.870238\pi$
$102$		0			0
$103$	−11.7082			−1.15364			−0.576822	−	0.816870i	$-0.695707\pi$
$103$	−11.7082			−1.15364			−0.576822	+	0.816870i	$0.695707\pi$
$104$	−10.1651	+	0.818750i	−0.996772	+	0.0802851i
$105$		0			0
$106$	−4.91735	+	15.1340i	−0.477615	+	1.46995i
$107$		−	2.65626i		−	0.256791i	−0.991723	−	0.128395i	$-0.959017\pi$
$107$		−	2.65626i		−	0.256791i	0.991723	−	0.128395i	$-0.0409826\pi$
$108$		0			0
$109$	12.1564			1.16437			0.582184	−	0.813057i	$-0.302198\pi$
$109$	12.1564			1.16437			0.582184	+	0.813057i	$0.302198\pi$
$110$	−2.29753	−	0.746512i	−0.219061	−	0.0711772i
$111$		0			0
$112$	13.5912	−	4.41606i	1.28425	−	0.417278i
$113$	5.05251			0.475300			0.237650	−	0.971351i	$-0.423623\pi$
$113$	5.05251			0.475300			0.237650	+	0.971351i	$0.423623\pi$
$114$		0			0
$115$	13.5912			1.26739
$116$	−5.05251	+	6.95418i	−0.469114	+	0.645680i
$117$		0			0
$118$	13.0902	+	4.25325i	1.20505	+	0.391544i
$119$		−	18.0509i		−	1.65473i
$120$		0			0
$121$	−9.94427			−0.904025
$122$	−4.98752	+	15.3500i	−0.451549	+	1.38973i
$123$		0			0
$124$	−9.39130	+	12.9260i	−0.843364	+	1.16079i
$125$	12.0300			1.07600
$126$		0			0
$127$	−6.00000			−0.532414			−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000			−0.532414			−0.266207	+	0.963916i	$0.585770\pi$
$128$	6.65003	−	9.15298i	0.587785	−	0.809017i
$129$		0			0
$130$	6.43599	−	5.51727i	0.564473	−	0.483897i
$131$			16.5646i			1.44726i	0.690189	+	0.723629i	$0.257527\pi$
$131$			16.5646i			1.44726i	−0.690189	+	0.723629i	$0.742473\pi$
$132$		0			0
$133$		−	4.14725i		−	0.359612i
$134$	9.73645	+	3.16356i	0.841101	+	0.273290i
$135$		0			0
$136$	−8.39984	−	11.5614i	−0.720280	−	0.991381i
$137$			13.6020i			1.16209i	0.813870	+	0.581047i	$0.197357\pi$
$137$			13.6020i			1.16209i	−0.813870	+	0.581047i	$0.802643\pi$
$138$		0			0
$139$		−	19.3642i		−	1.64245i	−0.570607	−	0.821223i	$-0.693292\pi$
$139$		−	19.3642i		−	1.64245i	0.570607	−	0.821223i	$-0.306708\pi$
$140$	−6.98240	+	9.61045i	−0.590120	+	0.812231i
$141$		0			0
$142$	−4.61803	+	14.2128i	−0.387537	+	1.19272i
$143$	−1.92989	+	3.16228i	−0.161385	+	0.264443i
$144$		0			0
$145$		−	7.14533i		−	0.593387i
$146$	1.92989	−	5.93958i	0.159719	−	0.491563i
$147$		0			0
$148$	−15.9129	−	11.5614i	−1.30803	−	0.950340i
$149$	−7.04250			−0.576944			−0.288472	−	0.957488i	$-0.593147\pi$
$149$	−7.04250			−0.576944			−0.288472	+	0.957488i	$0.593147\pi$
$150$		0			0
$151$			3.57266i			0.290739i	0.989377	+	0.145370i	$0.0464371\pi$
$151$			3.57266i			0.290739i	−0.989377	+	0.145370i	$0.953563\pi$
$152$	−1.92989	−	2.65626i	−0.156535	−	0.215451i
$153$		0			0
$154$	1.60423	−	4.93730i	0.129272	−	0.397859i
$155$		−	13.2813i		−	1.06678i
$156$		0			0
$157$		−	0.555029i		−	0.0442961i	−0.999755	−	0.0221481i	$-0.992949\pi$
$157$		−	0.555029i		−	0.0442961i	0.999755	−	0.0221481i	$-0.00705053\pi$
$158$	9.34003	+	3.03476i	0.743052	+	0.241432i
$159$		0			0
$160$			9.40456i			0.743496i
$161$			29.2070i			2.30183i
$162$		0			0
$163$	−7.23901			−0.567003			−0.283501	−	0.958972i	$-0.591496\pi$
$163$	−7.23901			−0.567003			−0.283501	+	0.958972i	$0.591496\pi$
$164$	1.90506	−	2.62210i	0.148761	−	0.204751i
$165$		0			0
$166$	3.09017	+	1.00406i	0.239844	+	0.0779299i
$167$		−	7.32611i		−	0.566911i	−0.958985	−	0.283456i	$-0.908519\pi$
$167$		−	7.32611i		−	0.566911i	0.958985	−	0.283456i	$-0.0914809\pi$
$168$		0			0
$169$	−5.94427	−	11.5614i	−0.457252	−	0.889337i
$170$	11.2978	+	3.67086i	0.866498	+	0.281542i
$171$		0			0
$172$	2.76393	−	3.80423i	0.210748	−	0.290070i
$173$		−	11.2521i		−	0.855482i	−0.903901	−	0.427741i	$-0.859310\pi$
$173$		−	11.2521i		−	0.855482i	0.903901	−	0.427741i	$-0.140690\pi$
$174$		0			0
$175$			7.98872i			0.603891i
$176$	−1.27004	−	3.90879i	−0.0957331	−	0.294636i
$177$		0			0
$178$	3.94427	−	12.1392i	0.295636	−	0.909873i
$179$		−	21.8772i		−	1.63518i	−0.575804	−	0.817588i	$-0.695311\pi$
$179$		−	21.8772i		−	1.63518i	0.575804	−	0.817588i	$-0.304689\pi$
$180$		0			0
$181$			12.8658i			0.956305i	0.878277	+	0.478152i	$0.158693\pi$
$181$			12.8658i			0.956305i	−0.878277	+	0.478152i	$0.841307\pi$
$182$	11.8564	+	13.8307i	0.878855	+	1.02520i
$183$		0			0
$184$	13.5912	+	18.7067i	1.00196	+	1.37908i
$185$	16.3503			1.20210
$186$		0			0
$187$	−5.19139			−0.379632
$188$	9.09747	−	12.5216i	0.663501	−	0.913231i
$189$		0			0
$190$	2.59569	+	0.843392i	0.188311	+	0.0611861i
$191$	−1.92989			−0.139642			−0.0698209	−	0.997560i	$-0.522243\pi$
$191$	−1.92989			−0.139642			−0.0698209	+	0.997560i	$0.522243\pi$
$192$		0			0
$193$			4.41606i			0.317875i	0.987289	+	0.158937i	$0.0508068\pi$
$193$			4.41606i			0.317875i	−0.987289	+	0.158937i	$0.949193\pi$
$194$	5.05251	−	15.5500i	0.362749	−	1.11643i
$195$		0			0
$196$	−9.32624	−	6.77591i	−0.666160	−	0.483993i
$197$	23.1825			1.65168			0.825841	−	0.563903i	$-0.190701\pi$
$197$	23.1825			1.65168			0.825841	+	0.563903i	$0.190701\pi$
$198$		0			0
$199$	18.0000			1.27599			0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000			1.27599			0.637993	+	0.770042i	$0.279765\pi$
$200$	3.71748	+	5.11667i	0.262866	+	0.361803i
$201$		0			0
$202$	3.48249	−	10.7180i	0.245027	−	0.754115i
$203$	15.3550			1.07771
$204$		0			0
$205$			2.69417i			0.188169i
$206$	15.7475	+	5.11667i	1.09718	+	0.356495i
$207$		0			0
$208$	14.0299	+	3.34110i	0.972796	+	0.231664i
$209$	−1.19274			−0.0825033
$210$		0			0
$211$		−	10.5146i		−	0.723856i	−0.932206	−	0.361928i	$-0.882119\pi$
$211$		−	10.5146i		−	0.723856i	0.932206	−	0.361928i	$-0.117881\pi$
$212$	13.2276	−	18.2063i	0.908478	−	1.25041i
$213$		0			0
$214$	−1.16083	+	3.57266i	−0.0793526	+	0.244222i
$215$			3.90879i			0.266577i
$216$		0			0
$217$	28.5410			1.93749
$218$	−16.3503	−	5.31252i	−1.10738	−	0.359810i
$219$		0			0
$220$	2.76393	+	2.00811i	0.186344	+	0.135387i
$221$	9.48993	−	15.5500i	0.638362	−	1.04601i
$222$		0			0
$223$		−	7.98872i		−	0.534964i	−0.963563	−	0.267482i	$-0.913808\pi$
$223$		−	7.98872i		−	0.534964i	0.963563	−	0.267482i	$-0.0861916\pi$
$224$	−20.2100			−1.35034
$225$		0			0
$226$	−6.79561	−	2.20803i	−0.452037	−	0.146876i
$227$	4.35250			0.288886			0.144443	−	0.989513i	$-0.453861\pi$
$227$	4.35250			0.288886			0.144443	+	0.989513i	$0.453861\pi$
$228$		0			0
$229$	−1.43486			−0.0948185			−0.0474092	−	0.998876i	$-0.515096\pi$
$229$	−1.43486			−0.0948185			−0.0474092	+	0.998876i	$0.515096\pi$
$230$	−18.2802	−	5.93958i	−1.20536	−	0.391644i
$231$		0			0
$232$	9.83470	−	7.14533i	0.645680	−	0.469114i
$233$	16.3503			1.07114			0.535571	−	0.844490i	$-0.320096\pi$
$233$	16.3503			1.07114			0.535571	+	0.844490i	$0.320096\pi$
$234$		0			0
$235$			12.8658i			0.839270i
$236$	−15.7475	−	11.4412i	−1.02507	−	0.744761i
$237$		0			0
$238$	−7.88854	+	24.2784i	−0.511338	+	1.57374i
$239$			7.07107i			0.457389i	0.973498	+	0.228695i	$0.0734457\pi$
$239$			7.07107i			0.457389i	−0.973498	+	0.228695i	$0.926554\pi$
$240$		0			0
$241$		−	15.9774i		−	1.02920i	−0.857431	−	0.514599i	$-0.827941\pi$
$241$		−	15.9774i		−	1.02920i	0.857431	−	0.514599i	$-0.172059\pi$
$242$	13.3750	+	4.34581i	0.859779	+	0.279359i
$243$		0			0
$244$	13.4164	−	18.4661i	0.858898	−	1.18217i
$245$	9.58258			0.612209
$246$		0			0
$247$	2.18034	−	3.57266i	0.138732	−	0.227323i
$248$	18.2802	−	13.2813i	1.16079	−	0.843364i
$249$		0			0
$250$	−16.1803	−	5.25731i	−1.02333	−	0.332502i
$251$			22.5042i			1.42045i	0.703973	+	0.710227i	$0.251408\pi$
$251$			22.5042i			1.42045i	−0.703973	+	0.710227i	$0.748592\pi$
$252$		0			0
$253$	8.39984			0.528093
$254$	8.06998	+	2.62210i	0.506356	+	0.164525i
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	22.5955			1.40947			0.704735	−	0.709471i	$-0.251066\pi$
$257$	22.5955			1.40947			0.704735	+	0.709471i	$0.251066\pi$
$258$		0			0
$259$			35.1361i			2.18325i
$260$	−11.0675	+	4.60809i	−0.686378	+	0.285781i
$261$		0			0
$262$	7.23901	−	22.2794i	0.447227	−	1.37642i
$263$	−16.3503			−1.00820			−0.504100	−	0.863645i	$-0.668176\pi$
$263$	−16.3503			−1.00820			−0.504100	+	0.863645i	$0.668176\pi$
$264$		0			0
$265$			18.7067i			1.14914i
$266$	−1.81242	+	5.57804i	−0.111126	+	0.342012i
$267$		0			0
$268$	−11.7130	−	8.50997i	−0.715483	−	0.519829i
$269$		−	7.58124i		−	0.462237i	−0.972926	−	0.231118i	$-0.925762\pi$
$269$		−	7.58124i		−	0.462237i	0.972926	−	0.231118i	$-0.0742384\pi$
$270$		0			0
$271$		−	7.98872i		−	0.485280i	−0.970116	−	0.242640i	$-0.921987\pi$
$271$		−	7.98872i		−	0.485280i	0.970116	−	0.242640i	$-0.0780134\pi$
$272$	6.24525	+	19.2209i	0.378674	+	1.16544i
$273$		0			0
$274$	5.94427	−	18.2946i	0.359107	−	1.10522i
$275$	2.29753			0.138546
$276$		0			0
$277$			18.1231i			1.08891i	0.838790	+	0.544455i	$0.183263\pi$
$277$			18.1231i			1.08891i	−0.838790	+	0.544455i	$0.816737\pi$
$278$	−8.46245	+	26.0447i	−0.507544	+	1.56206i
$279$		0			0
$280$	13.5912	−	9.87460i	0.812231	−	0.590120i
$281$		−	6.19704i		−	0.369684i	−0.982768	−	0.184842i	$-0.940823\pi$
$281$		−	6.19704i		−	0.369684i	0.982768	−	0.184842i	$-0.0591774\pi$
$282$		0			0
$283$		−	2.90617i		−	0.172754i	−0.996263	−	0.0863769i	$-0.972471\pi$
$283$		−	2.90617i		−	0.172754i	0.996263	−	0.0863769i	$-0.0275289\pi$
$284$	12.4225	−	17.0981i	0.737139	−	1.01458i
$285$		0			0
$286$	3.97766	−	3.40986i	0.235204	−	0.201629i
$287$	−5.78966			−0.341753
$288$		0			0
$289$	8.52786			0.501639
$290$	−3.12262	+	9.61045i	−0.183367	+	0.564345i
$291$		0			0
$292$	−5.19139	+	7.14533i	−0.303803	+	0.418149i
$293$	−0.877578			−0.0512687			−0.0256343	−	0.999671i	$-0.508161\pi$
$293$	−0.877578			−0.0512687			−0.0256343	+	0.999671i	$0.508161\pi$
$294$		0			0
$295$	16.1803			0.942056
$296$	16.3503	+	22.5042i	0.950340	+	1.30803i
$297$		0			0
$298$	9.47214	+	3.07768i	0.548706	+	0.178285i
$299$	−15.3550	+	25.1605i	−0.888005	+	1.45507i
$300$		0			0
$301$	−8.39984			−0.484159
$302$	1.56131	−	4.80522i	0.0898434	−	0.276510i
$303$		0			0
$304$	1.43486	+	4.41606i	0.0822951	+	0.253278i
$305$			18.9737i			1.08643i
$306$		0			0
$307$	14.7521			0.841944			0.420972	−	0.907074i	$-0.361689\pi$
$307$	14.7521			0.841944			0.420972	+	0.907074i	$0.361689\pi$
$308$	−4.31536	+	5.93958i	−0.245890	+	0.338439i
$309$		0			0
$310$	−5.80415	+	17.8633i	−0.329653	+	1.01457i
$311$	−24.5254			−1.39071			−0.695354	−	0.718667i	$-0.744752\pi$
$311$	−24.5254			−1.39071			−0.695354	+	0.718667i	$0.744752\pi$
$312$		0			0
$313$	6.47214			0.365827			0.182913	−	0.983129i	$-0.441447\pi$
$313$	6.47214			0.365827			0.182913	+	0.983129i	$0.441447\pi$
$314$	−0.242557	+	0.746512i	−0.0136883	+	0.0421281i
$315$		0			0
$316$	−11.2361	−	8.16348i	−0.632078	−	0.459232i
$317$	−31.1025			−1.74689			−0.873446	−	0.486921i	$-0.838120\pi$
$317$	−31.1025			−1.74689			−0.873446	+	0.486921i	$0.838120\pi$
$318$		0			0
$319$		−	4.41606i		−	0.247252i
$320$	4.10995	−	12.6491i	0.229753	−	0.707107i
$321$		0			0
$322$	12.7639	−	39.2833i	0.711306	−	2.18918i
$323$	5.86510			0.326343
$324$		0			0
$325$	−4.19992	+	6.88191i	−0.232970	+	0.381740i
$326$	9.73645	+	3.16356i	0.539252	+	0.175214i
$327$		0			0
$328$	−3.70820	+	2.69417i	−0.204751	+	0.148761i
$329$	−27.6480			−1.52428
$330$		0			0
$331$	10.9955			0.604369			0.302185	−	0.953249i	$-0.402284\pi$
$331$	10.9955			0.604369			0.302185	+	0.953249i	$0.402284\pi$
$332$	−3.71748	−	2.70091i	−0.204023	−	0.148232i
$333$		0			0
$334$	−3.20163	+	9.85359i	−0.175185	+	0.539165i
$335$	12.0349			0.657537
$336$		0			0
$337$	5.52786			0.301122			0.150561	−	0.988601i	$-0.451892\pi$
$337$	5.52786			0.301122			0.150561	+	0.988601i	$0.451892\pi$
$338$	2.94252	+	18.1478i	0.160052	+	0.987109i
$339$		0			0
$340$	−13.5912	−	9.87460i	−0.737088	−	0.535525i
$341$		−	8.20830i		−	0.444504i
$342$		0			0
$343$		−	4.41606i		−	0.238445i
$344$	−5.37999	+	3.90879i	−0.290070	+	0.210748i
$345$		0			0
$346$	−4.91735	+	15.1340i	−0.264358	+	0.813612i
$347$		−	20.4750i		−	1.09916i	−0.835442	−	0.549578i	$-0.814788\pi$
$347$		−	20.4750i		−	1.09916i	0.835442	−	0.549578i	$-0.185212\pi$
$348$		0			0
$349$	13.5912			0.727522			0.363761	−	0.931492i	$-0.381493\pi$
$349$	13.5912			0.727522			0.363761	+	0.931492i	$0.381493\pi$
$350$	3.49120	−	10.7448i	0.186612	−	0.574334i
$351$		0			0
$352$			5.81234i			0.309799i
$353$		−	20.5942i		−	1.09612i	−0.836439	−	0.548060i	$-0.815367\pi$
$353$		−	20.5942i		−	1.09612i	0.836439	−	0.548060i	$-0.184633\pi$
$354$		0			0
$355$			17.5680i			0.932415i
$356$	−10.6101	+	14.6035i	−0.562332	+	0.773984i
$357$		0			0
$358$	−9.56067	+	29.4247i	−0.505297	+	1.55514i
$359$		−	32.1142i		−	1.69493i	−0.530855	−	0.847463i	$-0.678129\pi$
$359$		−	32.1142i		−	1.69493i	0.530855	−	0.847463i	$-0.321871\pi$
$360$		0			0
$361$	−17.6525			−0.929078
$362$	5.62254	−	17.3044i	0.295514	−	0.909500i
$363$		0			0
$364$	−9.90260	−	23.7837i	−0.519037	−	1.24660i
$365$		−	7.34173i		−	0.384284i
$366$		0			0
$367$	31.1246			1.62469			0.812346	−	0.583176i	$-0.198190\pi$
$367$	31.1246			1.62469			0.812346	+	0.583176i	$0.198190\pi$
$368$	−10.1050	−	31.1001i	−0.526761	−	1.62120i
$369$		0			0
$370$	−21.9911	−	7.14533i	−1.14326	−	0.371468i
$371$	−40.2000			−2.08708
$372$		0			0
$373$		−	26.0746i		−	1.35009i	−0.737777	−	0.675045i	$-0.764124\pi$
$373$		−	26.0746i		−	1.35009i	0.737777	−	0.675045i	$-0.235876\pi$
$374$	6.98240	+	2.26872i	0.361051	+	0.117313i
$375$		0			0
$376$	−17.7082	+	12.8658i	−0.913231	+	0.663501i
$377$	13.2276	+	8.07262i	0.681258	+	0.415761i
$378$		0			0
$379$	19.3954			0.996273			0.498137	−	0.867099i	$-0.334018\pi$
$379$	19.3954			0.996273			0.498137	+	0.867099i	$0.334018\pi$
$380$	−3.12262	−	2.26872i	−0.160187	−	0.116383i
$381$		0			0
$382$	2.59569	+	0.843392i	0.132807	+	0.0431517i
$383$			3.57494i			0.182671i	0.995820	+	0.0913354i	$0.0291135\pi$
$383$			3.57494i			0.182671i	−0.995820	+	0.0913354i	$0.970886\pi$
$384$		0			0
$385$		−	6.10284i		−	0.311030i
$386$	1.92989	−	5.93958i	0.0982287	−	0.302317i
$387$		0			0
$388$	−13.5912	+	18.7067i	−0.689990	+	0.949690i
$389$			35.3980i			1.79475i	0.441270	+	0.897374i	$0.354528\pi$
$389$			35.3980i			1.79475i	−0.441270	+	0.897374i	$0.645472\pi$
$390$		0			0
$391$	−41.3050			−2.08888
$392$	9.58258	+	13.1893i	0.483993	+	0.666160i
$393$		0			0
$394$	−31.1803	−	10.1311i	−1.57084	−	0.510398i
$395$	11.5449			0.580887
$396$		0			0
$397$	20.5562			1.03169			0.515843	−	0.856683i	$-0.327479\pi$
$397$	20.5562			1.03169			0.515843	+	0.856683i	$0.327479\pi$
$398$	−24.2099	−	7.86629i	−1.21353	−	0.394301i
$399$		0			0
$400$	−2.76393	−	8.50651i	−0.138197	−	0.425325i
$401$		−	30.4149i		−	1.51885i	−0.650596	−	0.759424i	$-0.725481\pi$
$401$		−	30.4149i		−	1.51885i	0.650596	−	0.759424i	$-0.274519\pi$
$402$		0			0
$403$	24.5868	+	15.0049i	1.22475	+	0.747447i
$404$	−9.36787	+	12.8938i	−0.466069	+	0.641489i
$405$		0			0
$406$	−20.6525	−	6.71040i	−1.02497	−	0.333031i
$407$	10.1050			0.500887
$408$		0			0
$409$		−	34.6842i		−	1.71502i	−0.514466	−	0.857511i	$-0.672010\pi$
$409$		−	34.6842i		−	1.71502i	0.514466	−	0.857511i	$-0.327990\pi$
$410$	1.17739	−	3.62365i	0.0581474	−	0.178959i
$411$		0			0
$412$	−18.9443	−	13.7638i	−0.933317	−	0.678095i
$413$			34.7709i			1.71097i
$414$		0			0
$415$	3.81966			0.187500
$416$	−17.4100	−	10.6250i	−0.853596	−	0.520936i
$417$		0			0
$418$	1.60423	+	0.521245i	0.0784653	+	0.0254949i
$419$		−	7.96879i		−	0.389301i	−0.980873	−	0.194650i	$-0.937643\pi$
$419$		−	7.96879i		−	0.389301i	0.980873	−	0.194650i	$-0.0623572\pi$
$420$		0			0
$421$	−8.39984			−0.409383			−0.204692	−	0.978827i	$-0.565619\pi$
$421$	−8.39984			−0.409383			−0.204692	+	0.978827i	$0.565619\pi$
$422$	−4.59506	+	14.1421i	−0.223684	+	0.688428i
$423$		0			0
$424$	−25.7476	+	18.7067i	−1.25041	+	0.908478i
$425$	−11.2978			−0.548022
$426$		0			0
$427$	−40.7737			−1.97318
$428$	3.12262	−	4.29792i	0.150938	−	0.207748i
$429$		0			0
$430$	1.70820	−	5.25731i	0.0823769	−	0.253530i
$431$			1.82688i			0.0879975i	0.999032	+	0.0439987i	$0.0140098\pi$
$431$			1.82688i			0.0879975i	−0.999032	+	0.0439987i	$0.985990\pi$
$432$		0			0
$433$	−8.18034			−0.393122			−0.196561	−	0.980492i	$-0.562977\pi$
$433$	−8.18034			−0.393122			−0.196561	+	0.980492i	$0.562977\pi$
$434$	−38.3876	−	12.4729i	−1.84266	−	0.598718i
$435$		0			0
$436$	19.6694	+	14.2907i	0.941994	+	0.684398i
$437$	−9.48993			−0.453965
$438$		0			0
$439$	1.81966			0.0868476			0.0434238	−	0.999057i	$-0.486173\pi$
$439$	1.81966			0.0868476			0.0434238	+	0.999057i	$0.486173\pi$
$440$	−2.83990	−	3.90879i	−0.135387	−	0.186344i
$441$		0			0
$442$	−19.5595	+	16.7675i	−0.930353	+	0.797548i
$443$			11.2521i			0.534604i	0.963613	+	0.267302i	$0.0861321\pi$
$443$			11.2521i			0.534604i	−0.963613	+	0.267302i	$0.913868\pi$
$444$		0			0
$445$		−	15.0049i		−	0.711301i
$446$	−3.49120	+	10.7448i	−0.165313	+	0.508781i
$447$		0			0
$448$	27.1824	+	8.83211i	1.28425	+	0.417278i
$449$			4.86163i			0.229435i	0.993398	+	0.114717i	$0.0365962\pi$
$449$			4.86163i			0.229435i	−0.993398	+	0.114717i	$0.963404\pi$
$450$		0			0
$451$			1.66509i			0.0784059i
$452$	8.17513	+	5.93958i	0.384526	+	0.279374i
$453$		0			0
$454$	−5.85410	−	1.90211i	−0.274747	−	0.0892706i
$455$	18.2802	+	11.1561i	0.856987	+	0.523005i
$456$		0			0
$457$			17.0199i			0.796159i	0.917351	+	0.398079i	$0.130323\pi$
$457$			17.0199i			0.796159i	−0.917351	+	0.398079i	$0.869677\pi$
$458$	1.92989	+	0.627058i	0.0901777	+	0.0293005i
$459$		0			0
$460$	21.9911	+	15.9774i	1.02534	+	0.744952i
$461$	−21.9124			−1.02056			−0.510282	−	0.860007i	$-0.670459\pi$
$461$	−21.9124			−1.02056			−0.510282	+	0.860007i	$0.670459\pi$
$462$		0			0
$463$		−	3.57266i		−	0.166036i	−0.996548	−	0.0830179i	$-0.973544\pi$
$463$		−	3.57266i		−	0.166036i	0.996548	−	0.0830179i	$-0.0264559\pi$
$464$	−16.3503	+	5.31252i	−0.759042	+	0.246628i
$465$		0			0
$466$	−21.9911	−	7.14533i	−1.01872	−	0.331001i
$467$		−	8.59584i		−	0.397768i	−0.980023	−	0.198884i	$-0.936268\pi$
$467$		−	8.59584i		−	0.397768i	0.980023	−	0.198884i	$-0.0637317\pi$
$468$		0			0
$469$			25.8626i			1.19422i
$470$	5.62254	−	17.3044i	0.259349	−	0.798193i
$471$		0			0
$472$	16.1803	+	22.2703i	0.744761	+	1.02507i
$473$			2.41577i			0.111077i
$474$		0			0
$475$	−2.59569			−0.119099
$476$	21.2201	−	29.2070i	0.972623	−	1.33870i
$477$		0			0
$478$	3.09017	−	9.51057i	0.141341	−	0.435003i
$479$		−	29.2858i		−	1.33810i	−0.743216	−	0.669052i	$-0.766700\pi$
$479$		−	29.2858i		−	1.33810i	0.743216	−	0.669052i	$-0.233300\pi$
$480$		0			0
$481$	−18.4721	+	30.2681i	−0.842257	+	1.38011i
$482$	−6.98240	+	21.4896i	−0.318040	+	0.978825i
$483$		0			0
$484$	−16.0902	−	11.6902i	−0.731371	−	0.531372i
$485$		−	19.2209i		−	0.872776i
$486$		0			0
$487$		−	0.843392i		−	0.0382177i	−0.999817	−	0.0191089i	$-0.993917\pi$
$487$		−	0.843392i		−	0.0382177i	0.999817	−	0.0191089i	$-0.00608291\pi$
$488$	−26.1150	+	18.9737i	−1.18217	+	0.858898i
$489$		0			0
$490$	−12.8885	−	4.18774i	−0.582245	−	0.189183i
$491$		−	2.02920i		−	0.0915767i	−0.998951	−	0.0457883i	$-0.985420\pi$
$491$		−	2.02920i		−	0.0915767i	0.998951	−	0.0457883i	$-0.0145800\pi$
$492$		0			0
$493$			21.7153i			0.978008i
$494$	−4.49386	+	3.85238i	−0.202188	+	0.173327i
$495$		0			0
$496$	−30.3909	+	9.87460i	−1.36459	+	0.443383i
$497$	−37.7530			−1.69346
$498$		0			0
$499$	33.5347			1.50122			0.750609	−	0.660747i	$-0.229760\pi$
$499$	33.5347			1.50122			0.750609	+	0.660747i	$0.229760\pi$
$500$	19.4650	+	14.1421i	0.870500	+	0.632456i
$501$		0			0
$502$	9.83470	−	30.2681i	0.438944	−	1.35093i
$503$	−6.24525			−0.278462			−0.139231	−	0.990260i	$-0.544463\pi$
$503$	−6.24525			−0.278462			−0.139231	+	0.990260i	$0.544463\pi$
$504$		0			0
$505$		−	13.2482i		−	0.589536i
$506$	−11.2978	−	3.67086i	−0.502247	−	0.163190i
$507$		0			0
$508$	−9.70820	−	7.05342i	−0.430732	−	0.312945i
$509$	9.58258			0.424740			0.212370	−	0.977189i	$-0.431882\pi$
$509$	9.58258			0.424740			0.212370	+	0.977189i	$0.431882\pi$
$510$		0			0
$511$	15.7771			0.697937
$512$	21.5200	−	6.99226i	0.951057	−	0.309017i
$513$		0			0
$514$	−30.3909	−	9.87460i	−1.34049	−	0.435550i
$515$	19.4650			0.857729
$516$		0			0
$517$			7.95148i			0.349706i
$518$	15.3550	−	47.2579i	0.674661	−	2.07639i
$519$		0			0
$520$	16.8996	−	1.36118i	0.741096	−	0.0596916i
$521$	−1.19274			−0.0522547			−0.0261274	−	0.999659i	$-0.508318\pi$
$521$	−1.19274			−0.0522547			−0.0261274	+	0.999659i	$0.508318\pi$
$522$		0			0
$523$		−	24.0664i		−	1.05235i	−0.850376	−	0.526176i	$-0.823625\pi$
$523$		−	24.0664i		−	1.05235i	0.850376	−	0.526176i	$-0.176375\pi$
$524$	−19.4729	+	26.8021i	−0.850677	+	1.17086i
$525$		0			0
$526$	21.9911	+	7.14533i	0.958856	+	0.311551i
$527$			40.3631i			1.75824i
$528$		0			0
$529$	43.8328			1.90577
$530$	8.17513	−	25.1605i	0.355105	−	1.09290i
$531$		0			0
$532$	4.87539	−	6.71040i	0.211375	−	0.290933i
$533$	−4.98752	−	3.04381i	−0.216034	−	0.131842i
$534$		0			0
$535$			4.41606i			0.190923i
$536$	12.0349	+	16.5646i	0.519829	+	0.715483i
$537$		0			0
$538$	−3.31312	+	10.1967i	−0.142839	+	0.439613i
$539$	5.92236			0.255094
$540$		0			0
$541$	−3.20845			−0.137942			−0.0689711	−	0.997619i	$-0.521972\pi$
$541$	−3.20845			−0.137942			−0.0689711	+	0.997619i	$0.521972\pi$
$542$	−3.49120	+	10.7448i	−0.149960	+	0.461529i
$543$		0			0
$544$		−	28.5813i		−	1.22541i
$545$	−20.2100			−0.865703
$546$		0			0
$547$			42.1895i			1.80389i	0.431847	+	0.901947i	$0.357862\pi$
$547$			42.1895i			1.80389i	−0.431847	+	0.901947i	$0.642138\pi$
$548$	−15.9901	+	22.0084i	−0.683061	+	0.940153i
$549$		0			0
$550$	−3.09017	−	1.00406i	−0.131765	−	0.0428131i
$551$			4.98915i			0.212545i
$552$		0			0
$553$			24.8096i			1.05501i
$554$	7.92007	−	24.3755i	0.336492	−	1.03561i
$555$		0			0
$556$	22.7639	−	31.3319i	0.965406	−	1.32877i
$557$	24.4525			1.03609			0.518043	−	0.855355i	$-0.326661\pi$
$557$	24.4525			1.03609			0.518043	+	0.855355i	$0.326661\pi$
$558$		0			0
$559$	−7.23607	−	4.41606i	−0.306053	−	0.186779i
$560$	−22.5955	+	7.34173i	−0.954835	+	0.310245i
$561$		0			0
$562$	−2.70820	+	8.33499i	−0.114239	+	0.351591i
$563$		−	0.627058i		−	0.0264274i	−0.999913	−	0.0132137i	$-0.995794\pi$
$563$		−	0.627058i		−	0.0264274i	0.999913	−	0.0132137i	$-0.00420617\pi$
$564$		0			0
$565$	−8.39984			−0.353384
$566$	−1.27004	+	3.90879i	−0.0533839	+	0.164299i
$567$		0			0
$568$	−24.1803	+	17.5680i	−1.01458	+	0.737139i
$569$	−26.4553			−1.10906			−0.554532	−	0.832163i	$-0.687103\pi$
$569$	−26.4553			−1.10906			−0.554532	+	0.832163i	$0.687103\pi$
$570$		0			0
$571$		−	24.6215i		−	1.03038i	−0.857077	−	0.515188i	$-0.827722\pi$
$571$		−	24.6215i		−	1.03038i	0.857077	−	0.515188i	$-0.172278\pi$
$572$	−6.84010	+	2.84795i	−0.285999	+	0.119079i
$573$		0			0
$574$	7.78708	+	2.53018i	0.325026	+	0.105607i
$575$	18.2802			0.762335
$576$		0			0
$577$		−	44.5588i		−	1.85501i	−0.373816	−	0.927503i	$-0.621951\pi$
$577$		−	44.5588i		−	1.85501i	0.373816	−	0.927503i	$-0.378049\pi$
$578$	−11.4700	−	3.72681i	−0.477087	−	0.155015i
$579$		0			0
$580$	8.39984	−	11.5614i	0.348784	−	0.480060i
$581$			8.20830i			0.340538i
$582$		0			0
$583$			11.5614i			0.478824i
$584$	10.1050	−	7.34173i	0.418149	−	0.303803i
$585$		0			0
$586$	1.18034	+	0.383516i	0.0487594	+	0.0158429i
$587$	−10.5174			−0.434100			−0.217050	−	0.976160i	$-0.569644\pi$
$587$	−10.5174			−0.434100			−0.217050	+	0.976160i	$0.569644\pi$
$588$		0			0
$589$			9.27354i			0.382110i
$590$	−21.7625	−	7.07107i	−0.895948	−	0.291111i
$591$		0			0
$592$	−12.1564	−	37.4134i	−0.499623	−	1.53768i
$593$		−	26.6637i		−	1.09495i	−0.836823	−	0.547474i	$-0.815589\pi$
$593$		−	26.6637i		−	1.09495i	0.836823	−	0.547474i	$-0.184411\pi$
$594$		0			0
$595$			30.0098i			1.23028i
$596$	−11.3950	−	8.27895i	−0.466757	−	0.339119i
$597$		0			0
$598$	31.6480	−	27.1304i	1.29418	−	1.10944i
$599$	46.6653			1.90669			0.953347	−	0.301877i	$-0.0976132\pi$
$599$	46.6653			1.90669			0.953347	+	0.301877i	$0.0976132\pi$
$600$		0			0
$601$	19.4164			0.792012			0.396006	−	0.918248i	$-0.370396\pi$
$601$	19.4164			0.792012			0.396006	+	0.918248i	$0.370396\pi$
$602$	11.2978	+	3.67086i	0.460462	+	0.149613i
$603$		0			0
$604$	−4.19992	+	5.78069i	−0.170892	+	0.235213i
$605$	16.5324			0.672139
$606$		0			0
$607$	15.7082			0.637576			0.318788	−	0.947826i	$-0.396724\pi$
$607$	15.7082			0.637576			0.318788	+	0.947826i	$0.396724\pi$
$608$		−	6.56664i		−	0.266312i
$609$		0			0
$610$	8.29180	−	25.5195i	0.335725	−	1.03326i
$611$	−23.8175	−	14.5354i	−0.963552	−	0.588040i
$612$		0			0
$613$	6.96497			0.281313			0.140656	−	0.990058i	$-0.455079\pi$
$613$	6.96497			0.281313			0.140656	+	0.990058i	$0.455079\pi$
$614$	−19.8415	−	6.44688i	−0.800736	−	0.260175i
$615$		0			0
$616$	8.39984	−	6.10284i	0.338439	−	0.245890i
$617$		−	3.36861i		−	0.135615i	−0.997698	−	0.0678075i	$-0.978400\pi$
$617$		−	3.36861i		−	0.135615i	0.997698	−	0.0678075i	$-0.0216004\pi$
$618$		0			0
$619$	−29.7781			−1.19688			−0.598442	−	0.801166i	$-0.704213\pi$
$619$	−29.7781			−1.19688			−0.598442	+	0.801166i	$0.704213\pi$
$620$	15.6131	−	21.4896i	0.627038	−	0.863044i
$621$		0			0
$622$	32.9866	+	10.7180i	1.32264	+	0.429752i
$623$	32.2450			1.29187
$624$		0			0
$625$	−8.81966			−0.352786
$626$	−8.70500	−	2.82843i	−0.347922	−	0.113047i
$627$		0			0
$628$	0.652476	−	0.898056i	0.0260366	−	0.0358363i
$629$	−49.6899			−1.98127
$630$		0			0
$631$			22.2794i			0.886928i	0.896292	+	0.443464i	$0.146251\pi$
$631$			22.2794i			0.886928i	−0.896292	+	0.443464i	$0.853749\pi$
$632$	11.5449	+	15.8902i	0.459232	+	0.632078i
$633$		0			0
$634$	41.8328	+	13.5923i	1.66139	+	0.539819i
$635$	9.97505			0.395848
$636$		0			0
$637$	−10.8262	+	17.7396i	−0.428948	+	0.702867i
$638$	−1.92989	+	5.93958i	−0.0764050	+	0.235150i
$639$		0			0
$640$	−11.0557	+	15.2169i	−0.437016	+	0.601501i
$641$	43.9983			1.73783			0.868914	−	0.494963i	$-0.164818\pi$
$641$	43.9983			1.73783			0.868914	+	0.494963i	$0.164818\pi$
$642$		0			0
$643$	−33.8734			−1.33584			−0.667918	−	0.744235i	$-0.732814\pi$
$643$	−33.8734			−1.33584			−0.667918	+	0.744235i	$0.732814\pi$
$644$	−34.3349	+	47.2579i	−1.35298	+	1.86222i
$645$		0			0
$646$	−7.88854	−	2.56314i	−0.310371	−	0.100846i
$647$	−36.5603			−1.43733			−0.718667	−	0.695354i	$-0.755247\pi$
$647$	−36.5603			−1.43733			−0.718667	+	0.695354i	$0.755247\pi$
$648$		0			0
$649$	10.0000			0.392534
$650$	8.65638	−	7.42072i	0.339531	−	0.291064i
$651$		0			0
$652$	−11.7130	−	8.50997i	−0.458715	−	0.333276i
$653$			33.3688i			1.30582i	0.757435	+	0.652911i	$0.226452\pi$
$653$			33.3688i			1.30582i	−0.757435	+	0.652911i	$0.773548\pi$
$654$		0			0
$655$		−	27.5388i		−	1.07603i
$656$	6.16492	−	2.00310i	0.240700	−	0.0782080i
$657$		0			0
$658$	37.1865	+	12.0826i	1.44968	+	0.471030i
$659$			36.4126i			1.41843i	0.704991	+	0.709216i	$0.250951\pi$
$659$			36.4126i			1.41843i	−0.704991	+	0.709216i	$0.749049\pi$
$660$		0			0
$661$	46.3038			1.80101			0.900504	−	0.434847i	$-0.143198\pi$
$661$	46.3038			1.80101			0.900504	+	0.434847i	$0.143198\pi$
$662$	−14.7890	−	4.80522i	−0.574789	−	0.186760i
$663$		0			0
$664$	3.81966	+	5.25731i	0.148232	+	0.204023i
$665$			6.89484i			0.267370i
$666$		0			0
$667$		−	35.1361i		−	1.36047i
$668$	8.61236	−	11.8539i	0.333222	−	0.458641i
$669$		0			0
$670$	−16.1869	−	5.25945i	−0.625355	−	0.203190i
$671$			11.7264i			0.452692i
$672$		0			0
$673$	−12.6525			−0.487717			−0.243859	−	0.969811i	$-0.578413\pi$
$673$	−12.6525			−0.487717			−0.243859	+	0.969811i	$0.578413\pi$
$674$	−7.43496	−	2.41577i	−0.286384	−	0.0930518i
$675$		0			0
$676$	3.97319	−	25.6946i	0.152815	−	0.988255i
$677$		−	17.8187i		−	0.684830i	−0.939549	−	0.342415i	$-0.888755\pi$
$677$		−	17.8187i		−	0.684830i	0.939549	−	0.342415i	$-0.111245\pi$
$678$		0			0
$679$	41.3050			1.58514
$680$	13.9648	+	19.2209i	0.535525	+	0.737088i
$681$		0			0
$682$	−3.58716	+	11.0401i	−0.137359	+	0.422749i
$683$	−10.5174			−0.402438			−0.201219	−	0.979546i	$-0.564490\pi$
$683$	−10.5174			−0.402438			−0.201219	+	0.979546i	$0.564490\pi$
$684$		0			0
$685$		−	22.6134i		−	0.864012i
$686$	−1.92989	+	5.93958i	−0.0736834	+	0.226774i
$687$		0			0
$688$	8.94427	−	2.90617i	0.340997	−	0.110797i
$689$	−34.6304	−	21.1344i	−1.31931	−	0.805156i
$690$		0			0
$691$	4.91735			0.187065			0.0935324	−	0.995616i	$-0.470184\pi$
$691$	4.91735			0.187065			0.0935324	+	0.995616i	$0.470184\pi$
$692$	13.2276	−	18.2063i	0.502840	−	0.692099i
$693$		0			0
$694$	−8.94791	+	27.5388i	−0.339658	+	1.04536i
$695$			32.1931i			1.22115i
$696$		0			0
$697$		−	8.18782i		−	0.310136i
$698$	−18.2802	−	5.93958i	−0.691914	−	0.224817i
$699$		0			0
$700$	−9.39130	+	12.9260i	−0.354958	+	0.488558i
$701$		−	26.8021i		−	1.01230i	−0.862445	−	0.506151i	$-0.831068\pi$
$701$		−	26.8021i		−	1.01230i	0.862445	−	0.506151i	$-0.168932\pi$
$702$		0			0
$703$	−11.4164			−0.430578
$704$	2.54009	−	7.81758i	0.0957331	−	0.294636i
$705$		0			0
$706$	−9.00000	+	27.6992i	−0.338719	+	1.04247i
$707$	28.4698			1.07072
$708$		0			0
$709$	8.39984			0.315463			0.157731	−	0.987482i	$-0.449582\pi$
$709$	8.39984			0.315463			0.157731	+	0.987482i	$0.449582\pi$
$710$	7.67752	−	23.6290i	0.288132	−	0.886779i
$711$		0			0
$712$	20.6525	−	15.0049i	0.773984	−	0.562332i
$713$		−	65.3089i		−	2.44584i
$714$		0			0
$715$	3.20845	−	5.25731i	0.119989	−	0.196612i
$716$	25.7181	−	35.3980i	0.961132	−	1.32288i
$717$		0			0
$718$	−14.0344	+	43.1936i	−0.523761	+	1.61197i
$719$	−12.0349			−0.448826			−0.224413	−	0.974494i	$-0.572047\pi$
$719$	−12.0349			−0.448826			−0.224413	+	0.974494i	$0.572047\pi$
$720$		0			0
$721$			41.8295i			1.55781i
$722$	23.7425	+	7.71441i	0.883605	+	0.287101i
$723$		0			0
$724$	−15.1246	+	20.8172i	−0.562102	+	0.773667i
$725$		−	9.61045i		−	0.356923i
$726$		0			0
$727$	−22.3607			−0.829312			−0.414656	−	0.909978i	$-0.636098\pi$
$727$	−22.3607			−0.829312			−0.414656	+	0.909978i	$0.636098\pi$
$728$	2.92512	+	36.3166i	0.108412	+	1.34598i
$729$		0			0
$730$	−3.20845	+	9.87460i	−0.118750	+	0.365475i
$731$		−	11.8792i		−	0.439367i
$732$		0			0
$733$	−46.8519			−1.73051			−0.865256	−	0.501330i	$-0.832844\pi$
$733$	−46.8519			−1.73051			−0.865256	+	0.501330i	$0.832844\pi$
$734$	−41.8625	−	13.6020i	−1.54517	−	0.502057i
$735$		0			0
$736$			46.2455i			1.70463i
$737$	7.43798			0.273982
$738$		0			0
$739$	19.9434			0.733631			0.366816	−	0.930294i	$-0.380448\pi$
$739$	19.9434			0.733631			0.366816	+	0.930294i	$0.380448\pi$
$740$	26.4553	+	19.2209i	0.972516	+	0.706574i
$741$		0			0
$742$	54.0689	+	17.5680i	1.98493	+	0.644943i
$743$			47.5918i			1.74597i	0.487744	+	0.872987i	$0.337820\pi$
$743$			47.5918i			1.74597i	−0.487744	+	0.872987i	$0.662180\pi$
$744$		0			0
$745$	11.7082			0.428955
$746$	−11.3950	+	35.0702i	−0.417201	+	1.28401i
$747$		0			0
$748$	−8.39984	−	6.10284i	−0.307129	−	0.223142i
$749$	−9.48993			−0.346755
$750$		0			0
$751$	−54.5410			−1.99023			−0.995115	−	0.0987222i	$-0.968524\pi$
$751$	−54.5410			−1.99023			−0.995115	+	0.0987222i	$0.968524\pi$
$752$	29.4400	−	9.56564i	1.07357	−	0.348823i
$753$		0			0
$754$	−14.2633	−	16.6383i	−0.519438	−	0.605933i
$755$		−	5.93958i		−	0.216164i
$756$		0			0
$757$		−	8.50651i		−	0.309174i	−0.987979	−	0.154587i	$-0.950595\pi$
$757$		−	8.50651i		−	0.309174i	0.987979	−	0.154587i	$-0.0494047\pi$
$758$	−26.0867	−	8.47609i	−0.947512	−	0.307865i
$759$		0			0
$760$	3.20845	+	4.41606i	0.116383	+	0.160187i
$761$		−	16.4304i		−	0.595601i	−0.954628	−	0.297800i	$-0.903747\pi$
$761$		−	16.4304i		−	0.595601i	0.954628	−	0.297800i	$-0.0962530\pi$
$762$		0			0
$763$		−	43.4306i		−	1.57229i
$764$	−3.12262	−	2.26872i	−0.112973	−	0.0820794i
$765$		0			0
$766$	1.56231	−	4.80828i	0.0564484	−	0.173730i
$767$	−18.2802	+	29.9535i	−0.660058	+	1.08156i
$768$		0			0
$769$		−	24.1653i		−	0.871422i	−0.900087	−	0.435711i	$-0.856497\pi$
$769$		−	24.1653i		−	0.871422i	0.900087	−	0.435711i	$-0.143503\pi$
$770$	−2.66704	+	8.20830i	−0.0961134	+	0.295807i
$771$		0			0
$772$	−5.19139	+	7.14533i	−0.186842	+	0.257166i
$773$	−30.3176			−1.09045			−0.545224	−	0.838290i	$-0.683555\pi$
$773$	−30.3176			−1.09045			−0.545224	+	0.838290i	$0.683555\pi$
$774$		0			0
$775$		−	17.8633i		−	0.641669i
$776$	26.4553	−	19.2209i	0.949690	−	0.689990i
$777$		0			0
$778$	15.4695	−	47.6102i	0.554608	−	1.70691i
$779$		−	1.88118i		−	0.0674001i
$780$		0			0
$781$			10.8576i			0.388517i
$782$	55.5550	+	18.0509i	1.98664	+	0.645500i
$783$		0			0
$784$	−7.12461	−	21.9273i	−0.254450	−	0.783118i
$785$			0.922740i			0.0329340i
$786$		0			0
$787$	40.4996			1.44366			0.721828	−	0.692072i	$-0.243302\pi$
$787$	40.4996			1.44366			0.721828	+	0.692072i	$0.243302\pi$
$788$	37.5100	+	27.2526i	1.33624	+	0.970834i
$789$		0			0
$790$	−15.5279	−	5.04531i	−0.552457	−	0.179504i
$791$

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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} -1.66251 q^{5} -3.57266i q^{7} +(-1.66251 - 2.28825i) q^{8} +O(q^{10})\)	\(q+(-1.34500 - 0.437016i) q^{2} +(1.61803 + 1.17557i) q^{4} -1.66251 q^{5} -3.57266i q^{7} +(-1.66251 - 2.28825i) q^{8} +(2.23607 + 0.726543i) q^{10} -1.02749 q^{11} +(1.87826 - 3.07768i) q^{13} +(-1.56131 + 4.80522i) q^{14} +(1.23607 + 3.80423i) q^{16} +5.05251 q^{17} +1.16083 q^{19} +(-2.68999 - 1.95440i) q^{20} +(1.38197 + 0.449028i) q^{22} -8.17513 q^{23} -2.23607 q^{25} +(-3.87125 + 3.31865i) q^{26} +(4.19992 - 5.78069i) q^{28} +4.29792i q^{29} +7.98872i q^{31} -5.65685i q^{32} +(-6.79561 - 2.20803i) q^{34} +5.93958i q^{35} -9.83470 q^{37} +(-1.56131 - 0.507301i) q^{38} +(2.76393 + 3.80423i) q^{40} -1.62054i q^{41} -2.35114i q^{43} +(-1.66251 - 1.20788i) q^{44} +(10.9955 + 3.57266i) q^{46} -7.73877i q^{47} -5.76393 q^{49} +(3.00750 + 0.977198i) q^{50} +(6.65712 - 2.77177i) q^{52} -11.2521i q^{53} +1.70820 q^{55} +(-8.17513 + 5.93958i) q^{56} +(1.87826 - 5.78069i) q^{58} -9.73249 q^{59} -11.4127i q^{61} +(3.49120 - 10.7448i) q^{62} +(-2.47214 + 7.60845i) q^{64} +(-3.12262 + 5.11667i) q^{65} -7.23901 q^{67} +(8.17513 + 5.93958i) q^{68} +(2.59569 - 7.98872i) q^{70} -10.5672i q^{71} +4.41606i q^{73} +(13.2276 + 4.29792i) q^{74} +(1.87826 + 1.36464i) q^{76} +3.67086i q^{77} -6.94427 q^{79} +(-2.05497 - 6.32456i) q^{80} +(-0.708204 + 2.17963i) q^{82} -2.29753 q^{83} -8.39984 q^{85} +(-1.02749 + 3.16228i) q^{86} +(1.70820 + 2.35114i) q^{88} +9.02546i q^{89} +(-10.9955 - 6.71040i) q^{91} +(-13.2276 - 9.61045i) q^{92} +(-3.38197 + 10.4086i) q^{94} -1.92989 q^{95} +11.5614i q^{97} +(7.75247 + 2.51893i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94}+O(q^{100}) \) 16 * q + 8 * q^4 - 16 * q^16 + 40 * q^22 + 80 * q^40 - 128 * q^49 + 40 * q^52 - 80 * q^55 + 32 * q^64 + 32 * q^79 + 96 * q^82 - 80 * q^88 - 72 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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