LMFDB - Embedded newform 1134.2.f.k.757.1

Newspace parameters

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.f (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			757.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1134.757
Dual form			1134.2.f.k.379.1

$q$-expansion

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +(-2.50000 + 4.33013i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.00000 q^{17} +2.00000 q^{19} +(-4.50000 + 7.79423i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{26} +1.00000 q^{28} +(-1.50000 - 2.59808i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{34} +2.00000 q^{37} +(1.00000 + 1.73205i) q^{38} +(-3.00000 + 5.19615i) q^{41} +(0.500000 + 0.866025i) q^{43} -9.00000 q^{46} +(-3.00000 - 5.19615i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-2.50000 + 4.33013i) q^{50} +(-2.50000 - 4.33013i) q^{52} -3.00000 q^{53} +(0.500000 + 0.866025i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -5.00000 q^{62} +1.00000 q^{64} +(6.50000 - 11.2583i) q^{67} +(1.50000 - 2.59808i) q^{68} -9.00000 q^{71} +2.00000 q^{73} +(1.00000 + 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(5.00000 + 8.66025i) q^{79} -6.00000 q^{82} +(-6.00000 - 10.3923i) q^{83} +(-0.500000 + 0.866025i) q^{86} -15.0000 q^{89} +5.00000 q^{91} +(-4.50000 - 7.79423i) q^{92} +(3.00000 - 5.19615i) q^{94} +(-4.00000 - 6.92820i) q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - q^{7} - 2 q^{8} - 5 q^{13} + q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - 9 q^{23} + 5 q^{25} - 10 q^{26} + 2 q^{28} - 3 q^{29} - 5 q^{31} + q^{32} - 3 q^{34} + 4 q^{37} + 2 q^{38} - 6 q^{41} + q^{43} - 18 q^{46} - 6 q^{47} - q^{49} - 5 q^{50} - 5 q^{52} - 6 q^{53} + q^{56} + 3 q^{58} - 3 q^{59} + 10 q^{61} - 10 q^{62} + 2 q^{64} + 13 q^{67} + 3 q^{68} - 18 q^{71} + 4 q^{73} + 2 q^{74} - 2 q^{76} + 10 q^{79} - 12 q^{82} - 12 q^{83} - q^{86} - 30 q^{89} + 10 q^{91} - 9 q^{92} + 6 q^{94} - 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^7 - 2 * q^8 - 5 * q^13 + q^14 - q^16 - 6 * q^17 + 4 * q^19 - 9 * q^23 + 5 * q^25 - 10 * q^26 + 2 * q^28 - 3 * q^29 - 5 * q^31 + q^32 - 3 * q^34 + 4 * q^37 + 2 * q^38 - 6 * q^41 + q^43 - 18 * q^46 - 6 * q^47 - q^49 - 5 * q^50 - 5 * q^52 - 6 * q^53 + q^56 + 3 * q^58 - 3 * q^59 + 10 * q^61 - 10 * q^62 + 2 * q^64 + 13 * q^67 + 3 * q^68 - 18 * q^71 + 4 * q^73 + 2 * q^74 - 2 * q^76 + 10 * q^79 - 12 * q^82 - 12 * q^83 - q^86 - 30 * q^89 + 10 * q^91 - 9 * q^92 + 6 * q^94 - 8 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0.500000	−	0.866025i	0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.50000	+	7.79423i	−0.938315	+	1.62521i	−0.169701	+	0.985496i	$0.554280\pi$
$23$	−4.50000	+	7.79423i	−0.938315	+	1.62521i	−0.768613	+	0.639713i	$0.779053\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	−5.00000			−0.980581
$27$		0			0
$28$	1.00000			0.188982
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−1.50000	−	2.59808i	−0.257248	−	0.445566i
$35$		0			0
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	+	1.73205i	0.162221	+	0.280976i
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		0			0
$45$		0			0
$46$	−9.00000			−1.32698
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	0.559908	−	0.828554i	$-0.310836\pi$
$47$	−3.00000	−	5.19615i	−0.437595	−	0.757937i	−0.997503	+	0.0706177i	$0.977503\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$	−2.50000	+	4.33013i	−0.353553	+	0.612372i
$51$		0			0
$52$	−2.50000	−	4.33013i	−0.346688	−	0.600481i
$53$	−3.00000			−0.412082			−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000			−0.412082			−0.206041	+	0.978543i	$0.566058\pi$
$54$		0			0
$55$		0			0
$56$	0.500000	+	0.866025i	0.0668153	+	0.115728i
$57$		0			0
$58$	1.50000	−	2.59808i	0.196960	−	0.341144i
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	−0.946993	−	0.321253i	$-0.895896\pi$
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	0.751710	+	0.659494i	$0.229229\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$	−5.00000			−0.635001
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	−0.129307	−	0.991605i	$-0.541275\pi$
$67$	6.50000	−	11.2583i	0.794101	−	1.37542i	0.923408	−	0.383819i	$-0.125391\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$		0			0
$70$		0			0
$71$	−9.00000			−1.06810			−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000			−1.06810			−0.534052	+	0.845452i	$0.679331\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$	1.00000	+	1.73205i	0.116248	+	0.201347i
$75$		0			0
$76$	−1.00000	+	1.73205i	−0.114708	+	0.198680i
$77$		0			0
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$	−6.00000			−0.662589
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$		0			0
$86$	−0.500000	+	0.866025i	−0.0539164	+	0.0933859i
$87$		0			0
$88$		0			0
$89$	−15.0000			−1.59000			−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000			−1.59000			−0.794998	+	0.606612i	$0.792528\pi$
$90$		0			0
$91$	5.00000			0.524142
$92$	−4.50000	−	7.79423i	−0.469157	−	0.812605i
$93$		0			0
$94$	3.00000	−	5.19615i	0.309426	−	0.535942i
$95$		0			0
$96$		0			0
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	$-0.299792\pi$
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	−0.994453	+	0.105180i	$0.966458\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	−5.00000			−0.500000
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	2.50000	−	4.33013i	0.245145	−	0.424604i
$105$		0			0
$106$	−1.50000	−	2.59808i	−0.145693	−	0.252347i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	0.866025i	−0.0472456	+	0.0818317i
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	−0.432670	−	0.901553i	$-0.642428\pi$
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	0.997102	−	0.0760733i	$-0.0242383\pi$
$114$		0			0
$115$		0			0
$116$	3.00000			0.278543
$117$		0			0
$118$	−3.00000			−0.276172
$119$	1.50000	+	2.59808i	0.137505	+	0.238165i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	−5.00000	+	8.66025i	−0.452679	+	0.784063i
$123$		0			0
$124$	−2.50000	−	4.33013i	−0.224507	−	0.388857i
$125$		0			0
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	−0.992865	−	0.119241i	$-0.961954\pi$
$131$	−4.50000	+	7.79423i	−0.393167	+	0.680985i	0.599699	+	0.800226i	$0.295287\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$	13.0000			1.12303
$135$		0			0
$136$	3.00000			0.257248
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	0.938148	+	0.346235i	$0.112540\pi$
$137$	9.00000	+	15.5885i	0.768922	+	1.33181i	−0.169226	+	0.985577i	$0.554127\pi$
$138$		0			0
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	0.399992	+	0.916519i	$0.369013\pi$
$139$	−7.00000	+	12.1244i	−0.593732	+	1.02837i	−0.993724	+	0.111856i	$0.964321\pi$
$140$		0			0
$141$		0			0
$142$	−4.50000	−	7.79423i	−0.377632	−	0.654077i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	1.00000	+	1.73205i	0.0827606	+	0.143346i
$147$		0			0
$148$	−1.00000	+	1.73205i	−0.0821995	+	0.142374i
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	−0.620701	−	0.784047i	$-0.713152\pi$
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	0.989355	+	0.145519i	$0.0464853\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$	−2.00000			−0.162221
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.5000	+	19.9186i	−0.917800	+	1.58968i	−0.115050	+	0.993360i	$0.536703\pi$
$157$	−11.5000	+	19.9186i	−0.917800	+	1.58968i	−0.802749	+	0.596316i	$0.796630\pi$
$158$	−5.00000	+	8.66025i	−0.397779	+	0.688973i
$159$		0			0
$160$		0			0
$161$	9.00000			0.709299
$162$		0			0
$163$	11.0000			0.861586			0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000			0.861586			0.430793	+	0.902451i	$0.358234\pi$
$164$	−3.00000	−	5.19615i	−0.234261	−	0.405751i
$165$		0			0
$166$	6.00000	−	10.3923i	0.465690	−	0.806599i
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$	−1.00000			−0.0762493
$173$	−12.0000	−	20.7846i	−0.912343	−	1.58022i	−0.810745	−	0.585399i	$-0.800938\pi$
$173$	−12.0000	−	20.7846i	−0.912343	−	1.58022i	−0.101598	−	0.994826i	$-0.532395\pi$
$174$		0			0
$175$	2.50000	−	4.33013i	0.188982	−	0.327327i
$176$		0			0
$177$		0			0
$178$	−7.50000	−	12.9904i	−0.562149	−	0.973670i
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	2.50000	+	4.33013i	0.185312	+	0.320970i
$183$		0			0
$184$	4.50000	−	7.79423i	0.331744	−	0.574598i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	6.00000			0.437595
$189$		0			0
$190$		0			0
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	0.997225	−	0.0744412i	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	−2.50000	+	4.33013i	−0.179954	+	0.311689i	−0.941865	−	0.335993i	$-0.890928\pi$
$193$	−2.50000	+	4.33013i	−0.179954	+	0.311689i	0.761911	+	0.647682i	$0.224262\pi$
$194$	4.00000	−	6.92820i	0.287183	−	0.497416i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.0357143	−	0.0618590i
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	11.0000			0.779769			0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000			0.779769			0.389885	+	0.920864i	$0.372515\pi$
$200$	−2.50000	−	4.33013i	−0.176777	−	0.306186i
$201$		0			0
$202$	−9.00000	+	15.5885i	−0.633238	+	1.09680i
$203$	−1.50000	+	2.59808i	−0.105279	+	0.182349i
$204$		0			0
$205$		0			0
$206$	13.0000			0.905753
$207$		0			0
$208$	5.00000			0.346688
$209$		0			0
$210$		0			0
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	0.998221	+	0.0596196i	$0.0189888\pi$
$212$	1.50000	−	2.59808i	0.103020	−	0.178437i
$213$		0			0
$214$	6.00000	+	10.3923i	0.410152	+	0.710403i
$215$		0			0
$216$		0			0
$217$	5.00000			0.339422
$218$	−8.00000	−	13.8564i	−0.541828	−	0.938474i
$219$		0			0
$220$		0			0
$221$	7.50000	−	12.9904i	0.504505	−	0.873828i
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$	12.0000			0.798228
$227$	4.50000	+	7.79423i	0.298675	+	0.517321i	0.975833	−	0.218517i	$-0.0701218\pi$
$227$	4.50000	+	7.79423i	0.298675	+	0.517321i	−0.677158	+	0.735838i	$0.736789\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	12.0000			0.786146			0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000			0.786146			0.393073	+	0.919507i	$0.371412\pi$
$234$		0			0
$235$		0			0
$236$	−1.50000	−	2.59808i	−0.0976417	−	0.169120i
$237$		0			0
$238$	−1.50000	+	2.59808i	−0.0972306	+	0.168408i
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	−0.604087	−	0.796918i	$-0.706462\pi$
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	0.992195	+	0.124696i	$0.0397955\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$	11.0000			0.707107
$243$		0			0
$244$	−10.0000			−0.640184
$245$		0			0
$246$		0			0
$247$	−5.00000	+	8.66025i	−0.318142	+	0.551039i
$248$	2.50000	−	4.33013i	0.158750	−	0.274963i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	10.0000	+	17.3205i	0.627456	+	1.08679i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	−0.435970	−	0.899961i	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		0			0
$259$	−1.00000	−	1.73205i	−0.0621370	−	0.107624i
$260$		0			0
$261$		0			0
$262$	−9.00000			−0.556022
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	0.970758	−	0.240059i	$-0.0771668\pi$
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	−0.693276	+	0.720672i	$0.743833\pi$
$264$		0			0
$265$		0			0
$266$	1.00000	−	1.73205i	0.0613139	−	0.106199i
$267$		0			0
$268$	6.50000	+	11.2583i	0.397051	+	0.687712i
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$	−7.00000			−0.425220			−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000			−0.425220			−0.212610	+	0.977137i	$0.568196\pi$
$272$	1.50000	+	2.59808i	0.0909509	+	0.157532i
$273$		0			0
$274$	−9.00000	+	15.5885i	−0.543710	+	0.941733i
$275$		0			0
$276$		0			0
$277$	−4.00000	−	6.92820i	−0.240337	−	0.416275i	0.720473	−	0.693482i	$-0.243925\pi$
$277$	−4.00000	−	6.92820i	−0.240337	−	0.416275i	−0.960810	+	0.277207i	$0.910591\pi$
$278$	−14.0000			−0.839664
$279$		0			0
$280$		0			0
$281$	3.00000	+	5.19615i	0.178965	+	0.309976i	0.941526	−	0.336939i	$-0.109392\pi$
$281$	3.00000	+	5.19615i	0.178965	+	0.309976i	−0.762561	+	0.646916i	$0.776058\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	4.50000	−	7.79423i	0.267026	−	0.462502i
$285$		0			0
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$	−1.00000	+	1.73205i	−0.0585206	+	0.101361i
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	−0.940252	−	0.340480i	$-0.889411\pi$
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	0.764990	+	0.644042i	$0.222744\pi$
$294$		0			0
$295$		0			0
$296$	−2.00000			−0.116248
$297$		0			0
$298$	9.00000			0.521356
$299$	−22.5000	−	38.9711i	−1.30121	−	2.25376i
$300$		0			0
$301$	0.500000	−	0.866025i	0.0288195	−	0.0499169i
$302$	−5.00000	+	8.66025i	−0.287718	+	0.498342i
$303$		0			0
$304$	−1.00000	−	1.73205i	−0.0573539	−	0.0993399i
$305$		0			0
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$	−23.0000			−1.29797
$315$		0			0
$316$	−10.0000			−0.562544
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	0.887788	+	0.460252i	$0.152241\pi$
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	−0.0453045	+	0.998973i	$0.514426\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	4.50000	+	7.79423i	0.250775	+	0.434355i
$323$	−6.00000			−0.333849
$324$		0			0
$325$	−25.0000			−1.38675
$326$	5.50000	+	9.52628i	0.304617	+	0.527612i
$327$		0			0
$328$	3.00000	−	5.19615i	0.165647	−	0.286910i
$329$	−3.00000	+	5.19615i	−0.165395	+	0.286473i
$330$		0			0
$331$	9.50000	+	16.4545i	0.522167	+	0.904420i	0.999667	+	0.0257885i	$0.00820965\pi$
$331$	9.50000	+	16.4545i	0.522167	+	0.904420i	−0.477500	+	0.878632i	$0.658457\pi$
$332$	12.0000			0.658586
$333$		0			0
$334$	12.0000			0.656611
$335$		0			0
$336$		0			0
$337$	6.50000	−	11.2583i	0.354078	−	0.613280i	−0.632882	−	0.774248i	$-0.718128\pi$
$337$	6.50000	−	11.2583i	0.354078	−	0.613280i	0.986960	+	0.160968i	$0.0514616\pi$
$338$	6.00000	−	10.3923i	0.326357	−	0.565267i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.00000			0.0539949
$344$	−0.500000	−	0.866025i	−0.0269582	−	0.0466930i
$345$		0			0
$346$	12.0000	−	20.7846i	0.645124	−	1.11739i
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	0.110885	+	0.993833i	$0.464631\pi$
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	−0.916127	+	0.400887i	$0.868702\pi$
$348$		0			0
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	0.879097	−	0.476642i	$-0.158146\pi$
$349$	0.500000	+	0.866025i	0.0267644	+	0.0463573i	−0.852333	+	0.523000i	$0.824813\pi$
$350$	5.00000			0.267261
$351$		0			0
$352$		0			0
$353$	−13.5000	−	23.3827i	−0.718532	−	1.24453i	−0.961581	−	0.274521i	$-0.911481\pi$
$353$	−13.5000	−	23.3827i	−0.718532	−	1.24453i	0.243049	−	0.970014i	$-0.421853\pi$
$354$		0			0
$355$		0			0
$356$	7.50000	−	12.9904i	0.397499	−	0.688489i
$357$		0			0
$358$	12.0000	+	20.7846i	0.634220	+	1.09850i
$359$	3.00000			0.158334			0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000			0.158334			0.0791670	+	0.996861i	$0.474774\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	−3.50000	−	6.06218i	−0.183956	−	0.318621i
$363$		0			0
$364$	−2.50000	+	4.33013i	−0.131036	+	0.226960i
$365$		0			0
$366$		0			0
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	0.999989	−	0.00473247i	$-0.00150640\pi$
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	−0.504093	+	0.863649i	$0.668173\pi$
$368$	9.00000			0.469157
$369$		0			0
$370$		0			0
$371$	1.50000	+	2.59808i	0.0778761	+	0.134885i
$372$		0			0
$373$	−7.00000	+	12.1244i	−0.362446	+	0.627775i	−0.988363	−	0.152115i	$-0.951392\pi$
$373$	−7.00000	+	12.1244i	−0.362446	+	0.627775i	0.625917	+	0.779890i	$0.284725\pi$
$374$		0			0
$375$		0			0
$376$	3.00000	+	5.19615i	0.154713	+	0.267971i
$377$	15.0000			0.772539
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$		0			0
$381$		0			0
$382$	−6.00000	+	10.3923i	−0.306987	+	0.531717i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$	−5.00000			−0.254493
$387$		0			0
$388$	8.00000			0.406138
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	0.998763	−	0.0497253i	$-0.0158346\pi$
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	−0.542445	+	0.840091i	$0.682501\pi$
$390$		0			0
$391$	13.5000	−	23.3827i	0.682724	−	1.18251i
$392$	0.500000	−	0.866025i	0.0252538	−	0.0437409i
$393$		0			0
$394$	−9.00000	−	15.5885i	−0.453413	−	0.785335i
$395$		0			0
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$	5.50000	+	9.52628i	0.275690	+	0.477509i
$399$		0			0
$400$	2.50000	−	4.33013i	0.125000	−	0.216506i
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	−12.5000	−	21.6506i	−0.622669	−	1.07849i
$404$	−18.0000			−0.895533
$405$		0			0
$406$	−3.00000			−0.148888
$407$		0			0
$408$		0			0
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	−0.985558	−	0.169338i	$-0.945837\pi$
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	0.639430	+	0.768849i	$0.279170\pi$
$410$		0			0
$411$		0			0
$412$	6.50000	+	11.2583i	0.320232	+	0.554658i
$413$	3.00000			0.147620
$414$		0			0
$415$		0			0
$416$	2.50000	+	4.33013i	0.122573	+	0.212302i
$417$		0			0
$418$		0			0
$419$	−7.50000	+	12.9904i	−0.366399	+	0.634622i	−0.989000	−	0.147918i	$-0.952743\pi$
$419$	−7.50000	+	12.9904i	−0.366399	+	0.634622i	0.622601	+	0.782540i	$0.286076\pi$
$420$		0			0
$421$	5.00000	+	8.66025i	0.243685	+	0.422075i	0.961761	−	0.273890i	$-0.0883103\pi$
$421$	5.00000	+	8.66025i	0.243685	+	0.422075i	−0.718076	+	0.695965i	$0.754977\pi$
$422$	13.0000			0.632830
$423$		0			0
$424$	3.00000			0.145693
$425$	−7.50000	−	12.9904i	−0.363803	−	0.630126i
$426$		0			0
$427$	5.00000	−	8.66025i	0.241967	−	0.419099i
$428$	−6.00000	+	10.3923i	−0.290021	+	0.502331i
$429$		0			0
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$	2.50000	+	4.33013i	0.120004	+	0.207853i
$435$		0			0
$436$	8.00000	−	13.8564i	0.383131	−	0.663602i
$437$	−9.00000	+	15.5885i	−0.430528	+	0.745697i
$438$		0			0
$439$	−17.5000	−	30.3109i	−0.835229	−	1.44666i	−0.893843	−	0.448379i	$-0.852001\pi$
$439$	−17.5000	−	30.3109i	−0.835229	−	1.44666i	0.0586141	−	0.998281i	$-0.481332\pi$
$440$		0			0
$441$		0			0
$442$	15.0000			0.713477
$443$	−3.00000	−	5.19615i	−0.142534	−	0.246877i	0.785916	−	0.618333i	$-0.212192\pi$
$443$	−3.00000	−	5.19615i	−0.142534	−	0.246877i	−0.928450	+	0.371457i	$0.878858\pi$
$444$		0			0
$445$		0			0
$446$	4.00000	−	6.92820i	0.189405	−	0.328060i
$447$		0			0
$448$	−0.500000	−	0.866025i	−0.0236228	−	0.0409159i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	6.00000	+	10.3923i	0.282216	+	0.488813i
$453$		0			0
$454$	−4.50000	+	7.79423i	−0.211195	+	0.365801i
$455$		0			0
$456$		0			0
$457$	−17.5000	−	30.3109i	−0.818615	−	1.41788i	−0.906702	−	0.421771i	$-0.861409\pi$
$457$	−17.5000	−	30.3109i	−0.818615	−	1.41788i	0.0880870	−	0.996113i	$-0.471925\pi$
$458$	−14.0000			−0.654177
$459$		0			0
$460$		0			0
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	−0.968945	−	0.247276i	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		0			0
$463$	11.0000	−	19.0526i	0.511213	−	0.885448i	−0.488702	−	0.872451i	$-0.662530\pi$
$463$	11.0000	−	19.0526i	0.511213	−	0.885448i	0.999916	−	0.0129968i	$-0.00413714\pi$
$464$	−1.50000	+	2.59808i	−0.0696358	+	0.120613i
$465$		0			0
$466$	6.00000	+	10.3923i	0.277945	+	0.481414i
$467$	−24.0000			−1.11059			−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		0			0
$469$	−13.0000			−0.600284
$470$		0			0
$471$		0			0
$472$	1.50000	−	2.59808i	0.0690431	−	0.119586i
$473$		0			0
$474$		0			0
$475$	5.00000	+	8.66025i	0.229416	+	0.397360i
$476$	−3.00000			−0.137505
$477$		0			0
$478$	12.0000			0.548867
$479$	18.0000	+	31.1769i	0.822441	+	1.42451i	0.903859	+	0.427830i	$0.140722\pi$
$479$	18.0000	+	31.1769i	0.822441	+	1.42451i	−0.0814184	+	0.996680i	$0.525945\pi$
$480$		0			0
$481$	−5.00000	+	8.66025i	−0.227980	+	0.394874i
$482$	−5.00000	+	8.66025i	−0.227744	+	0.394464i
$483$		0			0
$484$	5.50000	+	9.52628i	0.250000	+	0.433013i
$485$		0			0
$486$		0			0
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$	−5.00000	−	8.66025i	−0.226339	−	0.392031i
$489$		0			0
$490$		0			0
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		0			0
$493$	4.50000	+	7.79423i	0.202670	+	0.351034i
$494$	−10.0000			−0.449921
$495$		0			0
$496$	5.00000			0.224507
$497$	4.50000	+	7.79423i	0.201853	+	0.349619i
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−10.0000	+	17.3205i	−0.443678	+	0.768473i
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	−1.00000	−	1.73205i	−0.0442374	−	0.0766214i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	18.0000			0.793946
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.00000	−	1.73205i	0.0439375	−	0.0761019i
$519$		0			0
$520$		0			0
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	−4.50000	−	7.79423i	−0.196583	−	0.340492i
$525$		0			0
$526$	−4.50000	+	7.79423i	−0.196209	+	0.339845i
$527$	7.50000	−	12.9904i	0.326705	−	0.565870i
$528$		0			0
$529$	−29.0000	−	50.2295i	−1.26087	−	2.18389i
$530$		0			0
$531$		0			0
$532$	2.00000			0.0867110
$533$	−15.0000	−	25.9808i	−0.649722	−	1.12535i
$534$		0			0
$535$		0			0
$536$	−6.50000	+	11.2583i	−0.280757	+	0.486286i
$537$		0			0
$538$	12.0000	+	20.7846i	0.517357	+	0.896088i
$539$		0			0
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$	−3.50000	−	6.06218i	−0.150338	−	0.260393i
$543$		0			0
$544$	−1.50000	+	2.59808i	−0.0643120	+	0.111392i
$545$		0			0
$546$		0			0
$547$	−22.0000	−	38.1051i	−0.940652	−	1.62926i	−0.764231	−	0.644942i	$-0.776881\pi$
$547$	−22.0000	−	38.1051i	−0.940652	−	1.62926i	−0.176421	−	0.984315i	$-0.556452\pi$
$548$	−18.0000			−0.768922
$549$		0			0
$550$		0			0
$551$	−3.00000	−	5.19615i	−0.127804	−	0.221364i
$552$		0			0
$553$	5.00000	−	8.66025i	0.212622	−	0.368271i
$554$	4.00000	−	6.92820i	0.169944	−	0.294351i
$555$		0			0
$556$	−7.00000	−	12.1244i	−0.296866	−	0.514187i
$557$	−21.0000			−0.889799			−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000			−0.889799			−0.444899	+	0.895581i	$0.646761\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$		0			0
$562$	−3.00000	+	5.19615i	−0.126547	+	0.219186i
$563$	−1.50000	+	2.59808i	−0.0632175	+	0.109496i	−0.895902	−	0.444252i	$-0.853470\pi$
$563$	−1.50000	+	2.59808i	−0.0632175	+	0.109496i	0.832684	+	0.553748i	$0.186803\pi$
$564$		0			0
$565$		0			0
$566$	4.00000			0.168133
$567$		0			0
$568$	9.00000			0.377632
$569$	−12.0000	−	20.7846i	−0.503066	−	0.871336i	−0.999994	−	0.00354413i	$-0.998872\pi$
$569$	−12.0000	−	20.7846i	−0.503066	−	0.871336i	0.496928	−	0.867792i	$-0.334461\pi$
$570$		0			0
$571$	−2.50000	+	4.33013i	−0.104622	+	0.181210i	−0.913584	−	0.406651i	$-0.866697\pi$
$571$	−2.50000	+	4.33013i	−0.104622	+	0.181210i	0.808962	+	0.587861i	$0.200030\pi$
$572$		0			0
$573$		0			0
$574$	3.00000	+	5.19615i	0.125218	+	0.216883i
$575$	−45.0000			−1.87663
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	−4.00000	−	6.92820i	−0.166378	−	0.288175i
$579$		0			0
$580$		0			0
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$		0			0
$584$	−2.00000			−0.0827606
$585$		0			0
$586$	−6.00000			−0.247858
$587$	16.5000	+	28.5788i	0.681028	+	1.17957i	0.974668	+	0.223659i	$0.0718001\pi$
$587$	16.5000	+	28.5788i	0.681028	+	1.17957i	−0.293640	+	0.955916i	$0.594867\pi$
$588$		0			0
$589$	−5.00000	+	8.66025i	−0.206021	+	0.356840i
$590$		0			0
$591$		0			0
$592$	−1.00000	−	1.73205i	−0.0410997	−	0.0711868i
$593$	−42.0000			−1.72473			−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000			−1.72473			−0.862367	+	0.506284i	$0.831019\pi$
$594$		0			0
$595$		0			0
$596$	4.50000	+	7.79423i	0.184327	+	0.319264i
$597$		0			0
$598$	22.5000	−	38.9711i	0.920093	−	1.59365i
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	−0.895042	−	0.445983i	$-0.852854\pi$
$599$	−1.50000	+	2.59808i	−0.0612883	+	0.106155i	0.833753	+	0.552137i	$0.186188\pi$
$600$		0			0
$601$	−4.00000	−	6.92820i	−0.163163	−	0.282607i	0.772838	−	0.634603i	$-0.218836\pi$
$601$	−4.00000	−	6.92820i	−0.163163	−	0.282607i	−0.936002	+	0.351996i	$0.885503\pi$
$602$	1.00000			0.0407570
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$	−2.50000	+	4.33013i	−0.101472	+	0.175754i	−0.912291	−	0.409542i	$-0.865689\pi$
$607$	−2.50000	+	4.33013i	−0.101472	+	0.175754i	0.810819	+	0.585296i	$0.199022\pi$
$608$	1.00000	−	1.73205i	0.0405554	−	0.0702439i
$609$		0			0
$610$		0			0
$611$	30.0000			1.21367
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$	1.00000	+	1.73205i	0.0403567	+	0.0698999i
$615$		0			0
$616$		0			0
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	0.0398207	+	0.999207i	$0.487321\pi$
$617$	−21.0000	+	36.3731i	−0.845428	+	1.46432i	−0.885249	+	0.465118i	$0.846012\pi$
$618$		0			0
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	0.948840	−	0.315757i	$-0.102258\pi$
$619$	5.00000	+	8.66025i	0.200967	+	0.348085i	−0.747873	+	0.663842i	$0.768925\pi$
$620$		0			0
$621$		0			0
$622$	24.0000			0.962312
$623$	7.50000	+	12.9904i	0.300481	+	0.520449i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$	−5.00000	+	8.66025i	−0.199840	+	0.346133i
$627$		0			0
$628$	−11.5000	−	19.9186i	−0.458900	−	0.794838i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−5.00000	−	8.66025i	−0.198889	−	0.344486i
$633$		0			0
$634$	−15.0000	+	25.9808i	−0.595726	+	1.03183i
$635$		0			0
$636$		0			0
$637$	−2.50000	−	4.33013i	−0.0990536	−	0.171566i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0000	+	31.1769i	0.710957	+	1.23141i	0.964498	+	0.264089i	$0.0850714\pi$
$641$	18.0000	+	31.1769i	0.710957	+	1.23141i	−0.253541	+	0.967325i	$0.581595\pi$
$642$		0			0
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	0.926750	−	0.375680i	$-0.122591\pi$
$644$	−4.50000	+	7.79423i	−0.177325	+	0.307136i
$645$		0			0
$646$	−3.00000	−	5.19615i	−0.118033	−	0.204440i
$647$	−30.0000			−1.17942			−0.589711	−	0.807614i	$-0.700758\pi$
$647$	−30.0000			−1.17942			−0.589711	+	0.807614i	$0.700758\pi$
$648$		0			0
$649$		0			0
$650$	−12.5000	−	21.6506i	−0.490290	−	0.849208i
$651$		0			0
$652$	−5.50000	+	9.52628i	−0.215397	+	0.373078i
$653$	−4.50000	+	7.79423i	−0.176099	+	0.305012i	−0.940541	−	0.339680i	$-0.889681\pi$
$653$	−4.50000	+	7.79423i	−0.176099	+	0.305012i	0.764442	+	0.644692i	$0.223014\pi$
$654$		0			0
$655$		0			0
$656$	6.00000			0.234261
$657$		0			0
$658$	−6.00000			−0.233904
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$	−9.50000	+	16.4545i	−0.369228	+	0.639522i
$663$		0			0
$664$	6.00000	+	10.3923i	0.232845	+	0.403300i
$665$		0			0
$666$		0			0
$667$	27.0000			1.04544
$668$	6.00000	+	10.3923i	0.232147	+	0.402090i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	0.875501	−	0.483216i	$-0.160531\pi$
$673$	0.500000	+	0.866025i	0.0192736	+	0.0333828i	−0.856228	+	0.516599i	$0.827198\pi$
$674$	13.0000			0.500741
$675$		0			0
$676$	12.0000			0.461538
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	0.957984	−	0.286820i	$-0.0925982\pi$
$677$	6.00000	+	10.3923i	0.230599	+	0.399409i	−0.727386	+	0.686229i	$0.759265\pi$
$678$		0			0
$679$	−4.00000	+	6.92820i	−0.153506	+	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$	0.500000	+	0.866025i	0.0190901	+	0.0330650i
$687$		0			0
$688$	0.500000	−	0.866025i	0.0190623	−	0.0330169i
$689$	7.50000	−	12.9904i	0.285727	−	0.494894i
$690$		0			0
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	0.999276	+	0.0380440i	$0.0121127\pi$
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	−0.466691	+	0.884420i	$0.654554\pi$
$692$	24.0000			0.912343
$693$		0			0
$694$	−30.0000			−1.13878
$695$		0			0
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$	−0.500000	+	0.866025i	−0.0189253	+	0.0327795i
$699$		0			0
$700$	2.50000	+	4.33013i	0.0944911	+	0.163663i
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$		0			0
$705$		0			0
$706$	13.5000	−	23.3827i	0.508079	−	0.880019i
$707$	9.00000	−	15.5885i	0.338480	−	0.586264i
$708$		0			0
$709$	14.0000	+	24.2487i	0.525781	+	0.910679i	0.999549	+	0.0300298i	$0.00956021\pi$
$709$	14.0000	+	24.2487i	0.525781	+	0.910679i	−0.473768	+	0.880650i	$0.657106\pi$
$710$		0			0
$711$		0			0
$712$	15.0000			0.562149
$713$	−22.5000	−	38.9711i	−0.842632	−	1.45948i
$714$		0			0
$715$		0			0
$716$	−12.0000	+	20.7846i	−0.448461	+	0.776757i
$717$		0			0
$718$	1.50000	+	2.59808i	0.0559795	+	0.0969593i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	−13.0000			−0.484145
$722$	−7.50000	−	12.9904i	−0.279121	−	0.483452i
$723$		0			0
$724$	3.50000	−	6.06218i	0.130076	−	0.225299i
$725$	7.50000	−	12.9904i	0.278543	−	0.482451i
$726$		0			0
$727$	−8.50000	−	14.7224i	−0.315248	−	0.546025i	0.664243	−	0.747517i	$-0.268754\pi$
$727$	−8.50000	−	14.7224i	−0.315248	−	0.546025i	−0.979490	+	0.201492i	$0.935421\pi$
$728$	−5.00000			−0.185312
$729$		0			0
$730$		0			0
$731$	−1.50000	−	2.59808i	−0.0554795	−	0.0960933i
$732$		0			0
$733$	−11.5000	+	19.9186i	−0.424762	+	0.735710i	−0.996398	−	0.0847976i	$-0.972976\pi$
$733$	−11.5000	+	19.9186i	−0.424762	+	0.735710i	0.571636	+	0.820507i	$0.306309\pi$
$734$	−9.50000	+	16.4545i	−0.350651	+	0.607346i
$735$		0			0
$736$	4.50000	+	7.79423i	0.165872	+	0.287299i
$737$		0			0
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$		0			0
$742$	−1.50000	+	2.59808i	−0.0550667	+	0.0953784i
$743$	1.50000	−	2.59808i	0.0550297	−	0.0953142i	−0.837198	−	0.546899i	$-0.815808\pi$
$743$	1.50000	−	2.59808i	0.0550297	−	0.0953142i	0.892228	+	0.451585i	$0.149141\pi$
$744$		0			0
$745$		0			0
$746$	−14.0000			−0.512576
$747$		0			0
$748$		0			0
$749$	−6.00000	−	10.3923i	−0.219235	−	0.379727i
$750$		0			0
$751$	−7.00000	+	12.1244i	−0.255434	+	0.442424i	−0.965013	−	0.262201i	$-0.915552\pi$
$751$	−7.00000	+	12.1244i	−0.255434	+	0.442424i	0.709580	+	0.704625i	$0.248885\pi$
$752$	−3.00000	+	5.19615i	−0.109399	+	0.189484i
$753$		0			0
$754$	7.50000	+	12.9904i	0.273134	+	0.473082i
$755$		0			0
$756$		0			0
$757$	−16.0000			−0.581530			−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000			−0.581530			−0.290765	+	0.956795i	$0.593910\pi$
$758$	−8.00000	−	13.8564i	−0.290573	−	0.503287i
$759$		0			0
$760$		0			0
$761$	4.50000	−	7.79423i	0.163125	−	0.282541i	−0.772863	−	0.634573i	$-0.781176\pi$
$761$	4.50000	−	7.79423i	0.163125	−	0.282541i	0.935988	+	0.352032i	$0.114509\pi$
$762$		0			0
$763$	8.00000	+	13.8564i	0.289619	+	0.501636i
$764$	−12.0000			−0.434145
$765$		0			0
$766$		0			0
$767$	−7.50000	−	12.9904i	−0.270809	−	0.469055i
$768$		0			0
$769$	20.0000	−	34.6410i	0.721218	−	1.24919i	−0.239293	−	0.970947i	$-0.576916\pi$
$769$	20.0000	−	34.6410i	0.721218	−	1.24919i	0.960512	−	0.278240i	$-0.0897509\pi$
$770$		0			0
$771$		0			0
$772$	−2.50000	−	4.33013i	−0.0899770	−	0.155845i
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		0			0
$775$	−25.0000			−0.898027
$776$	4.00000	+	6.92820i	0.143592	+	0.248708i
$777$		0			0
$778$	−9.00000	+	15.5885i	−0.322666	+	0.558873i
$779$	−6.00000	+	10.3923i	−0.214972	+	0.372343i
$780$		0			0
$781$		0			0
$782$	27.0000			0.965518
$783$		0			0
$784$	1.00000			0.0357143
$785$		0			0
$786$		0			0
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	−0.963394	−	0.268091i	$-0.913607\pi$
$787$	−7.00000	+	12.1244i	−0.249523	+	0.432187i	0.713871	+	0.700278i	$0.246941\pi$
$788$	9.00000	−	15.5885i	0.320612	−	0.555316i
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	−50.0000			−1.77555
$794$	1.00000	+	1.73205i	0.0354887	+	0.0614682i
$795$		0			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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +(-2.50000 + 4.33013i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.00000 q^{17} +2.00000 q^{19} +(-4.50000 + 7.79423i) q^{23} +(2.50000 + 4.33013i) q^{25} -5.00000 q^{26} +1.00000 q^{28} +(-1.50000 - 2.59808i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{34} +2.00000 q^{37} +(1.00000 + 1.73205i) q^{38} +(-3.00000 + 5.19615i) q^{41} +(0.500000 + 0.866025i) q^{43} -9.00000 q^{46} +(-3.00000 - 5.19615i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-2.50000 + 4.33013i) q^{50} +(-2.50000 - 4.33013i) q^{52} -3.00000 q^{53} +(0.500000 + 0.866025i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -5.00000 q^{62} +1.00000 q^{64} +(6.50000 - 11.2583i) q^{67} +(1.50000 - 2.59808i) q^{68} -9.00000 q^{71} +2.00000 q^{73} +(1.00000 + 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(5.00000 + 8.66025i) q^{79} -6.00000 q^{82} +(-6.00000 - 10.3923i) q^{83} +(-0.500000 + 0.866025i) q^{86} -15.0000 q^{89} +5.00000 q^{91} +(-4.50000 - 7.79423i) q^{92} +(3.00000 - 5.19615i) q^{94} +(-4.00000 - 6.92820i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - q^{7} - 2 q^{8} - 5 q^{13} + q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - 9 q^{23} + 5 q^{25} - 10 q^{26} + 2 q^{28} - 3 q^{29} - 5 q^{31} + q^{32} - 3 q^{34} + 4 q^{37} + 2 q^{38} - 6 q^{41} + q^{43} - 18 q^{46} - 6 q^{47} - q^{49} - 5 q^{50} - 5 q^{52} - 6 q^{53} + q^{56} + 3 q^{58} - 3 q^{59} + 10 q^{61} - 10 q^{62} + 2 q^{64} + 13 q^{67} + 3 q^{68} - 18 q^{71} + 4 q^{73} + 2 q^{74} - 2 q^{76} + 10 q^{79} - 12 q^{82} - 12 q^{83} - q^{86} - 30 q^{89} + 10 q^{91} - 9 q^{92} + 6 q^{94} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^7 - 2 * q^8 - 5 * q^13 + q^14 - q^16 - 6 * q^17 + 4 * q^19 - 9 * q^23 + 5 * q^25 - 10 * q^26 + 2 * q^28 - 3 * q^29 - 5 * q^31 + q^32 - 3 * q^34 + 4 * q^37 + 2 * q^38 - 6 * q^41 + q^43 - 18 * q^46 - 6 * q^47 - q^49 - 5 * q^50 - 5 * q^52 - 6 * q^53 + q^56 + 3 * q^58 - 3 * q^59 + 10 * q^61 - 10 * q^62 + 2 * q^64 + 13 * q^67 + 3 * q^68 - 18 * q^71 + 4 * q^73 + 2 * q^74 - 2 * q^76 + 10 * q^79 - 12 * q^82 - 12 * q^83 - q^86 - 30 * q^89 + 10 * q^91 - 9 * q^92 + 6 * q^94 - 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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