LMFDB - Embedded newform 1400.2.bh.f.849.2

Newspace parameters

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.bh (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			849.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1400.849
Dual form			1400.2.bh.f.249.2

$q$-expansion

$f(q)$	$=$	$q+(2.59808 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(3.00000 - 5.19615i) q^{9} +O(q^{10})$	$q+(2.59808 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(3.00000 - 5.19615i) q^{9} +(0.500000 + 0.866025i) q^{11} +2.00000i q^{13} +(-2.59808 + 1.50000i) q^{17} +(2.50000 - 4.33013i) q^{19} +(1.50000 - 7.79423i) q^{21} +(2.59808 + 1.50000i) q^{23} -9.00000i q^{27} +6.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.59808 + 1.50000i) q^{33} +(-4.33013 - 2.50000i) q^{37} +(3.00000 + 5.19615i) q^{39} -10.0000 q^{41} -4.00000i q^{43} +(0.866025 + 0.500000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{51} +(-7.79423 + 4.50000i) q^{53} -15.0000i q^{57} +(1.50000 + 2.59808i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-5.19615 - 15.0000i) q^{63} +(-9.52628 + 5.50000i) q^{67} +9.00000 q^{69} +16.0000 q^{71} +(6.06218 - 3.50000i) q^{73} +(2.59808 + 0.500000i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-4.50000 - 7.79423i) q^{81} -4.00000i q^{83} +(15.5885 - 9.00000i) q^{87} +(-4.50000 + 7.79423i) q^{89} +(4.00000 + 3.46410i) q^{91} +(2.59808 + 1.50000i) q^{93} -6.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4q + 12q^{9} + O(q^{10}) $	$ 4q + 12q^{9} + 2q^{11} + 10q^{19} + 6q^{21} + 24q^{29} + 2q^{31} + 12q^{39} - 40q^{41} - 4q^{49} - 18q^{51} + 6q^{59} - 6q^{61} + 36q^{69} + 64q^{71} - 22q^{79} - 18q^{81} - 18q^{89} + 16q^{91} + 24q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	2.59808	−	1.50000i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$3$	2.59808	−	1.50000i	1.50000	−	0.866025i	1.00000			$0$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$7$	1.73205	−	2.00000i	0.654654	−	0.755929i
$8$		0			0
$9$	3.00000	−	5.19615i	1.00000	−	1.73205i
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.59808	+	1.50000i	−0.630126	+	0.363803i	−0.780801	−	0.624780i	$-0.785189\pi$
$17$	−2.59808	+	1.50000i	−0.630126	+	0.363803i	0.150675	+	0.988583i	$0.451855\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$	1.50000	−	7.79423i	0.327327	−	1.70084i
$22$		0			0
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	0.745782	−	0.666190i	$-0.232076\pi$
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	−0.204046	+	0.978961i	$0.565409\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	9.00000i		−	1.73205i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$	2.59808	+	1.50000i	0.452267	+	0.261116i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.33013	−	2.50000i	−0.711868	−	0.410997i	0.0998840	−	0.994999i	$-0.468153\pi$
$37$	−4.33013	−	2.50000i	−0.711868	−	0.410997i	−0.811752	+	0.584002i	$0.801486\pi$
$38$		0			0
$39$	3.00000	+	5.19615i	0.480384	+	0.832050i
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.866025	+	0.500000i	0.126323	+	0.0729325i	0.561830	−	0.827253i	$-0.310098\pi$
$47$	0.866025	+	0.500000i	0.126323	+	0.0729325i	−0.435507	+	0.900185i	$0.643431\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$		0			0
$51$	−4.50000	+	7.79423i	−0.630126	+	1.09141i
$52$		0			0
$53$	−7.79423	+	4.50000i	−1.07062	+	0.618123i	−0.928351	−	0.371706i	$-0.878773\pi$
$53$	−7.79423	+	4.50000i	−1.07062	+	0.618123i	−0.142269	+	0.989828i	$0.545440\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	15.0000i		−	1.98680i
$58$		0			0
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	−0.945931	−	0.324367i	$-0.894849\pi$
$61$	−1.50000	+	2.59808i	−0.192055	+	0.332650i	0.753876	+	0.657017i	$0.228182\pi$
$62$		0			0
$63$	−5.19615	−	15.0000i	−0.654654	−	1.88982i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	−9.52628	+	5.50000i	−1.16382	+	0.671932i	−0.211604	+	0.977356i	$0.567869\pi$
$68$		0			0
$69$	9.00000			1.08347
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$		0			0
$73$	6.06218	−	3.50000i	0.709524	−	0.409644i	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	6.06218	−	3.50000i	0.709524	−	0.409644i	0.810885	+	0.585206i	$0.198986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.59808	+	0.500000i	0.296078	+	0.0569803i
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	0.370907	+	0.928670i	$0.379047\pi$
$79$	−5.50000	+	9.52628i	−0.618798	+	1.07179i	−0.989705	+	0.143120i	$0.954286\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	15.5885	−	9.00000i	1.67126	−	0.964901i
$88$		0			0
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	0.522654	+	0.852545i	$0.324942\pi$
$90$		0			0
$91$	4.00000	+	3.46410i	0.419314	+	0.363137i
$92$		0			0
$93$	2.59808	+	1.50000i	0.269408	+	0.155543i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	6.50000	+	11.2583i	0.646774	+	1.12025i	0.983889	+	0.178782i	$0.0572157\pi$
$101$	6.50000	+	11.2583i	0.646774	+	1.12025i	−0.337115	+	0.941464i	$0.609451\pi$
$102$		0			0
$103$	−4.33013	−	2.50000i	−0.426660	−	0.246332i	0.271263	−	0.962505i	$-0.412559\pi$
$103$	−4.33013	−	2.50000i	−0.426660	−	0.246332i	−0.697923	+	0.716173i	$0.745892\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.59808	−	1.50000i	−0.251166	−	0.145010i	0.369132	−	0.929377i	$-0.379655\pi$
$107$	−2.59808	−	1.50000i	−0.251166	−	0.145010i	−0.620298	+	0.784366i	$0.712988\pi$
$108$		0			0
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	0.999512	+	0.0312328i	$0.00994332\pi$
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	−0.472708	+	0.881219i	$0.656723\pi$
$110$		0			0
$111$	−15.0000			−1.42374
$112$		0			0
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	10.3923	+	6.00000i	0.960769	+	0.554700i
$118$		0			0
$119$	−1.50000	+	7.79423i	−0.137505	+	0.714496i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$	−25.9808	+	15.0000i	−2.34261	+	1.35250i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$	−6.00000	−	10.3923i	−0.528271	−	0.914991i
$130$		0			0
$131$	−8.50000	+	14.7224i	−0.742648	+	1.28630i	0.208637	+	0.977993i	$0.433097\pi$
$131$	−8.50000	+	14.7224i	−0.742648	+	1.28630i	−0.951285	+	0.308312i	$0.900236\pi$
$132$		0			0
$133$	−4.33013	−	12.5000i	−0.375470	−	1.08389i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.59808	+	1.50000i	−0.221969	+	0.128154i	−0.606861	−	0.794808i	$-0.707572\pi$
$137$	−2.59808	+	1.50000i	−0.221969	+	0.128154i	0.384893	+	0.922961i	$0.374238\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$	−1.73205	+	1.00000i	−0.144841	+	0.0836242i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−12.9904	−	16.5000i	−1.07143	−	1.36090i
$148$		0			0
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	−0.376061	−	0.926595i	$-0.622722\pi$
$149$	7.50000	−	12.9904i	0.614424	−	1.06421i	0.990485	−	0.137619i	$-0.0439449\pi$
$150$		0			0
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	−0.991183	−	0.132502i	$-0.957699\pi$
$151$	−7.50000	−	12.9904i	−0.610341	−	1.05714i	0.380841	−	0.924640i	$-0.375634\pi$
$152$		0			0
$153$			18.0000i			1.45521i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−12.9904	+	7.50000i	−1.03675	+	0.598565i	−0.918910	−	0.394468i	$-0.870929\pi$
$157$	−12.9904	+	7.50000i	−1.03675	+	0.598565i	−0.117836	+	0.993033i	$0.537596\pi$
$158$		0			0
$159$	−13.5000	+	23.3827i	−1.07062	+	1.85437i
$160$		0			0
$161$	7.50000	−	2.59808i	0.591083	−	0.204757i
$162$		0			0
$163$	−7.79423	−	4.50000i	−0.610491	−	0.352467i	0.162667	−	0.986681i	$-0.447991\pi$
$163$	−7.79423	−	4.50000i	−0.610491	−	0.352467i	−0.773158	+	0.634214i	$0.781324\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			20.0000i			1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$			20.0000i			1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−15.0000	−	25.9808i	−1.14708	−	1.98680i
$172$		0			0
$173$	18.1865	+	10.5000i	1.38270	+	0.798300i	0.992478	−	0.122422i	$-0.0390662\pi$
$173$	18.1865	+	10.5000i	1.38270	+	0.798300i	0.390218	+	0.920722i	$0.372399\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.79423	+	4.50000i	0.585850	+	0.338241i
$178$		0			0
$179$	−0.500000	−	0.866025i	−0.0373718	−	0.0647298i	0.846735	−	0.532016i	$-0.178565\pi$
$179$	−0.500000	−	0.866025i	−0.0373718	−	0.0647298i	−0.884106	+	0.467286i	$0.845232\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$			9.00000i			0.665299i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.59808	−	1.50000i	−0.189990	−	0.109691i
$188$		0			0
$189$	−18.0000	−	15.5885i	−1.30931	−	1.13389i
$190$		0			0
$191$	−8.50000	+	14.7224i	−0.615038	+	1.06528i	0.375339	+	0.926887i	$0.377526\pi$
$191$	−8.50000	+	14.7224i	−0.615038	+	1.06528i	−0.990378	+	0.138390i	$0.955807\pi$
$192$		0			0
$193$	−4.33013	+	2.50000i	−0.311689	+	0.179954i	−0.647682	−	0.761911i	$-0.724262\pi$
$193$	−4.33013	+	2.50000i	−0.311689	+	0.179954i	0.335993	+	0.941865i	$0.390928\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	−4.50000	−	7.79423i	−0.318997	−	0.552518i	0.661282	−	0.750137i	$-0.270013\pi$
$199$	−4.50000	−	7.79423i	−0.318997	−	0.552518i	−0.980279	+	0.197619i	$0.936679\pi$
$200$		0			0
$201$	−16.5000	+	28.5788i	−1.16382	+	2.01580i
$202$		0			0
$203$	10.3923	−	12.0000i	0.729397	−	0.842235i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	15.5885	−	9.00000i	1.08347	−	0.625543i
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$	41.5692	−	24.0000i	2.84828	−	1.64445i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.59808	+	0.500000i	0.176369	+	0.0339422i
$218$		0			0
$219$	10.5000	−	18.1865i	0.709524	−	1.22893i
$220$		0			0
$221$	−3.00000	−	5.19615i	−0.201802	−	0.349531i
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.06218	+	3.50000i	−0.402361	+	0.232303i	−0.687502	−	0.726182i	$-0.741293\pi$
$227$	−6.06218	+	3.50000i	−0.402361	+	0.232303i	0.285141	+	0.958485i	$0.407959\pi$
$228$		0			0
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	3.50000	−	6.06218i	0.231287	−	0.400600i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$	7.50000	−	2.59808i	0.493464	−	0.170941i
$232$		0			0
$233$	11.2583	+	6.50000i	0.737558	+	0.425829i	0.821181	−	0.570668i	$-0.193316\pi$
$233$	11.2583	+	6.50000i	0.737558	+	0.425829i	−0.0836229	+	0.996497i	$0.526649\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			33.0000i			2.14358i
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	0.998443	+	0.0557856i	$0.0177663\pi$
$241$	8.50000	+	14.7224i	0.547533	+	0.948355i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.66025	+	5.00000i	0.551039	+	0.318142i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.2583	−	6.50000i	−0.702275	−	0.405459i	0.105919	−	0.994375i	$-0.466222\pi$
$257$	−11.2583	−	6.50000i	−0.702275	−	0.405459i	−0.808194	+	0.588916i	$0.799555\pi$
$258$		0			0
$259$	−12.5000	+	4.33013i	−0.776712	+	0.269061i
$260$		0			0
$261$	18.0000	−	31.1769i	1.11417	−	1.92980i
$262$		0			0
$263$	2.59808	−	1.50000i	0.160204	−	0.0924940i	−0.417755	−	0.908560i	$-0.637183\pi$
$263$	2.59808	−	1.50000i	0.160204	−	0.0924940i	0.577959	+	0.816066i	$0.303849\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			27.0000i			1.65237i
$268$		0			0
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	−0.999775	−	0.0212079i	$-0.993249\pi$
$269$	−8.50000	−	14.7224i	−0.518254	−	0.897643i	0.481521	−	0.876435i	$-0.340085\pi$
$270$		0			0
$271$	1.50000	−	2.59808i	0.0911185	−	0.157822i	−0.816864	−	0.576831i	$-0.804289\pi$
$271$	1.50000	−	2.59808i	0.0911185	−	0.157822i	0.907982	+	0.419009i	$0.137622\pi$
$272$		0			0
$273$	15.5885	+	3.00000i	0.943456	+	0.181568i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.06218	+	3.50000i	−0.364241	+	0.210295i	−0.670940	−	0.741512i	$-0.734109\pi$
$277$	−6.06218	+	3.50000i	−0.364241	+	0.210295i	0.306699	+	0.951807i	$0.400776\pi$
$278$		0			0
$279$	6.00000			0.359211
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−14.7224	+	8.50000i	−0.875158	+	0.505273i	−0.869059	−	0.494709i	$-0.835275\pi$
$283$	−14.7224	+	8.50000i	−0.875158	+	0.505273i	−0.00609896	+	0.999981i	$0.501941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−17.3205	+	20.0000i	−1.02240	+	1.18056i
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$		0			0
$291$	−9.00000	−	15.5885i	−0.527589	−	0.913812i
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	7.79423	−	4.50000i	0.452267	−	0.261116i
$298$		0			0
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−8.00000	−	6.92820i	−0.461112	−	0.399335i
$302$		0			0
$303$	33.7750	+	19.5000i	1.94032	+	1.12025i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	−15.0000			−0.853320
$310$		0			0
$311$	−5.50000	−	9.52628i	−0.311876	−	0.540186i	0.666892	−	0.745154i	$-0.267624\pi$
$311$	−5.50000	−	9.52628i	−0.311876	−	0.540186i	−0.978769	+	0.204968i	$0.934291\pi$
$312$		0			0
$313$	−26.8468	−	15.5000i	−1.51747	−	0.876112i	−0.999789	−	0.0205381i	$-0.993462\pi$
$313$	−26.8468	−	15.5000i	−1.51747	−	0.876112i	−0.517681	−	0.855574i	$-0.673205\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	23.3827	+	13.5000i	1.31330	+	0.758236i	0.982642	−	0.185514i	$-0.0593950\pi$
$317$	23.3827	+	13.5000i	1.31330	+	0.758236i	0.330661	+	0.943750i	$0.392728\pi$
$318$		0			0
$319$	3.00000	+	5.19615i	0.167968	+	0.290929i
$320$		0			0
$321$	−9.00000			−0.502331
$322$		0			0
$323$			15.0000i			0.834622i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	28.5788	+	16.5000i	1.58041	+	0.912452i
$328$		0			0
$329$	2.50000	−	0.866025i	0.137829	−	0.0477455i
$330$		0			0
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	−0.753660	−	0.657264i	$-0.771714\pi$
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	0.946038	+	0.324057i	$0.105047\pi$
$332$		0			0
$333$	−25.9808	+	15.0000i	−1.42374	+	0.821995i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	14.0000i		−	0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$		−	14.0000i		−	0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$		0			0
$339$	−15.0000	−	25.9808i	−0.814688	−	1.41108i
$340$		0			0
$341$	−0.500000	+	0.866025i	−0.0270765	+	0.0468979i
$342$		0			0
$343$	−15.5885	−	10.0000i	−0.841698	−	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.59808	+	1.50000i	−0.139472	+	0.0805242i	−0.568112	−	0.822951i	$-0.692326\pi$
$347$	−2.59808	+	1.50000i	−0.139472	+	0.0805242i	0.428640	+	0.903475i	$0.358993\pi$
$348$		0			0
$349$	6.00000			0.321173			0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000			0.321173			0.160586	+	0.987022i	$0.448662\pi$
$350$		0			0
$351$	18.0000			0.960769
$352$		0			0
$353$	−4.33013	+	2.50000i	−0.230469	+	0.133062i	−0.610789	−	0.791794i	$-0.709147\pi$
$353$	−4.33013	+	2.50000i	−0.230469	+	0.133062i	0.380319	+	0.924855i	$0.375814\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	7.79423	+	22.5000i	0.412514	+	1.19083i
$358$		0			0
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	−0.993207	−	0.116358i	$-0.962878\pi$
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	0.597372	+	0.801964i	$0.296211\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$		−	30.0000i		−	1.57459i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	−0.863649	−	0.504093i	$-0.831827\pi$
$367$	−16.4545	+	9.50000i	−0.858917	+	0.495896i	0.00473247	+	0.999989i	$0.498494\pi$
$368$		0			0
$369$	−30.0000	+	51.9615i	−1.56174	+	2.70501i
$370$		0			0
$371$	−4.50000	+	23.3827i	−0.233628	+	1.21397i
$372$		0			0
$373$	−16.4545	−	9.50000i	−0.851981	−	0.491891i	0.00933789	−	0.999956i	$-0.497028\pi$
$373$	−16.4545	−	9.50000i	−0.851981	−	0.491891i	−0.861319	+	0.508065i	$0.830361\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$	12.0000	+	20.7846i	0.614779	+	1.06483i
$382$		0			0
$383$	−7.79423	−	4.50000i	−0.398266	−	0.229939i	0.287469	−	0.957790i	$-0.407186\pi$
$383$	−7.79423	−	4.50000i	−0.398266	−	0.229939i	−0.685736	+	0.727851i	$0.740519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−20.7846	−	12.0000i	−1.05654	−	0.609994i
$388$		0			0
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	0.999779	−	0.0210389i	$-0.00669738\pi$
$389$	9.50000	+	16.4545i	0.481669	+	0.834275i	−0.518110	+	0.855314i	$0.673364\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$		0			0
$393$			51.0000i			2.57261i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−14.7224	−	8.50000i	−0.738898	−	0.426603i	0.0827707	−	0.996569i	$-0.473623\pi$
$397$	−14.7224	−	8.50000i	−0.738898	−	0.426603i	−0.821668	+	0.569966i	$0.806956\pi$
$398$		0			0
$399$	−30.0000	−	25.9808i	−1.50188	−	1.30066i
$400$		0			0
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	−0.901046	−	0.433724i	$-0.857199\pi$
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	0.826139	+	0.563466i	$0.190532\pi$
$402$		0			0
$403$	−1.73205	+	1.00000i	−0.0862796	+	0.0498135i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.00000i		−	0.247841i
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$		0			0
$411$	−4.50000	+	7.79423i	−0.221969	+	0.384461i
$412$		0			0
$413$	7.79423	+	1.50000i	0.383529	+	0.0738102i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−10.3923	+	6.00000i	−0.508913	+	0.293821i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	5.19615	−	3.00000i	0.252646	−	0.145865i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.59808	+	7.50000i	0.125730	+	0.362950i
$428$		0			0
$429$	−3.00000	+	5.19615i	−0.144841	+	0.250873i
$430$		0			0
$431$	20.5000	+	35.5070i	0.987450	+	1.71031i	0.630497	+	0.776192i	$0.282851\pi$
$431$	20.5000	+	35.5070i	0.987450	+	1.71031i	0.356953	+	0.934122i	$0.383815\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.9904	−	7.50000i	0.621414	−	0.358774i
$438$		0			0
$439$	−7.50000	+	12.9904i	−0.357955	+	0.619997i	−0.987619	−	0.156871i	$-0.949859\pi$
$439$	−7.50000	+	12.9904i	−0.357955	+	0.619997i	0.629664	+	0.776868i	$0.283193\pi$
$440$		0			0
$441$	−39.0000	−	15.5885i	−1.85714	−	0.742307i
$442$		0			0
$443$	23.3827	+	13.5000i	1.11094	+	0.641404i	0.939074	−	0.343715i	$-0.111685\pi$
$443$	23.3827	+	13.5000i	1.11094	+	0.641404i	0.171871	+	0.985119i	$0.445019\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	45.0000i		−	2.12843i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	−5.00000	−	8.66025i	−0.235441	−	0.407795i
$452$		0			0
$453$	−38.9711	−	22.5000i	−1.83102	−	1.05714i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−14.7224	−	8.50000i	−0.688686	−	0.397613i	0.114433	−	0.993431i	$-0.463495\pi$
$457$	−14.7224	−	8.50000i	−0.688686	−	0.397613i	−0.803120	+	0.595818i	$0.796828\pi$
$458$		0			0
$459$	13.5000	+	23.3827i	0.630126	+	1.09141i
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	21.6506	+	12.5000i	1.00187	+	0.578431i	0.908802	−	0.417229i	$-0.136999\pi$
$467$	21.6506	+	12.5000i	1.00187	+	0.578431i	0.0930703	+	0.995660i	$0.470332\pi$
$468$		0			0
$469$	−5.50000	+	28.5788i	−0.253966	+	1.31965i
$470$		0			0
$471$	−22.5000	+	38.9711i	−1.03675	+	1.79570i
$472$		0			0
$473$	3.46410	−	2.00000i	0.159280	−	0.0919601i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			54.0000i			2.47249i
$478$		0			0
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	0.519973	−	0.854183i	$-0.325942\pi$
$479$	−10.5000	−	18.1865i	−0.479757	−	0.830964i	−0.999730	+	0.0232187i	$0.992609\pi$
$480$		0			0
$481$	5.00000	−	8.66025i	0.227980	−	0.394874i
$482$		0			0
$483$	15.5885	−	18.0000i	0.709299	−	0.819028i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	11.2583	−	6.50000i	0.510164	−	0.294543i	−0.222737	−	0.974879i	$-0.571499\pi$
$487$	11.2583	−	6.50000i	0.510164	−	0.294543i	0.732901	+	0.680335i	$0.238166\pi$
$488$		0			0
$489$	−27.0000			−1.22098
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−15.5885	+	9.00000i	−0.702069	+	0.405340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	27.7128	−	32.0000i	1.24309	−	1.43540i
$498$		0			0
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	−0.933670	−	0.358134i	$-0.883413\pi$
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	0.776989	+	0.629515i	$0.216746\pi$
$500$		0			0
$501$	30.0000	+	51.9615i	1.34030	+	2.32147i
$502$		0			0
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	23.3827	−	13.5000i	1.03846	−	0.599556i
$508$		0			0
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	−0.777973	−	0.628297i	$-0.783752\pi$
$509$	3.50000	−	6.06218i	0.155135	−	0.268701i	0.933108	+	0.359596i	$0.117085\pi$
$510$		0			0
$511$	3.50000	−	18.1865i	0.154831	−	0.804525i
$512$		0			0
$513$	−38.9711	−	22.5000i	−1.72062	−	0.993399i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			1.00000i			0.0439799i
$518$		0			0
$519$	63.0000			2.76539
$520$		0			0
$521$	−7.50000	−	12.9904i	−0.328581	−	0.569119i	0.653650	−	0.756797i	$-0.273237\pi$
$521$	−7.50000	−	12.9904i	−0.328581	−	0.569119i	−0.982231	+	0.187678i	$0.939904\pi$
$522$		0			0
$523$	−11.2583	−	6.50000i	−0.492292	−	0.284225i	0.233233	−	0.972421i	$-0.425070\pi$
$523$	−11.2583	−	6.50000i	−0.492292	−	0.284225i	−0.725525	+	0.688196i	$0.758403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.59808	−	1.50000i	−0.113174	−	0.0653410i
$528$		0			0
$529$	−7.00000	−	12.1244i	−0.304348	−	0.527146i
$530$		0			0
$531$	18.0000			0.781133
$532$		0			0
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.59808	−	1.50000i	−0.112115	−	0.0647298i
$538$		0			0
$539$	5.50000	−	4.33013i	0.236902	−	0.186512i
$540$		0			0
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	−0.461625	−	0.887075i	$-0.652733\pi$
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	0.999042	−	0.0437584i	$-0.0139332\pi$
$542$		0			0
$543$	57.1577	−	33.0000i	2.45287	−	1.41617i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	9.00000	+	15.5885i	0.384111	+	0.665299i
$550$		0			0
$551$	15.0000	−	25.9808i	0.639021	−	1.10682i
$552$		0			0
$553$	9.52628	+	27.5000i	0.405099	+	1.16942i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.52628	+	5.50000i	−0.403641	+	0.233042i	−0.688054	−	0.725660i	$-0.741535\pi$
$557$	−9.52628	+	5.50000i	−0.403641	+	0.233042i	0.284413	+	0.958702i	$0.408201\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−9.00000			−0.379980
$562$		0			0
$563$	9.52628	−	5.50000i	0.401485	−	0.231797i	−0.285640	−	0.958337i	$-0.592206\pi$
$563$	9.52628	−	5.50000i	0.401485	−	0.231797i	0.687124	+	0.726540i	$0.258873\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.3827	−	4.50000i	−0.981981	−	0.188982i
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	−0.876316	−	0.481737i	$-0.840006\pi$
$569$	−0.500000	+	0.866025i	−0.0209611	+	0.0363057i	0.855355	+	0.518043i	$0.173339\pi$
$570$		0			0
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	0.987240	−	0.159240i	$-0.0509044\pi$
$571$	8.50000	+	14.7224i	0.355714	+	0.616115i	−0.631526	+	0.775355i	$0.717571\pi$
$572$		0			0
$573$			51.0000i			2.13056i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−26.8468	+	15.5000i	−1.11765	+	0.645273i	−0.940799	−	0.338965i	$-0.889923\pi$
$577$	−26.8468	+	15.5000i	−1.11765	+	0.645273i	−0.176847	+	0.984238i	$0.556590\pi$
$578$		0			0
$579$	−7.50000	+	12.9904i	−0.311689	+	0.539862i
$580$		0			0
$581$	−8.00000	−	6.92820i	−0.331896	−	0.287430i
$582$		0			0
$583$	−7.79423	−	4.50000i	−0.322804	−	0.186371i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	5.00000			0.206021
$590$		0			0
$591$	−27.0000	−	46.7654i	−1.11063	−	1.92367i
$592$		0			0
$593$	−37.2391	−	21.5000i	−1.52923	−	0.882899i	−0.999394	−	0.0347964i	$-0.988922\pi$
$593$	−37.2391	−	21.5000i	−1.52923	−	0.882899i	−0.529832	−	0.848103i	$-0.677745\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−23.3827	−	13.5000i	−0.956990	−	0.552518i
$598$		0			0
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	0.567768	−	0.823189i	$-0.307807\pi$
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	−0.996786	+	0.0801071i	$0.974474\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$			66.0000i			2.68773i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−6.06218	−	3.50000i	−0.246056	−	0.142061i	0.371901	−	0.928272i	$-0.378706\pi$
$607$	−6.06218	−	3.50000i	−0.246056	−	0.142061i	−0.617957	+	0.786212i	$0.712039\pi$
$608$		0			0
$609$	9.00000	−	46.7654i	0.364698	−	1.89503i
$610$		0			0
$611$	−1.00000	+	1.73205i	−0.0404557	+	0.0700713i
$612$		0			0
$613$	−18.1865	+	10.5000i	−0.734547	+	0.424091i	−0.820083	−	0.572244i	$-0.806073\pi$
$613$	−18.1865	+	10.5000i	−0.734547	+	0.424091i	0.0855362	+	0.996335i	$0.472740\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		0			0
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	0.811400	−	0.584491i	$-0.198706\pi$
$619$	−2.50000	−	4.33013i	−0.100483	−	0.174042i	−0.911884	+	0.410448i	$0.865372\pi$
$620$		0			0
$621$	13.5000	−	23.3827i	0.541736	−	0.938315i
$622$		0			0
$623$	7.79423	+	22.5000i	0.312269	+	0.901443i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	12.9904	−	7.50000i	0.518786	−	0.299521i
$628$		0			0
$629$	15.0000			0.598089
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	31.1769	−	18.0000i	1.23917	−	0.715436i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	13.8564	−	2.00000i	0.549011	−	0.0792429i
$638$		0			0
$639$	48.0000	−	83.1384i	1.89885	−	3.28891i
$640$		0			0
$641$	−7.50000	−	12.9904i	−0.296232	−	0.513089i	0.679039	−	0.734103i	$-0.262397\pi$
$641$	−7.50000	−	12.9904i	−0.296232	−	0.513089i	−0.975271	+	0.221013i	$0.929064\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−37.2391	+	21.5000i	−1.46402	+	0.845252i	−0.999194	−	0.0401498i	$-0.987216\pi$
$647$	−37.2391	+	21.5000i	−1.46402	+	0.845252i	−0.464826	+	0.885402i	$0.653883\pi$
$648$		0			0
$649$	−1.50000	+	2.59808i	−0.0588802	+	0.101983i
$650$		0			0
$651$	7.50000	−	2.59808i	0.293948	−	0.101827i
$652$		0			0
$653$	4.33013	+	2.50000i	0.169451	+	0.0978326i	0.582327	−	0.812955i	$-0.302142\pi$
$653$	4.33013	+	2.50000i	0.169451	+	0.0978326i	−0.412876	+	0.910787i	$0.635476\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	42.0000i		−	1.63858i
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	0.875585	−	0.483063i	$-0.160476\pi$
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	−0.856138	+	0.516748i	$0.827143\pi$
$662$		0			0
$663$	−15.5885	−	9.00000i	−0.605406	−	0.349531i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	15.5885	+	9.00000i	0.603587	+	0.348481i
$668$		0			0
$669$	36.0000	+	62.3538i	1.39184	+	2.41074i
$670$		0			0
$671$	−3.00000			−0.115814
$672$		0			0
$673$		−	2.00000i		−	0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$		−	2.00000i		−	0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.79423	−	4.50000i	−0.299557	−	0.172949i	0.342687	−	0.939450i	$-0.388663\pi$
$677$	−7.79423	−	4.50000i	−0.299557	−	0.172949i	−0.642244	+	0.766501i	$0.721996\pi$
$678$		0			0
$679$	−12.0000	−	10.3923i	−0.460518	−	0.398820i
$680$		0			0
$681$	−10.5000	+	18.1865i	−0.402361	+	0.696909i
$682$		0			0
$683$	33.7750	−	19.5000i	1.29236	−	0.746147i	0.313291	−	0.949657i	$-0.398568\pi$
$683$	33.7750	−	19.5000i	1.29236	−	0.746147i	0.979073	+	0.203510i	$0.0652350\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	21.0000i		−	0.801200i
$688$		0			0
$689$	−9.00000	−	15.5885i	−0.342873	−	0.593873i
$690$		0			0
$691$	23.5000	−	40.7032i	0.893982	−	1.54842i	0.0589228	−	0.998263i	$-0.481233\pi$
$691$	23.5000	−	40.7032i	0.893982	−	1.54842i	0.835059	−	0.550160i	$-0.185433\pi$
$692$		0			0
$693$	10.3923	−	12.0000i	0.394771	−	0.455842i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	25.9808	−	15.0000i	0.984092	−	0.568166i
$698$		0			0
$699$	39.0000			1.47512
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−21.6506	+	12.5000i	−0.816569	+	0.471446i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.7750	+	6.50000i	1.27024	+	0.244458i
$708$		0			0
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	−0.492964	−	0.870050i	$-0.664087\pi$
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	0.999967	−	0.00810550i	$-0.00258009\pi$
$710$		0			0
$711$	33.0000	+	57.1577i	1.23760	+	2.14358i
$712$		0			0
$713$			3.00000i			0.112351i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.3923	−	6.00000i	0.388108	−	0.224074i
$718$		0			0
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	−0.458102	−	0.888900i	$-0.651471\pi$
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	0.998861	−	0.0477220i	$-0.0151961\pi$
$720$		0			0
$721$	−12.5000	+	4.33013i	−0.465524	+	0.161262i
$722$		0			0
$723$	44.1673	+	25.5000i	1.64260	+	0.948355i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	6.00000	+	10.3923i	0.221918	+	0.384373i
$732$		0			0
$733$	−9.52628	−	5.50000i	−0.351861	−	0.203147i	0.313644	−	0.949541i	$-0.398450\pi$
$733$	−9.52628	−	5.50000i	−0.351861	−	0.203147i	−0.665505	+	0.746394i	$0.731784\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.52628	−	5.50000i	−0.350905	−	0.202595i
$738$		0			0
$739$	−20.5000	−	35.5070i	−0.754105	−	1.30615i	−0.945818	−	0.324697i	$-0.894738\pi$
$739$	−20.5000	−	35.5070i	−0.754105	−	1.30615i	0.191714	−	0.981451i	$-0.438596\pi$
$740$		0			0
$741$	30.0000			1.10208
$742$		0			0
$743$			32.0000i			1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$			32.0000i			1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−20.7846	−	12.0000i	−0.760469	−	0.439057i
$748$		0			0
$749$	−7.50000	+	2.59808i	−0.274044	+	0.0949316i
$750$		0			0
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	−0.0167534	−	0.999860i	$-0.505333\pi$
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	0.874281	−	0.485421i	$-0.161334\pi$
$752$		0			0
$753$	62.3538	−	36.0000i	2.27230	−	1.31191i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	34.0000i		−	1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$		−	34.0000i		−	1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$		0			0
$759$	4.50000	+	7.79423i	0.163340	+	0.282913i
$760$		0			0
$761$	−13.5000	+	23.3827i	−0.489375	+	0.847622i	−0.999925	−	0.0122260i	$-0.996108\pi$
$761$	−13.5000	+	23.3827i	−0.489375	+	0.847622i	0.510551	+	0.859848i	$0.329442\pi$
$762$		0			0
$763$	28.5788	+	5.50000i	1.03462	+	0.199113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.19615	+	3.00000i	−0.187622	+	0.108324i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	−39.0000			−1.40455
$772$		0			0
$773$	30.3109	−	17.5000i	1.09021	−	0.629431i	0.156575	−	0.987666i	$-0.449955\pi$
$773$	30.3109	−	17.5000i	1.09021	−	0.629431i	0.933632	+	0.358235i	$0.116621\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−25.9808	+	30.0000i	−0.932055	+	1.07624i
$778$		0			0
$779$	−25.0000	+	43.3013i	−0.895718	+	1.55143i
$780$		0			0
$781$	8.00000	+	13.8564i	0.286263	+	0.495821i
$782$		0			0
$783$		−	54.0000i		−	1.92980i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	11.2583	−	6.50000i	0.401316	−	0.231700i	−0.285736	−	0.958308i	$-0.592238\pi$
$787$	11.2583	−	6.50000i	0.401316	−	0.231700i	0.687052	+	0.726609i	$0.258905\pi$
$788$		0			0
$789$	4.50000	−	7.79423i	0.160204	−	0.277482i
$790$		0			0
$791$	−20.0000	−	17.3205i	−0.711118	−	0.615846i
$792$		0			0
$793$	−5.19615	−	3.00000i	−0.184521	−	0.106533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	42.0000i		−	1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$		−	42.0000i		−	1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$		0			0
$799$	−3.00000			−0.106132

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.59808 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(3.00000 - 5.19615i) q^{9} +O(q^{10})\)	\(q+(2.59808 - 1.50000i) q^{3} +(1.73205 - 2.00000i) q^{7} +(3.00000 - 5.19615i) q^{9} +(0.500000 + 0.866025i) q^{11} +2.00000i q^{13} +(-2.59808 + 1.50000i) q^{17} +(2.50000 - 4.33013i) q^{19} +(1.50000 - 7.79423i) q^{21} +(2.59808 + 1.50000i) q^{23} -9.00000i q^{27} +6.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(2.59808 + 1.50000i) q^{33} +(-4.33013 - 2.50000i) q^{37} +(3.00000 + 5.19615i) q^{39} -10.0000 q^{41} -4.00000i q^{43} +(0.866025 + 0.500000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(-4.50000 + 7.79423i) q^{51} +(-7.79423 + 4.50000i) q^{53} -15.0000i q^{57} +(1.50000 + 2.59808i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-5.19615 - 15.0000i) q^{63} +(-9.52628 + 5.50000i) q^{67} +9.00000 q^{69} +16.0000 q^{71} +(6.06218 - 3.50000i) q^{73} +(2.59808 + 0.500000i) q^{77} +(-5.50000 + 9.52628i) q^{79} +(-4.50000 - 7.79423i) q^{81} -4.00000i q^{83} +(15.5885 - 9.00000i) q^{87} +(-4.50000 + 7.79423i) q^{89} +(4.00000 + 3.46410i) q^{91} +(2.59808 + 1.50000i) q^{93} -6.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q + 12q^{9} + O(q^{10}) \)	\( 4q + 12q^{9} + 2q^{11} + 10q^{19} + 6q^{21} + 24q^{29} + 2q^{31} + 12q^{39} - 40q^{41} - 4q^{49} - 18q^{51} + 6q^{59} - 6q^{61} + 36q^{69} + 64q^{71} - 22q^{79} - 18q^{81} - 18q^{89} + 16q^{91} + 24q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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