LMFDB - Embedded newform 315.2.d.c.64.2

Newspace parameters

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.2.d.c.64.1

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000i q^{8} +(2.00000 - 1.00000i) q^{10} +6.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +6.00000 q^{19} +(-1.00000 - 2.00000i) q^{20} +6.00000i q^{22} +(-3.00000 + 4.00000i) q^{25} -2.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} +4.00000 q^{34} +(-2.00000 + 1.00000i) q^{35} -4.00000i q^{37} +6.00000i q^{38} +(6.00000 - 3.00000i) q^{40} -2.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} -1.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +2.00000i q^{52} +6.00000i q^{53} +(-6.00000 - 12.0000i) q^{55} +3.00000 q^{56} -2.00000i q^{58} -8.00000 q^{59} -2.00000 q^{61} -10.0000i q^{62} -7.00000 q^{64} +(4.00000 - 2.00000i) q^{65} -16.0000i q^{67} -4.00000i q^{68} +(-1.00000 - 2.00000i) q^{70} -10.0000 q^{71} +6.00000i q^{73} +4.00000 q^{74} +6.00000 q^{76} -6.00000i q^{77} -4.00000 q^{79} +(1.00000 + 2.00000i) q^{80} -2.00000i q^{82} +8.00000i q^{83} +(-8.00000 + 4.00000i) q^{85} -4.00000 q^{86} +18.0000i q^{88} +6.00000 q^{89} +2.00000 q^{91} +(-6.00000 - 12.0000i) q^{95} -2.00000i q^{97} -1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5	$ 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} + 12 q^{11} + 2 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 4 q^{29} - 20 q^{31} + 8 q^{34} - 4 q^{35} + 12 q^{40} - 4 q^{41} + 12 q^{44} - 2 q^{49} - 8 q^{50} - 12 q^{55} + 6 q^{56} - 16 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} - 2 q^{70} - 20 q^{71} + 8 q^{74} + 12 q^{76} - 8 q^{79} + 2 q^{80} - 16 q^{85} - 8 q^{86} + 12 q^{89} + 4 q^{91} - 12 q^{95}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 + 12 * q^11 + 2 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 4 * q^29 - 20 * q^31 + 8 * q^34 - 4 * q^35 + 12 * q^40 - 4 * q^41 + 12 * q^44 - 2 * q^49 - 8 * q^50 - 12 * q^55 + 6 * q^56 - 16 * q^59 - 4 * q^61 - 14 * q^64 + 8 * q^65 - 2 * q^70 - 20 * q^71 + 8 * q^74 + 12 * q^76 - 8 * q^79 + 2 * q^80 - 16 * q^85 - 8 * q^86 + 12 * q^89 + 4 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			3.00000i			1.06066i
$9$		0			0
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	−1.00000			−0.250000
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−	2.00000i	−0.223607	−	0.447214i
$21$		0			0
$22$			6.00000i			1.27920i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$	−2.00000			−0.392232
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−10.0000			−1.79605			−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000			−1.79605			−0.898027	+	0.439941i	$0.854999\pi$
$32$			5.00000i			0.883883i
$33$		0			0
$34$	4.00000			0.685994
$35$	−2.00000	+	1.00000i	−0.338062	+	0.169031i
$36$		0			0
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$			6.00000i			0.973329i
$39$		0			0
$40$	6.00000	−	3.00000i	0.948683	−	0.474342i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	6.00000			0.904534
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$			2.00000i			0.277350i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	−6.00000	−	12.0000i	−0.809040	−	1.61808i
$56$	3.00000			0.400892
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		−	10.0000i		−	1.27000i
$63$		0			0
$64$	−7.00000			−0.875000
$65$	4.00000	−	2.00000i	0.496139	−	0.248069i
$66$		0			0
$67$		−	16.0000i		−	1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$		−	16.0000i		−	1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$		−	4.00000i		−	0.485071i
$69$		0			0
$70$	−1.00000	−	2.00000i	−0.119523	−	0.239046i
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	4.00000			0.464991
$75$		0			0
$76$	6.00000			0.688247
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	+	2.00000i	0.111803	+	0.223607i
$81$		0			0
$82$		−	2.00000i		−	0.220863i
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$	−8.00000	+	4.00000i	−0.867722	+	0.433861i
$86$	−4.00000			−0.431331
$87$		0			0
$88$			18.0000i			1.91881i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000	−	12.0000i	−0.615587	−	1.23117i
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	−6.00000			−0.582772
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$	12.0000	−	6.00000i	1.14416	−	0.572078i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		0			0
$116$	−2.00000			−0.185695
$117$		0			0
$118$		−	8.00000i		−	0.736460i
$119$	−4.00000			−0.366679
$120$		0			0
$121$	25.0000			2.27273
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$	−10.0000			−0.898027
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$			3.00000i			0.265165i
$129$		0			0
$130$	2.00000	+	4.00000i	0.175412	+	0.350823i
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		−	6.00000i		−	0.520266i
$134$	16.0000			1.38219
$135$		0			0
$136$	12.0000			1.02899
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$	−2.00000	+	1.00000i	−0.169031	+	0.0845154i
$141$		0			0
$142$		−	10.0000i		−	0.839181i
$143$			12.0000i			1.00349i
$144$		0			0
$145$	2.00000	+	4.00000i	0.166091	+	0.332182i
$146$	−6.00000			−0.496564
$147$		0			0
$148$		−	4.00000i		−	0.328798i
$149$	−14.0000			−1.14692			−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000			−1.14692			−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$			18.0000i			1.45999i
$153$		0			0
$154$	6.00000			0.483494
$155$	10.0000	+	20.0000i	0.803219	+	1.60644i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$		−	4.00000i		−	0.318223i
$159$		0			0
$160$	10.0000	−	5.00000i	0.790569	−	0.395285i
$161$		0			0
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−2.00000			−0.156174
$165$		0			0
$166$	−8.00000			−0.620920
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	−4.00000	−	8.00000i	−0.306786	−	0.613572i
$171$		0			0
$172$			4.00000i			0.304997i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	4.00000	+	3.00000i	0.302372	+	0.226779i
$176$	−6.00000			−0.452267
$177$		0			0
$178$			6.00000i			0.449719i
$179$	−14.0000			−1.04641			−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000			−1.04641			−0.523205	+	0.852207i	$0.675264\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$			2.00000i			0.148250i
$183$		0			0
$184$		0			0
$185$	−8.00000	+	4.00000i	−0.588172	+	0.294086i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		0			0
$189$		0			0
$190$	12.0000	−	6.00000i	0.870572	−	0.435286i
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$			8.00000i			0.575853i	0.957653	+	0.287926i	$0.0929658\pi$
$193$			8.00000i			0.575853i	−0.957653	+	0.287926i	$0.907034\pi$
$194$	2.00000			0.143592
$195$		0			0
$196$	−1.00000			−0.0714286
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	−14.0000			−0.992434			−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000			−0.992434			−0.496217	+	0.868199i	$0.665278\pi$
$200$	−12.0000	−	9.00000i	−0.848528	−	0.636396i
$201$		0			0
$202$			6.00000i			0.422159i
$203$			2.00000i			0.140372i
$204$		0			0
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$	8.00000			0.557386
$207$		0			0
$208$		−	2.00000i		−	0.138675i
$209$	36.0000			2.49017
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$			6.00000i			0.412082i
$213$		0			0
$214$	4.00000			0.273434
$215$	8.00000	−	4.00000i	0.545595	−	0.272798i
$216$		0			0
$217$			10.0000i			0.678844i
$218$		−	2.00000i		−	0.135457i
$219$		0			0
$220$	−6.00000	−	12.0000i	−0.404520	−	0.809040i
$221$	8.00000			0.538138
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	5.00000			0.334077
$225$		0			0
$226$	−6.00000			−0.399114
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		−	6.00000i		−	0.393919i
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		0			0
$235$		0			0
$236$	−8.00000			−0.520756
$237$		0			0
$238$		−	4.00000i		−	0.259281i
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$			25.0000i			1.60706i
$243$		0			0
$244$	−2.00000			−0.128037
$245$	1.00000	+	2.00000i	0.0638877	+	0.127775i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		−	30.0000i		−	1.90500i
$249$		0			0
$250$	−2.00000	+	11.0000i	−0.126491	+	0.695701i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−20.0000			−1.25491
$255$		0			0
$256$	−17.0000			−1.06250
$257$		−	16.0000i		−	0.998053i	−0.866587	−	0.499026i	$-0.833691\pi$
$257$		−	16.0000i		−	0.998053i	0.866587	−	0.499026i	$-0.166309\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$	4.00000	−	2.00000i	0.248069	−	0.124035i
$261$		0			0
$262$		−	4.00000i		−	0.247121i
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	12.0000	−	6.00000i	0.737154	−	0.368577i
$266$	6.00000			0.367884
$267$		0			0
$268$		−	16.0000i		−	0.977356i
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$			4.00000i			0.242536i
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−18.0000	+	24.0000i	−1.08544	+	1.44725i
$276$		0			0
$277$		−	28.0000i		−	1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$		−	28.0000i		−	1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$		−	2.00000i		−	0.119952i
$279$		0			0
$280$	−3.00000	−	6.00000i	−0.179284	−	0.358569i
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	−10.0000			−0.593391
$285$		0			0
$286$	−12.0000			−0.709575
$287$			2.00000i			0.118056i
$288$		0			0
$289$	1.00000			0.0588235
$290$	−4.00000	+	2.00000i	−0.234888	+	0.117444i
$291$		0			0
$292$			6.00000i			0.351123i
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		0			0
$295$	8.00000	+	16.0000i	0.465778	+	0.931556i
$296$	12.0000			0.697486
$297$		0			0
$298$		−	14.0000i		−	0.810998i
$299$		0			0
$300$		0			0
$301$	4.00000			0.230556
$302$			8.00000i			0.460348i
$303$		0			0
$304$	−6.00000			−0.344124
$305$	2.00000	+	4.00000i	0.114520	+	0.229039i
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		−	6.00000i		−	0.341882i
$309$		0			0
$310$	−20.0000	+	10.0000i	−1.13592	+	0.567962i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$		−	6.00000i		−	0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$		−	6.00000i		−	0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−18.0000			−1.01580
$315$		0			0
$316$	−4.00000			−0.225018
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$	7.00000	+	14.0000i	0.391312	+	0.782624i
$321$		0			0
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$	4.00000			0.221540
$327$		0			0
$328$		−	6.00000i		−	0.331295i
$329$		0			0
$330$		0			0
$331$	−24.0000			−1.31916			−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000			−1.31916			−0.659580	+	0.751635i	$0.729266\pi$
$332$			8.00000i			0.439057i
$333$		0			0
$334$	12.0000			0.656611
$335$	−32.0000	+	16.0000i	−1.74835	+	0.874173i
$336$		0			0
$337$		−	24.0000i		−	1.30736i	−0.756770	−	0.653682i	$-0.773224\pi$
$337$		−	24.0000i		−	1.30736i	0.756770	−	0.653682i	$-0.226776\pi$
$338$			9.00000i			0.489535i
$339$		0			0
$340$	−8.00000	+	4.00000i	−0.433861	+	0.216930i
$341$	−60.0000			−3.24918
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	−12.0000			−0.646997
$345$		0			0
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$	−3.00000	+	4.00000i	−0.160357	+	0.213809i
$351$		0			0
$352$			30.0000i			1.59901i
$353$			20.0000i			1.06449i	0.846590	+	0.532246i	$0.178652\pi$
$353$			20.0000i			1.06449i	−0.846590	+	0.532246i	$0.821348\pi$
$354$		0			0
$355$	10.0000	+	20.0000i	0.530745	+	1.06149i
$356$	6.00000			0.317999
$357$		0			0
$358$		−	14.0000i		−	0.739923i
$359$	22.0000			1.16112			0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000			1.16112			0.580558	+	0.814219i	$0.302835\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		−	6.00000i		−	0.315353i
$363$		0			0
$364$	2.00000			0.104828
$365$	12.0000	−	6.00000i	0.628109	−	0.314054i
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$	−4.00000	−	8.00000i	−0.207950	−	0.415900i
$371$	6.00000			0.311504
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	24.0000			1.24101
$375$		0			0
$376$		0			0
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	−6.00000	−	12.0000i	−0.307794	−	0.615587i
$381$		0			0
$382$			18.0000i			0.920960i
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$	−12.0000	+	6.00000i	−0.611577	+	0.305788i
$386$	−8.00000			−0.407189
$387$		0			0
$388$		−	2.00000i		−	0.101535i
$389$	26.0000			1.31825			0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000			1.31825			0.659126	+	0.752032i	$0.270926\pi$
$390$		0			0
$391$		0			0
$392$		−	3.00000i		−	0.151523i
$393$		0			0
$394$	2.00000			0.100759
$395$	4.00000	+	8.00000i	0.201262	+	0.402524i
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		−	14.0000i		−	0.701757i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		−	20.0000i		−	0.996271i
$404$	6.00000			0.298511
$405$		0			0
$406$	−2.00000			−0.0992583
$407$		−	24.0000i		−	1.18964i
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$	−4.00000	+	2.00000i	−0.197546	+	0.0987730i
$411$		0			0
$412$		−	8.00000i		−	0.394132i
$413$			8.00000i			0.393654i
$414$		0			0
$415$	16.0000	−	8.00000i	0.785409	−	0.392705i
$416$	−10.0000			−0.490290
$417$		0			0
$418$			36.0000i			1.76082i
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	18.0000			0.877266			0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000			0.877266			0.438633	+	0.898666i	$0.355463\pi$
$422$		−	16.0000i		−	0.778868i
$423$		0			0
$424$	−18.0000			−0.874157
$425$	16.0000	+	12.0000i	0.776114	+	0.582086i
$426$		0			0
$427$			2.00000i			0.0967868i
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$	4.00000	+	8.00000i	0.192897	+	0.385794i
$431$	14.0000			0.674356			0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000			0.674356			0.337178	+	0.941441i	$0.390528\pi$
$432$		0			0
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−10.0000			−0.480015
$435$		0			0
$436$	−2.00000			−0.0957826
$437$		0			0
$438$		0			0
$439$	−6.00000			−0.286364			−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000			−0.286364			−0.143182	+	0.989696i	$0.545733\pi$
$440$	36.0000	−	18.0000i	1.71623	−	0.858116i
$441$		0			0
$442$			8.00000i			0.380521i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$	−6.00000	−	12.0000i	−0.284427	−	0.568855i
$446$	−24.0000			−1.13643
$447$		0			0
$448$			7.00000i			0.330719i
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$			6.00000i			0.282216i
$453$		0			0
$454$	8.00000			0.375459
$455$	−2.00000	−	4.00000i	−0.0937614	−	0.187523i
$456$		0			0
$457$		−	20.0000i		−	0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$		−	20.0000i		−	0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$			10.0000i			0.467269i
$459$		0			0
$460$		0			0
$461$	−30.0000			−1.39724			−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000			−1.39724			−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	2.00000			0.0928477
$465$		0			0
$466$	26.0000			1.20443
$467$			24.0000i			1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$			24.0000i			1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$		0			0
$472$		−	24.0000i		−	1.10469i
$473$			24.0000i			1.10352i
$474$		0			0
$475$	−18.0000	+	24.0000i	−0.825897	+	1.10120i
$476$	−4.00000			−0.183340
$477$		0			0
$478$		−	6.00000i		−	0.274434i
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$			22.0000i			1.00207i
$483$		0			0
$484$	25.0000			1.13636
$485$	−4.00000	+	2.00000i	−0.181631	+	0.0908153i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		−	6.00000i		−	0.271607i
$489$		0			0
$490$	−2.00000	+	1.00000i	−0.0903508	+	0.0451754i
$491$	−10.0000			−0.451294			−0.225647	−	0.974209i	$-0.572450\pi$
$491$	−10.0000			−0.451294			−0.225647	+	0.974209i	$0.572450\pi$
$492$		0			0
$493$			8.00000i			0.360302i
$494$	−12.0000			−0.539906
$495$		0			0
$496$	10.0000			0.449013
$497$			10.0000i			0.448561i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$	11.0000	+	2.00000i	0.491935	+	0.0894427i
$501$		0			0
$502$		0			0
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$	−6.00000	−	12.0000i	−0.266996	−	0.533993i
$506$		0			0
$507$		0			0
$508$			20.0000i			0.887357i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		−	11.0000i		−	0.486136i
$513$		0			0
$514$	16.0000			0.705730
$515$	−16.0000	+	8.00000i	−0.705044	+	0.352522i
$516$		0			0
$517$		0			0
$518$		−	4.00000i		−	0.175750i
$519$		0			0
$520$	6.00000	+	12.0000i	0.263117	+	0.526235i
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−4.00000			−0.174741
$525$		0			0
$526$	24.0000			1.04645
$527$			40.0000i			1.74243i
$528$		0			0
$529$	23.0000			1.00000
$530$	6.00000	+	12.0000i	0.260623	+	0.521247i
$531$		0			0
$532$		−	6.00000i		−	0.260133i
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$	−8.00000	+	4.00000i	−0.345870	+	0.172935i
$536$	48.0000			2.07328
$537$		0			0
$538$		−	14.0000i		−	0.603583i
$539$	−6.00000			−0.258438
$540$		0			0
$541$	6.00000			0.257960			0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000			0.257960			0.128980	+	0.991647i	$0.458830\pi$
$542$		−	14.0000i		−	0.601351i
$543$		0			0
$544$	20.0000			0.857493
$545$	2.00000	+	4.00000i	0.0856706	+	0.171341i
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$			6.00000i			0.256307i
$549$		0			0
$550$	−24.0000	−	18.0000i	−1.02336	−	0.767523i
$551$	−12.0000			−0.511217
$552$		0			0
$553$			4.00000i			0.170097i
$554$	28.0000			1.18961
$555$		0			0
$556$	−2.00000			−0.0848189
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	2.00000	−	1.00000i	0.0845154	−	0.0422577i
$561$		0			0
$562$			2.00000i			0.0843649i
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$		0			0
$565$	12.0000	−	6.00000i	0.504844	−	0.252422i
$566$	20.0000			0.840663
$567$		0			0
$568$		−	30.0000i		−	1.25877i
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$	36.0000			1.50655			0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000			1.50655			0.753277	+	0.657704i	$0.228472\pi$
$572$			12.0000i			0.501745i
$573$		0			0
$574$	−2.00000			−0.0834784
$575$		0			0
$576$		0			0
$577$			14.0000i			0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$			14.0000i			0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$			1.00000i			0.0415945i
$579$		0			0
$580$	2.00000	+	4.00000i	0.0830455	+	0.166091i
$581$	8.00000			0.331896
$582$		0			0
$583$			36.0000i			1.49097i
$584$	−18.0000			−0.744845
$585$		0			0
$586$	−24.0000			−0.991431
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	−60.0000			−2.47226
$590$	−16.0000	+	8.00000i	−0.658710	+	0.329355i
$591$		0			0
$592$			4.00000i			0.164399i
$593$			44.0000i			1.80686i	0.428732	+	0.903432i	$0.358960\pi$
$593$			44.0000i			1.80686i	−0.428732	+	0.903432i	$0.641040\pi$
$594$		0			0
$595$	4.00000	+	8.00000i	0.163984	+	0.327968i
$596$	−14.0000			−0.573462
$597$		0			0
$598$		0			0
$599$	−2.00000			−0.0817178			−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000			−0.0817178			−0.0408589	+	0.999165i	$0.513009\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$			4.00000i			0.163028i
$603$		0			0
$604$	8.00000			0.325515
$605$	−25.0000	−	50.0000i	−1.01639	−	2.03279i
$606$		0			0
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$			30.0000i			1.21666i
$609$		0			0
$610$	−4.00000	+	2.00000i	−0.161955	+	0.0809776i
$611$		0			0
$612$		0			0
$613$		−	4.00000i		−	0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$		−	4.00000i		−	0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	−4.00000			−0.161427
$615$		0			0
$616$	18.0000			0.725241
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$	10.0000	+	20.0000i	0.401610	+	0.803219i
$621$		0			0
$622$			24.0000i			0.962312i
$623$		−	6.00000i		−	0.240385i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	6.00000			0.239808
$627$		0			0
$628$			18.0000i			0.718278i
$629$	−16.0000			−0.637962
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		−	12.0000i		−	0.477334i
$633$		0			0
$634$	−18.0000			−0.714871
$635$	40.0000	−	20.0000i	1.58735	−	0.793676i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		−	12.0000i		−	0.475085i
$639$		0			0
$640$	6.00000	−	3.00000i	0.237171	−	0.118585i
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$		0			0
$645$		0			0
$646$	24.0000			0.944267
$647$			20.0000i			0.786281i	0.919478	+	0.393141i	$0.128611\pi$
$647$			20.0000i			0.786281i	−0.919478	+	0.393141i	$0.871389\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$	6.00000	−	8.00000i	0.235339	−	0.313786i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$	4.00000	+	8.00000i	0.156293	+	0.312586i
$656$	2.00000			0.0780869
$657$		0			0
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		−	24.0000i		−	0.932786i
$663$		0			0
$664$	−24.0000			−0.931381
$665$	−12.0000	+	6.00000i	−0.465340	+	0.232670i
$666$		0			0
$667$		0			0
$668$		−	12.0000i		−	0.464294i
$669$		0			0
$670$	−16.0000	−	32.0000i	−0.618134	−	1.23627i
$671$	−12.0000			−0.463255
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$	24.0000			0.924445
$675$		0			0
$676$	9.00000			0.346154
$677$		−	32.0000i		−	1.22986i	−0.788582	−	0.614930i	$-0.789184\pi$
$677$		−	32.0000i		−	1.22986i	0.788582	−	0.614930i	$-0.210816\pi$
$678$		0			0
$679$	−2.00000			−0.0767530
$680$	−12.0000	−	24.0000i	−0.460179	−	0.920358i
$681$		0			0
$682$		−	60.0000i		−	2.29752i
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$		0			0
$685$	12.0000	−	6.00000i	0.458496	−	0.229248i
$686$	−1.00000			−0.0381802
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	−12.0000			−0.457164
$690$		0			0
$691$	−50.0000			−1.90209			−0.951045	−	0.309053i	$-0.899988\pi$
$691$	−50.0000			−1.90209			−0.951045	+	0.309053i	$0.899988\pi$
$692$		0			0
$693$		0			0
$694$	12.0000			0.455514
$695$	2.00000	+	4.00000i	0.0758643	+	0.151729i
$696$		0			0
$697$			8.00000i			0.303022i
$698$			2.00000i			0.0757011i
$699$		0			0
$700$	4.00000	+	3.00000i	0.151186	+	0.113389i
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$	−42.0000			−1.58293
$705$		0			0
$706$	−20.0000			−0.752710
$707$		−	6.00000i		−	0.225653i
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$	−20.0000	+	10.0000i	−0.750587	+	0.375293i
$711$		0			0
$712$			18.0000i			0.674579i
$713$		0			0
$714$		0			0
$715$	24.0000	−	12.0000i	0.897549	−	0.448775i
$716$	−14.0000			−0.523205
$717$		0			0
$718$			22.0000i			0.821033i
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$			17.0000i			0.632674i
$723$		0			0
$724$	−6.00000			−0.222988
$725$	6.00000	−	8.00000i	0.222834	−	0.297113i
$726$		0			0
$727$		−	40.0000i		−	1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$		−	40.0000i		−	1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$			6.00000i			0.222375i
$729$		0			0
$730$	6.00000	+	12.0000i	0.222070	+	0.444140i
$731$	16.0000			0.591781
$732$		0			0
$733$		−	22.0000i		−	0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$		−	22.0000i		−	0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	96.0000i		−	3.53621i
$738$		0			0
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$	−8.00000	+	4.00000i	−0.294086	+	0.147043i
$741$		0			0
$742$			6.00000i			0.220267i
$743$			40.0000i			1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$			40.0000i			1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$		0			0
$745$	14.0000	+	28.0000i	0.512920	+	1.02584i
$746$		0			0
$747$		0			0
$748$		−	24.0000i		−	0.877527i
$749$	−4.00000			−0.146157
$750$		0			0
$751$	−12.0000			−0.437886			−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000			−0.437886			−0.218943	+	0.975738i	$0.570261\pi$
$752$		0			0
$753$		0			0
$754$	4.00000			0.145671
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$		−	40.0000i		−	1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$		−	40.0000i		−	1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$			28.0000i			1.01701i
$759$		0			0
$760$	36.0000	−	18.0000i	1.30586	−	0.652929i
$761$	22.0000			0.797499			0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000			0.797499			0.398750	+	0.917060i	$0.369444\pi$
$762$		0			0
$763$			2.00000i			0.0724049i
$764$	18.0000			0.651217
$765$		0			0
$766$	20.0000			0.722629
$767$		−	16.0000i		−	0.577727i
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$	−6.00000	−	12.0000i	−0.216225	−	0.432450i
$771$		0			0
$772$			8.00000i			0.287926i
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$		0			0
$775$	30.0000	−	40.0000i	1.07763	−	1.43684i
$776$	6.00000			0.215387
$777$		0			0
$778$			26.0000i			0.932145i
$779$	−12.0000			−0.429945
$780$		0			0
$781$	−60.0000			−2.14697
$782$		0			0
$783$		0			0
$784$	1.00000			0.0357143
$785$	36.0000	−	18.0000i	1.28490	−	0.642448i
$786$		0			0
$787$		−	4.00000i		−	0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$		−	4.00000i		−	0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$		−	2.00000i		−	0.0712470i
$789$		0			0
$790$	−8.00000	+	4.00000i	−0.284627	+	0.142314i
$791$	6.00000			0.213335
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$	−22.0000			−0.780751
$795$		0			0
$796$	−14.0000			−0.496217

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000i q^{8} +(2.00000 - 1.00000i) q^{10} +6.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +6.00000 q^{19} +(-1.00000 - 2.00000i) q^{20} +6.00000i q^{22} +(-3.00000 + 4.00000i) q^{25} -2.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} +4.00000 q^{34} +(-2.00000 + 1.00000i) q^{35} -4.00000i q^{37} +6.00000i q^{38} +(6.00000 - 3.00000i) q^{40} -2.00000 q^{41} +4.00000i q^{43} +6.00000 q^{44} -1.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +2.00000i q^{52} +6.00000i q^{53} +(-6.00000 - 12.0000i) q^{55} +3.00000 q^{56} -2.00000i q^{58} -8.00000 q^{59} -2.00000 q^{61} -10.0000i q^{62} -7.00000 q^{64} +(4.00000 - 2.00000i) q^{65} -16.0000i q^{67} -4.00000i q^{68} +(-1.00000 - 2.00000i) q^{70} -10.0000 q^{71} +6.00000i q^{73} +4.00000 q^{74} +6.00000 q^{76} -6.00000i q^{77} -4.00000 q^{79} +(1.00000 + 2.00000i) q^{80} -2.00000i q^{82} +8.00000i q^{83} +(-8.00000 + 4.00000i) q^{85} -4.00000 q^{86} +18.0000i q^{88} +6.00000 q^{89} +2.00000 q^{91} +(-6.00000 - 12.0000i) q^{95} -2.00000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5	\( 2 q + 2 q^{4} - 2 q^{5} + 4 q^{10} + 12 q^{11} + 2 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 4 q^{29} - 20 q^{31} + 8 q^{34} - 4 q^{35} + 12 q^{40} - 4 q^{41} + 12 q^{44} - 2 q^{49} - 8 q^{50} - 12 q^{55} + 6 q^{56} - 16 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} - 2 q^{70} - 20 q^{71} + 8 q^{74} + 12 q^{76} - 8 q^{79} + 2 q^{80} - 16 q^{85} - 8 q^{86} + 12 q^{89} + 4 q^{91} - 12 q^{95}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 + 4 * q^10 + 12 * q^11 + 2 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 4 * q^29 - 20 * q^31 + 8 * q^34 - 4 * q^35 + 12 * q^40 - 4 * q^41 + 12 * q^44 - 2 * q^49 - 8 * q^50 - 12 * q^55 + 6 * q^56 - 16 * q^59 - 4 * q^61 - 14 * q^64 + 8 * q^65 - 2 * q^70 - 20 * q^71 + 8 * q^74 + 12 * q^76 - 8 * q^79 + 2 * q^80 - 16 * q^85 - 8 * q^86 + 12 * q^89 + 4 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more