LMFDB - Embedded newform 78.2.b.a.25.2

Newspace parameters

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	78.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.622833135766$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	78.25
Dual form			78.2.b.a.25.1

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} -2.00000i q^{20} -2.00000i q^{21} +4.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} +1.00000 q^{27} +2.00000i q^{28} -10.0000 q^{29} -2.00000 q^{30} +10.0000i q^{31} +1.00000i q^{32} -2.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} +8.00000i q^{37} +6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} +2.00000 q^{40} +10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000i q^{45} +4.00000i q^{46} -12.0000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -2.00000 q^{51} +(3.00000 + 2.00000i) q^{52} -6.00000 q^{53} +1.00000i q^{54} -2.00000 q^{56} -6.00000i q^{57} -10.0000i q^{58} +4.00000i q^{59} -2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} +2.00000 q^{68} +4.00000 q^{69} +4.00000i q^{70} -1.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} +2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -4.00000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -10.0000 q^{87} -6.00000i q^{89} -2.00000 q^{90} +(-4.00000 + 6.00000i) q^{91} -4.00000 q^{92} +10.0000i q^{93} +12.0000 q^{94} +12.0000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 20 q^{29} - 4 q^{30} + 8 q^{35} - 2 q^{36} + 12 q^{38} - 6 q^{39} + 4 q^{40} + 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 4 q^{51} + 6 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} - 20 q^{62} - 2 q^{64} + 8 q^{65} + 4 q^{68} + 8 q^{69} - 16 q^{74} + 2 q^{75} + 4 q^{78} + 2 q^{81} - 20 q^{82} - 20 q^{87} - 4 q^{90} - 8 q^{91} - 8 q^{92} + 24 q^{94} + 24 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 4 * q^10 - 2 * q^12 - 6 * q^13 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 20 * q^29 - 4 * q^30 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 6 * q^39 + 4 * q^40 + 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 4 * q^51 + 6 * q^52 - 12 * q^53 - 4 * q^56 + 4 * q^61 - 20 * q^62 - 2 * q^64 + 8 * q^65 + 4 * q^68 + 8 * q^69 - 16 * q^74 + 2 * q^75 + 4 * q^78 + 2 * q^81 - 20 * q^82 - 20 * q^87 - 4 * q^90 - 8 * q^91 - 8 * q^92 + 24 * q^94 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$1$	$-1$

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
$2$			1.00000i			0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$			1.00000i			0.408248i
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	−2.00000			−0.632456
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−1.00000			−0.288675
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$	2.00000			0.534522
$15$			2.00000i			0.516398i
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$			1.00000i			0.235702i
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		−	2.00000i		−	0.447214i
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$		−	1.00000i		−	0.204124i
$25$	1.00000			0.200000
$26$	2.00000	−	3.00000i	0.392232	−	0.588348i
$27$	1.00000			0.192450
$28$			2.00000i			0.377964i
$29$	−10.0000			−1.85695			−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000			−1.85695			−0.928477	+	0.371391i	$0.878881\pi$
$30$	−2.00000			−0.365148
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	4.00000			0.676123
$36$	−1.00000			−0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	6.00000			0.973329
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$	2.00000			0.316228
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	2.00000			0.308607
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$			2.00000i			0.298142i
$46$			4.00000i			0.589768i
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	1.00000			0.144338
$49$	3.00000			0.428571
$50$			1.00000i			0.141421i
$51$	−2.00000			−0.280056
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$			1.00000i			0.136083i
$55$		0			0
$56$	−2.00000			−0.267261
$57$		−	6.00000i		−	0.794719i
$58$		−	10.0000i		−	1.31306i
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		−	2.00000i		−	0.258199i
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	−10.0000			−1.27000
$63$		−	2.00000i		−	0.251976i
$64$	−1.00000			−0.125000
$65$	4.00000	−	6.00000i	0.496139	−	0.744208i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	2.00000			0.242536
$69$	4.00000			0.481543
$70$			4.00000i			0.478091i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	1.00000i		−	0.117851i
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−8.00000			−0.929981
$75$	1.00000			0.115470
$76$			6.00000i			0.688247i
$77$		0			0
$78$	2.00000	−	3.00000i	0.226455	−	0.339683i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			2.00000i			0.223607i
$81$	1.00000			0.111111
$82$	−10.0000			−1.10432
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$			2.00000i			0.218218i
$85$		−	4.00000i		−	0.433861i
$86$			4.00000i			0.431331i
$87$	−10.0000			−1.07211
$88$		0			0
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	−2.00000			−0.210819
$91$	−4.00000	+	6.00000i	−0.419314	+	0.628971i
$92$	−4.00000			−0.417029
$93$			10.0000i			1.03695i
$94$	12.0000			1.23771
$95$	12.0000			1.23117
$96$			1.00000i			0.102062i
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$			3.00000i			0.303046i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	2.00000i		−	0.198030i
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	+	3.00000i	−0.196116	+	0.294174i
$105$	4.00000			0.390360
$106$		−	6.00000i		−	0.582772i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$	−1.00000			−0.0962250
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		−	2.00000i		−	0.188982i
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$	6.00000			0.561951
$115$			8.00000i			0.746004i
$116$	10.0000			0.928477
$117$	−3.00000	−	2.00000i	−0.277350	−	0.184900i
$118$	−4.00000			−0.368230
$119$			4.00000i			0.366679i
$120$	2.00000			0.182574
$121$	11.0000			1.00000
$122$			2.00000i			0.181071i
$123$			10.0000i			0.901670i
$124$		−	10.0000i		−	0.898027i
$125$			12.0000i			1.07331i
$126$	2.00000			0.178174
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	4.00000			0.352180
$130$	6.00000	+	4.00000i	0.526235	+	0.350823i
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$	−12.0000			−1.04053
$134$	2.00000			0.172774
$135$			2.00000i			0.172133i
$136$			2.00000i			0.171499i
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$			4.00000i			0.340503i
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.00000			−0.338062
$141$		−	12.0000i		−	1.01058i
$142$		0			0
$143$		0			0
$144$	1.00000			0.0833333
$145$		−	20.0000i		−	1.66091i
$146$	4.00000			0.331042
$147$	3.00000			0.247436
$148$		−	8.00000i		−	0.657596i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$			1.00000i			0.0816497i
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	−6.00000			−0.486664
$153$	−2.00000			−0.161690
$154$		0			0
$155$	−20.0000			−1.60644
$156$	3.00000	+	2.00000i	0.240192	+	0.160128i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$	−2.00000			−0.158114
$161$		−	8.00000i		−	0.630488i
$162$			1.00000i			0.0785674i
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$		−	10.0000i		−	0.780869i
$165$		0			0
$166$	4.00000			0.310460
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	−2.00000			−0.154303
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	4.00000			0.306786
$171$		−	6.00000i		−	0.458831i
$172$	−4.00000			−0.304997
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		−	10.0000i		−	0.758098i
$175$		−	2.00000i		−	0.151186i
$176$		0			0
$177$			4.00000i			0.300658i
$178$	6.00000			0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		−	2.00000i		−	0.149071i
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$	−6.00000	−	4.00000i	−0.444750	−	0.296500i
$183$	2.00000			0.147844
$184$		−	4.00000i		−	0.294884i
$185$	−16.0000			−1.17634
$186$	−10.0000			−0.733236
$187$		0			0
$188$			12.0000i			0.875190i
$189$		−	2.00000i		−	0.145479i
$190$			12.0000i			0.870572i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000			−0.0721688
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	12.0000			0.861550
$195$	4.00000	−	6.00000i	0.286446	−	0.429669i
$196$	−3.00000			−0.214286
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		−	2.00000i		−	0.141069i
$202$			2.00000i			0.140720i
$203$			20.0000i			1.40372i
$204$	2.00000			0.140028
$205$	−20.0000			−1.39686
$206$		−	16.0000i		−	1.11477i
$207$	4.00000			0.278019
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$		0			0
$210$			4.00000i			0.276026i
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$			8.00000i			0.546869i
$215$			8.00000i			0.545595i
$216$		−	1.00000i		−	0.0680414i
$217$	20.0000			1.35769
$218$	−4.00000			−0.270914
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	6.00000	+	4.00000i	0.403604	+	0.269069i
$222$	−8.00000			−0.536925
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	2.00000			0.133631
$225$	1.00000			0.0666667
$226$			14.0000i			0.931266i
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$			6.00000i			0.397360i
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	−8.00000			−0.527504
$231$		0			0
$232$			10.0000i			0.656532i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	−	3.00000i	0.130744	−	0.196116i
$235$	24.0000			1.56559
$236$		−	4.00000i		−	0.260378i
$237$		0			0
$238$	−4.00000			−0.259281
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$			2.00000i			0.129099i
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$			11.0000i			0.707107i
$243$	1.00000			0.0641500
$244$	−2.00000			−0.128037
$245$			6.00000i			0.383326i
$246$	−10.0000			−0.637577
$247$	−12.0000	+	18.0000i	−0.763542	+	1.14531i
$248$	10.0000			0.635001
$249$		−	4.00000i		−	0.253490i
$250$	−12.0000			−0.758947
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$			2.00000i			0.125988i
$253$		0			0
$254$			8.00000i			0.501965i
$255$		−	4.00000i		−	0.250490i
$256$	1.00000			0.0625000
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$			4.00000i			0.249029i
$259$	16.0000			0.994192
$260$	−4.00000	+	6.00000i	−0.248069	+	0.372104i
$261$	−10.0000			−0.618984
$262$		−	8.00000i		−	0.494242i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		−	12.0000i		−	0.735767i
$267$		−	6.00000i		−	0.367194i
$268$			2.00000i			0.122169i
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$	−2.00000			−0.121716
$271$		−	10.0000i		−	0.607457i	−0.952759	−	0.303728i	$-0.901768\pi$
$271$		−	10.0000i		−	0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$	−2.00000			−0.121268
$273$	−4.00000	+	6.00000i	−0.242091	+	0.363137i
$274$	2.00000			0.120824
$275$		0			0
$276$	−4.00000			−0.240772
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		−	20.0000i		−	1.19952i
$279$			10.0000i			0.598684i
$280$		−	4.00000i		−	0.239046i
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	12.0000			0.714590
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	12.0000			0.710819
$286$		0			0
$287$	20.0000			1.18056
$288$			1.00000i			0.0589256i
$289$	−13.0000			−0.764706
$290$	20.0000			1.17444
$291$		−	12.0000i		−	0.703452i
$292$			4.00000i			0.234082i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$			3.00000i			0.174964i
$295$	−8.00000			−0.465778
$296$	8.00000			0.464991
$297$		0			0
$298$	−14.0000			−0.810998
$299$	−12.0000	−	8.00000i	−0.693978	−	0.462652i
$300$	−1.00000			−0.0577350
$301$		−	8.00000i		−	0.461112i
$302$	−10.0000			−0.575435
$303$	2.00000			0.114897
$304$		−	6.00000i		−	0.344124i
$305$			4.00000i			0.229039i
$306$		−	2.00000i		−	0.114332i
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		−	20.0000i		−	1.13592i
$311$	−28.0000			−1.58773			−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000			−1.58773			−0.793867	+	0.608091i	$0.791935\pi$
$312$	−2.00000	+	3.00000i	−0.113228	+	0.169842i
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		−	2.00000i		−	0.112867i
$315$	4.00000			0.225374
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		−	6.00000i		−	0.336463i
$319$		0			0
$320$		−	2.00000i		−	0.111803i
$321$	8.00000			0.446516
$322$	8.00000			0.445823
$323$			12.0000i			0.667698i
$324$	−1.00000			−0.0555556
$325$	−3.00000	−	2.00000i	−0.166410	−	0.110940i
$326$	14.0000			0.775388
$327$			4.00000i			0.221201i
$328$	10.0000			0.552158
$329$	−24.0000			−1.32316
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$			4.00000i			0.219529i
$333$			8.00000i			0.438397i
$334$	12.0000			0.656611
$335$	4.00000			0.218543
$336$		−	2.00000i		−	0.109109i
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−12.0000	+	5.00000i	−0.652714	+	0.271964i
$339$	14.0000			0.760376
$340$			4.00000i			0.216930i
$341$		0			0
$342$	6.00000			0.324443
$343$		−	20.0000i		−	1.07990i
$344$		−	4.00000i		−	0.215666i
$345$			8.00000i			0.430706i
$346$		−	6.00000i		−	0.322562i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$	10.0000			0.536056
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	2.00000			0.106904
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$		0			0
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	−4.00000			−0.212598
$355$		0			0
$356$			6.00000i			0.317999i
$357$			4.00000i			0.211702i
$358$		0			0
$359$			4.00000i			0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$	2.00000			0.105409
$361$	−17.0000			−0.894737
$362$			22.0000i			1.15629i
$363$	11.0000			0.577350
$364$	4.00000	−	6.00000i	0.209657	−	0.314485i
$365$	8.00000			0.418739
$366$			2.00000i			0.104542i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000			0.208514
$369$			10.0000i			0.520579i
$370$		−	16.0000i		−	0.831800i
$371$			12.0000i			0.623009i
$372$		−	10.0000i		−	0.518476i
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$	−12.0000			−0.618853
$377$	30.0000	+	20.0000i	1.54508	+	1.03005i
$378$	2.00000			0.102869
$379$			34.0000i			1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	−12.0000			−0.615587
$381$	8.00000			0.409852
$382$			12.0000i			0.613973i
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	−16.0000			−0.814379
$387$	4.00000			0.203331
$388$			12.0000i			0.609208i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$	6.00000	+	4.00000i	0.303822	+	0.202548i
$391$	−8.00000			−0.404577
$392$		−	3.00000i		−	0.151523i
$393$	−8.00000			−0.403547
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$	−12.0000			−0.600751
$400$	1.00000			0.0500000
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	2.00000			0.0997509
$403$	20.0000	−	30.0000i	0.996271	−	1.49441i
$404$	−2.00000			−0.0995037
$405$			2.00000i			0.0993808i
$406$	−20.0000			−0.992583
$407$		0			0
$408$			2.00000i			0.0990148i
$409$			4.00000i			0.197787i	0.995098	+	0.0988936i	$0.0315304\pi$
$409$			4.00000i			0.197787i	−0.995098	+	0.0988936i	$0.968470\pi$
$410$		−	20.0000i		−	0.987730i
$411$		−	2.00000i		−	0.0986527i
$412$	16.0000			0.788263
$413$	8.00000			0.393654
$414$			4.00000i			0.196589i
$415$	8.00000			0.392705
$416$	2.00000	−	3.00000i	0.0980581	−	0.147087i
$417$	−20.0000			−0.979404
$418$		0			0
$419$	40.0000			1.95413			0.977064	−	0.212946i	$-0.0683059\pi$
$419$	40.0000			1.95413			0.977064	+	0.212946i	$0.0683059\pi$
$420$	−4.00000			−0.195180
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$			12.0000i			0.584151i
$423$		−	12.0000i		−	0.583460i
$424$			6.00000i			0.291386i
$425$	−2.00000			−0.0970143
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$	−8.00000			−0.386695
$429$		0			0
$430$	−8.00000			−0.385794
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$	1.00000			0.0481125
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$			20.0000i			0.960031i
$435$		−	20.0000i		−	0.958927i
$436$		−	4.00000i		−	0.191565i
$437$		−	24.0000i		−	1.14808i
$438$	4.00000			0.191127
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$	−4.00000	+	6.00000i	−0.190261	+	0.285391i
$443$	−16.0000			−0.760183			−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000			−0.760183			−0.380091	+	0.924949i	$0.624107\pi$
$444$		−	8.00000i		−	0.379663i
$445$	12.0000			0.568855
$446$	14.0000			0.662919
$447$			14.0000i			0.662177i
$448$			2.00000i			0.0944911i
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$			1.00000i			0.0471405i
$451$		0			0
$452$	−14.0000			−0.658505
$453$			10.0000i			0.469841i
$454$	−8.00000			−0.375459
$455$	−12.0000	−	8.00000i	−0.562569	−	0.375046i
$456$	−6.00000			−0.280976
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	−4.00000			−0.186908
$459$	−2.00000			−0.0933520
$460$		−	8.00000i		−	0.373002i
$461$		−	30.0000i		−	1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$		−	30.0000i		−	1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	−10.0000			−0.464238
$465$	−20.0000			−0.927478
$466$		−	6.00000i		−	0.277945i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$	3.00000	+	2.00000i	0.138675	+	0.0924500i
$469$	−4.00000			−0.184703
$470$			24.0000i			1.10704i
$471$	−2.00000			−0.0921551
$472$	4.00000			0.184115
$473$		0			0
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$		−	4.00000i		−	0.183340i
$477$	−6.00000			−0.274721
$478$	16.0000			0.731823
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	−2.00000			−0.0912871
$481$	16.0000	−	24.0000i	0.729537	−	1.09431i
$482$	−20.0000			−0.910975
$483$		−	8.00000i		−	0.364013i
$484$	−11.0000			−0.500000
$485$	24.0000			1.08978
$486$			1.00000i			0.0453609i
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		−	2.00000i		−	0.0905357i
$489$		−	14.0000i		−	0.633102i
$490$	−6.00000			−0.271052
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		−	10.0000i		−	0.450835i
$493$	20.0000			0.900755
$494$	−18.0000	−	12.0000i	−0.809858	−	0.539906i
$495$		0			0
$496$			10.0000i			0.449013i
$497$		0			0
$498$	4.00000			0.179244
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		−	12.0000i		−	0.536656i
$501$		−	12.0000i		−	0.536120i
$502$		−	28.0000i		−	1.24970i
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000			−0.0890871
$505$			4.00000i			0.177998i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$	−8.00000			−0.354943
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	4.00000			0.177123
$511$	−8.00000			−0.353899
$512$			1.00000i			0.0441942i
$513$		−	6.00000i		−	0.264906i
$514$			18.0000i			0.793946i
$515$		−	32.0000i		−	1.41009i
$516$	−4.00000			−0.176090
$517$		0			0
$518$			16.0000i			0.703000i
$519$	−6.00000			−0.263371
$520$	−6.00000	−	4.00000i	−0.263117	−	0.175412i
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		−	10.0000i		−	0.437688i
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	8.00000			0.349482
$525$		−	2.00000i		−	0.0872872i
$526$			24.0000i			1.04645i
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	12.0000			0.521247
$531$			4.00000i			0.173585i
$532$	12.0000			0.520266
$533$	20.0000	−	30.0000i	0.866296	−	1.29944i
$534$	6.00000			0.259645
$535$			16.0000i			0.691740i
$536$	−2.00000			−0.0863868
$537$		0			0
$538$		−	10.0000i		−	0.431131i
$539$		0			0
$540$		−	2.00000i		−	0.0860663i
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	10.0000			0.429537
$543$	22.0000			0.944110
$544$		−	2.00000i		−	0.0857493i
$545$	−8.00000			−0.342682
$546$	−6.00000	−	4.00000i	−0.256776	−	0.171184i
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$			2.00000i			0.0854358i
$549$	2.00000			0.0853579
$550$		0			0
$551$			60.0000i			2.55609i
$552$		−	4.00000i		−	0.170251i
$553$		0			0
$554$		−	2.00000i		−	0.0849719i
$555$	−16.0000			−0.679162
$556$	20.0000			0.848189
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	−10.0000			−0.423334
$559$	−12.0000	−	8.00000i	−0.507546	−	0.338364i
$560$	4.00000			0.169031
$561$		0			0
$562$	10.0000			0.421825
$563$	−16.0000			−0.674320			−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000			−0.674320			−0.337160	+	0.941447i	$0.609466\pi$
$564$			12.0000i			0.505291i
$565$			28.0000i			1.17797i
$566$			4.00000i			0.168133i
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$			12.0000i			0.502625i
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$			20.0000i			0.834784i
$575$	4.00000			0.166812
$576$	−1.00000			−0.0416667
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		−	13.0000i		−	0.540729i
$579$			16.0000i			0.664937i
$580$			20.0000i			0.830455i
$581$	−8.00000			−0.331896
$582$	12.0000			0.497416
$583$		0			0
$584$	−4.00000			−0.165521
$585$	4.00000	−	6.00000i	0.165380	−	0.248069i
$586$	14.0000			0.578335
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	−3.00000			−0.123718
$589$	60.0000			2.47226
$590$		−	8.00000i		−	0.329355i
$591$		−	22.0000i		−	0.904959i
$592$			8.00000i			0.328798i
$593$			26.0000i			1.06769i	0.845582	+	0.533846i	$0.179254\pi$
$593$			26.0000i			1.06769i	−0.845582	+	0.533846i	$0.820746\pi$
$594$		0			0
$595$	−8.00000			−0.327968
$596$		−	14.0000i		−	0.573462i
$597$		0			0
$598$	8.00000	−	12.0000i	0.327144	−	0.490716i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		−	1.00000i		−	0.0408248i
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$	8.00000			0.326056
$603$		−	2.00000i		−	0.0814463i
$604$		−	10.0000i		−	0.406894i
$605$			22.0000i			0.894427i
$606$			2.00000i			0.0812444i
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$	6.00000			0.243332
$609$			20.0000i			0.810441i
$610$	−4.00000			−0.161955
$611$	−24.0000	+	36.0000i	−0.970936	+	1.45640i
$612$	2.00000			0.0808452
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	2.00000			0.0807134
$615$	−20.0000			−0.806478
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		−	16.0000i		−	0.643614i
$619$		−	26.0000i		−	1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$		−	26.0000i		−	1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	20.0000			0.803219
$621$	4.00000			0.160514
$622$		−	28.0000i		−	1.12270i
$623$	−12.0000			−0.480770
$624$	−3.00000	−	2.00000i	−0.120096	−	0.0800641i
$625$	−19.0000			−0.760000
$626$		−	26.0000i		−	1.03917i
$627$		0			0
$628$	2.00000			0.0798087
$629$		−	16.0000i		−	0.637962i
$630$			4.00000i			0.159364i
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$	12.0000			0.476957
$634$	−18.0000			−0.714871
$635$			16.0000i			0.634941i
$636$	6.00000			0.237915
$637$	−9.00000	−	6.00000i	−0.356593	−	0.237729i
$638$		0			0
$639$		0			0
$640$	2.00000			0.0790569
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$			8.00000i			0.315735i
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$			8.00000i			0.315244i
$645$			8.00000i			0.315000i
$646$	−12.0000			−0.472134
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$	2.00000	−	3.00000i	0.0784465	−	0.117670i
$651$	20.0000			0.783862
$652$			14.0000i			0.548282i
$653$	−26.0000			−1.01746			−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000			−1.01746			−0.508729	+	0.860927i	$0.669885\pi$
$654$	−4.00000			−0.156412
$655$		−	16.0000i		−	0.625172i
$656$			10.0000i			0.390434i
$657$		−	4.00000i		−	0.156055i
$658$		−	24.0000i		−	0.935617i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$		−	40.0000i		−	1.55582i	−0.628376	−	0.777910i	$-0.716280\pi$
$661$		−	40.0000i		−	1.55582i	0.628376	−	0.777910i	$-0.283720\pi$
$662$	10.0000			0.388661
$663$	6.00000	+	4.00000i	0.233021	+	0.155347i
$664$	−4.00000			−0.155230
$665$		−	24.0000i		−	0.930680i
$666$	−8.00000			−0.309994
$667$	−40.0000			−1.54881
$668$			12.0000i			0.464294i
$669$		−	14.0000i		−	0.541271i
$670$			4.00000i			0.154533i
$671$		0			0
$672$	2.00000			0.0771517
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		−	2.00000i		−	0.0770371i
$675$	1.00000			0.0384900
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$			14.0000i			0.537667i
$679$	−24.0000			−0.921035
$680$	−4.00000			−0.153393
$681$			8.00000i			0.306561i
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$			6.00000i			0.229416i
$685$	4.00000			0.152832
$686$	20.0000			0.763604
$687$			4.00000i			0.152610i
$688$	4.00000			0.152499
$689$	18.0000	+	12.0000i	0.685745	+	0.457164i
$690$	−8.00000			−0.304555
$691$		−	10.0000i		−	0.380418i	−0.981744	−	0.190209i	$-0.939083\pi$
$691$		−	10.0000i		−	0.380418i	0.981744	−	0.190209i	$-0.0609166\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$		−	12.0000i		−	0.455514i
$695$		−	40.0000i		−	1.51729i
$696$			10.0000i			0.379049i
$697$		−	20.0000i		−	0.757554i
$698$	16.0000			0.605609
$699$	−6.00000			−0.226941
$700$			2.00000i			0.0755929i
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$	2.00000	−	3.00000i	0.0754851	−	0.113228i
$703$	48.0000			1.81035
$704$		0			0
$705$	24.0000			0.903892
$706$	−26.0000			−0.978523
$707$		−	4.00000i		−	0.150435i
$708$		−	4.00000i		−	0.150329i
$709$		−	36.0000i		−	1.35201i	−0.736898	−	0.676004i	$-0.763710\pi$
$709$		−	36.0000i		−	1.35201i	0.736898	−	0.676004i	$-0.236290\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000			−0.224860
$713$			40.0000i			1.49801i
$714$	−4.00000			−0.149696
$715$		0			0
$716$		0			0
$717$		−	16.0000i		−	0.597531i
$718$	−4.00000			−0.149279
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$			2.00000i			0.0745356i
$721$			32.0000i			1.19174i
$722$		−	17.0000i		−	0.632674i
$723$			20.0000i			0.743808i
$724$	−22.0000			−0.817624
$725$	−10.0000			−0.371391
$726$			11.0000i			0.408248i
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$	6.00000	+	4.00000i	0.222375	+	0.148250i
$729$	1.00000			0.0370370
$730$			8.00000i			0.296093i
$731$	−8.00000			−0.295891
$732$	−2.00000			−0.0739221
$733$		−	44.0000i		−	1.62518i	−0.582838	−	0.812589i	$-0.698058\pi$
$733$		−	44.0000i		−	1.62518i	0.582838	−	0.812589i	$-0.301942\pi$
$734$			8.00000i			0.295285i
$735$			6.00000i			0.221313i
$736$			4.00000i			0.147442i
$737$		0			0
$738$	−10.0000			−0.368105
$739$		−	26.0000i		−	0.956425i	−0.878244	−	0.478213i	$-0.841285\pi$
$739$		−	26.0000i		−	0.956425i	0.878244	−	0.478213i	$-0.158715\pi$
$740$	16.0000			0.588172
$741$	−12.0000	+	18.0000i	−0.440831	+	0.661247i
$742$	−12.0000			−0.440534
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	10.0000			0.366618
$745$	−28.0000			−1.02584
$746$		−	6.00000i		−	0.219676i
$747$		−	4.00000i		−	0.146352i
$748$		0			0
$749$		−	16.0000i		−	0.584627i
$750$	−12.0000			−0.438178
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$		−	12.0000i		−	0.437595i
$753$	−28.0000			−1.02038
$754$	−20.0000	+	30.0000i	−0.728357	+	1.09254i
$755$	−20.0000			−0.727875
$756$			2.00000i			0.0727393i
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	−34.0000			−1.23494
$759$		0			0
$760$		−	12.0000i		−	0.435286i
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$			8.00000i			0.289809i
$763$	8.00000			0.289619
$764$	−12.0000			−0.434145
$765$		−	4.00000i		−	0.144620i
$766$	4.00000			0.144526
$767$	8.00000	−	12.0000i	0.288863	−	0.433295i
$768$	1.00000			0.0360844
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		−	16.0000i		−	0.575853i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$			4.00000i			0.143777i
$775$			10.0000i			0.359211i
$776$	−12.0000			−0.430775
$777$	16.0000			0.573997
$778$			30.0000i			1.07555i
$779$	60.0000			2.14972
$780$	−4.00000	+	6.00000i	−0.143223	+	0.214834i
$781$		0			0
$782$		−	8.00000i		−	0.286079i
$783$	−10.0000			−0.357371
$784$	3.00000			0.107143
$785$		−	4.00000i		−	0.142766i
$786$		−	8.00000i		−	0.285351i
$787$			38.0000i			1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$			38.0000i			1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$			22.0000i			0.783718i
$789$	24.0000			0.854423
$790$		0			0
$791$		−	28.0000i		−	0.995565i
$792$		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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} -6.00000i q^{19} -2.00000i q^{20} -2.00000i q^{21} +4.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} +1.00000 q^{27} +2.00000i q^{28} -10.0000 q^{29} -2.00000 q^{30} +10.0000i q^{31} +1.00000i q^{32} -2.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} +8.00000i q^{37} +6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} +2.00000 q^{40} +10.0000i q^{41} +2.00000 q^{42} +4.00000 q^{43} +2.00000i q^{45} +4.00000i q^{46} -12.0000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} -2.00000 q^{51} +(3.00000 + 2.00000i) q^{52} -6.00000 q^{53} +1.00000i q^{54} -2.00000 q^{56} -6.00000i q^{57} -10.0000i q^{58} +4.00000i q^{59} -2.00000i q^{60} +2.00000 q^{61} -10.0000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} +2.00000 q^{68} +4.00000 q^{69} +4.00000i q^{70} -1.00000i q^{72} -4.00000i q^{73} -8.00000 q^{74} +1.00000 q^{75} +6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} +2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} -4.00000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -10.0000 q^{87} -6.00000i q^{89} -2.00000 q^{90} +(-4.00000 + 6.00000i) q^{91} -4.00000 q^{92} +10.0000i q^{93} +12.0000 q^{94} +12.0000 q^{95} +1.00000i q^{96} -12.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 20 q^{29} - 4 q^{30} + 8 q^{35} - 2 q^{36} + 12 q^{38} - 6 q^{39} + 4 q^{40} + 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 4 q^{51} + 6 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} - 20 q^{62} - 2 q^{64} + 8 q^{65} + 4 q^{68} + 8 q^{69} - 16 q^{74} + 2 q^{75} + 4 q^{78} + 2 q^{81} - 20 q^{82} - 20 q^{87} - 4 q^{90} - 8 q^{91} - 8 q^{92} + 24 q^{94} + 24 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 4 * q^10 - 2 * q^12 - 6 * q^13 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 20 * q^29 - 4 * q^30 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 6 * q^39 + 4 * q^40 + 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 4 * q^51 + 6 * q^52 - 12 * q^53 - 4 * q^56 + 4 * q^61 - 20 * q^62 - 2 * q^64 + 8 * q^65 + 4 * q^68 + 8 * q^69 - 16 * q^74 + 2 * q^75 + 4 * q^78 + 2 * q^81 - 20 * q^82 - 20 * q^87 - 4 * q^90 - 8 * q^91 - 8 * q^92 + 24 * q^94 + 24 * q^95

Introduction

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