LMFDB - Embedded newform 1216.2.h.d.1215.3

Newspace parameters

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14453810176.1
Defining polynomial:	$x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1215.3
Root			$1.06789 + 0.927153i$ of defining polynomial
Character	$\chi$	$=$	1216.1215
Dual form			1216.2.h.d.1215.4

$q$-expansion

$f(q)$	$=$	$q-1.19935 q^{3} -1.56155 q^{5} -0.868210i q^{7} -1.56155 q^{9} +O(q^{10})$	$q-1.19935 q^{3} -1.56155 q^{5} -0.868210i q^{7} -1.56155 q^{9} +3.09218i q^{11} -4.74990i q^{13} +1.87285 q^{15} -1.00000 q^{17} +(3.07221 - 3.09218i) q^{19} +1.04129i q^{21} +3.96039i q^{23} -2.56155 q^{25} +5.47091 q^{27} +8.45851i q^{29} +4.27156 q^{31} -3.70861i q^{33} +1.35576i q^{35} -3.70861i q^{37} +5.69681i q^{39} -3.70861i q^{41} +11.0129i q^{43} +2.43845 q^{45} +9.27653i q^{47} +6.24621 q^{49} +1.19935 q^{51} -1.04129i q^{53} -4.82860i q^{55} +(-3.68466 + 3.70861i) q^{57} +11.6153 q^{59} +0.684658 q^{61} +1.35576i q^{63} +7.41722i q^{65} -9.74247 q^{67} -4.74990i q^{69} +10.9418 q^{71} +8.12311 q^{73} +3.07221 q^{75} +2.68466 q^{77} +8.01726 q^{79} -1.87689 q^{81} +9.65719i q^{83} +1.56155 q^{85} -10.1447i q^{87} -5.79119i q^{89} -4.12391 q^{91} -5.12311 q^{93} +(-4.79741 + 4.82860i) q^{95} +16.9170i q^{97} -4.82860i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8q + 4q^{5} + 4q^{9} + O(q^{10}) $	$ 8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.19935			−0.692447			−0.346223	−	0.938152i	$-0.612536\pi$
$3$	−1.19935			−0.692447			−0.346223	+	0.938152i	$0.612536\pi$
$4$		0			0
$5$	−1.56155			−0.698348			−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155			−0.698348			−0.349174	+	0.937058i	$0.613538\pi$
$6$		0			0
$7$		−	0.868210i		−	0.328153i	−0.986448	−	0.164076i	$-0.947536\pi$
$7$		−	0.868210i		−	0.328153i	0.986448	−	0.164076i	$-0.0524643\pi$
$8$		0			0
$9$	−1.56155			−0.520518
$10$		0			0
$11$			3.09218i			0.932326i	0.884699	+	0.466163i	$0.154364\pi$
$11$			3.09218i			0.932326i	−0.884699	+	0.466163i	$0.845636\pi$
$12$		0			0
$13$		−	4.74990i		−	1.31739i	−0.752412	−	0.658693i	$-0.771110\pi$
$13$		−	4.74990i		−	1.31739i	0.752412	−	0.658693i	$-0.228890\pi$
$14$		0			0
$15$	1.87285			0.483569
$16$		0			0
$17$	−1.00000			−0.242536			−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000			−0.242536			−0.121268	+	0.992620i	$0.538696\pi$
$18$		0			0
$19$	3.07221	−	3.09218i	0.704812	−	0.709394i
$19$	3.07221	−	3.09218i	0.704812	−	0.709394i
$20$		0			0
$21$			1.04129i			0.227228i
$22$		0			0
$23$			3.96039i			0.825798i	0.910777	+	0.412899i	$0.135484\pi$
$23$			3.96039i			0.825798i	−0.910777	+	0.412899i	$0.864516\pi$
$24$		0			0
$25$	−2.56155			−0.512311
$26$		0			0
$27$	5.47091			1.05288
$28$		0			0
$29$			8.45851i			1.57071i	0.619048	+	0.785353i	$0.287519\pi$
$29$			8.45851i			1.57071i	−0.619048	+	0.785353i	$0.712481\pi$
$30$		0			0
$31$	4.27156			0.767195			0.383597	−	0.923500i	$-0.374685\pi$
$31$	4.27156			0.767195			0.383597	+	0.923500i	$0.374685\pi$
$32$		0			0
$33$		−	3.70861i		−	0.645586i
$34$		0			0
$35$			1.35576i			0.229165i
$36$		0			0
$37$		−	3.70861i		−	0.609692i	−0.952402	−	0.304846i	$-0.901395\pi$
$37$		−	3.70861i		−	0.609692i	0.952402	−	0.304846i	$-0.0986050\pi$
$38$		0			0
$39$			5.69681i			0.912219i
$40$		0			0
$41$		−	3.70861i		−	0.579188i	−0.957150	−	0.289594i	$-0.906480\pi$
$41$		−	3.70861i		−	0.579188i	0.957150	−	0.289594i	$-0.0935202\pi$
$42$		0			0
$43$			11.0129i			1.67946i	0.543005	+	0.839729i	$0.317286\pi$
$43$			11.0129i			1.67946i	−0.543005	+	0.839729i	$0.682714\pi$
$44$		0			0
$45$	2.43845			0.363502
$46$		0			0
$47$			9.27653i			1.35312i	0.736387	+	0.676560i	$0.236530\pi$
$47$			9.27653i			1.35312i	−0.736387	+	0.676560i	$0.763470\pi$
$48$		0			0
$49$	6.24621			0.892316
$50$		0			0
$51$	1.19935			0.167943
$52$		0			0
$53$		−	1.04129i		−	0.143032i	−0.997439	−	0.0715161i	$-0.977216\pi$
$53$		−	1.04129i		−	0.143032i	0.997439	−	0.0715161i	$-0.0227837\pi$
$54$		0			0
$55$		−	4.82860i		−	0.651088i
$56$		0			0
$57$	−3.68466	+	3.70861i	−0.488045	+	0.491217i
$58$		0			0
$59$	11.6153			1.51219			0.756093	−	0.654464i	$-0.227106\pi$
$59$	11.6153			1.51219			0.756093	+	0.654464i	$0.227106\pi$
$60$		0			0
$61$	0.684658			0.0876615			0.0438308	−	0.999039i	$-0.486044\pi$
$61$	0.684658			0.0876615			0.0438308	+	0.999039i	$0.486044\pi$
$62$		0			0
$63$			1.35576i			0.170809i
$64$		0			0
$65$			7.41722i			0.919993i
$66$		0			0
$67$	−9.74247			−1.19023			−0.595116	−	0.803640i	$-0.702894\pi$
$67$	−9.74247			−1.19023			−0.595116	+	0.803640i	$0.702894\pi$
$68$		0			0
$69$		−	4.74990i		−	0.571821i
$70$		0			0
$71$	10.9418			1.29856			0.649278	−	0.760551i	$-0.275071\pi$
$71$	10.9418			1.29856			0.649278	+	0.760551i	$0.275071\pi$
$72$		0			0
$73$	8.12311			0.950738			0.475369	−	0.879787i	$-0.342315\pi$
$73$	8.12311			0.950738			0.475369	+	0.879787i	$0.342315\pi$
$74$		0			0
$75$	3.07221			0.354748
$76$		0			0
$77$	2.68466			0.305945
$78$		0			0
$79$	8.01726			0.902013			0.451006	−	0.892521i	$-0.351065\pi$
$79$	8.01726			0.902013			0.451006	+	0.892521i	$0.351065\pi$
$80$		0			0
$81$	−1.87689			−0.208544
$82$		0			0
$83$			9.65719i			1.06001i	0.847993	+	0.530007i	$0.177811\pi$
$83$			9.65719i			1.06001i	−0.847993	+	0.530007i	$0.822189\pi$
$84$		0			0
$85$	1.56155			0.169374
$86$		0			0
$87$		−	10.1447i		−	1.08763i
$88$		0			0
$89$		−	5.79119i		−	0.613865i	−0.951731	−	0.306932i	$-0.900697\pi$
$89$		−	5.79119i		−	0.613865i	0.951731	−	0.306932i	$-0.0993026\pi$
$90$		0			0
$91$	−4.12391			−0.432303
$92$		0			0
$93$	−5.12311			−0.531241
$94$		0			0
$95$	−4.79741	+	4.82860i	−0.492204	+	0.495404i
$96$		0			0
$97$			16.9170i			1.71766i	0.512258	+	0.858832i	$0.328809\pi$
$97$			16.9170i			1.71766i	−0.512258	+	0.858832i	$0.671191\pi$
$98$		0			0
$99$		−	4.82860i		−	0.485292i
$100$		0			0
$101$	−8.24621			−0.820529			−0.410264	−	0.911967i	$-0.634564\pi$
$101$	−8.24621			−0.820529			−0.410264	+	0.911967i	$0.634564\pi$
$102$		0			0
$103$	−3.74571			−0.369075			−0.184538	−	0.982825i	$-0.559079\pi$
$103$	−3.74571			−0.369075			−0.184538	+	0.982825i	$0.559079\pi$
$104$		0			0
$105$		−	1.62603i		−	0.158684i
$106$		0			0
$107$	7.86962			0.760785			0.380392	−	0.924825i	$-0.375789\pi$
$107$	7.86962			0.760785			0.380392	+	0.924825i	$0.375789\pi$
$108$		0			0
$109$			15.8757i			1.52062i	0.649561	+	0.760310i	$0.274953\pi$
$109$			15.8757i			1.52062i	−0.649561	+	0.760310i	$0.725047\pi$
$110$		0			0
$111$			4.44793i			0.422179i
$112$		0			0
$113$			3.70861i			0.348877i	0.984668	+	0.174438i	$0.0558110\pi$
$113$			3.70861i			0.348877i	−0.984668	+	0.174438i	$0.944189\pi$
$114$		0			0
$115$		−	6.18435i		−	0.576694i
$116$		0			0
$117$			7.41722i			0.685722i
$118$		0			0
$119$			0.868210i			0.0795887i
$120$		0			0
$121$	1.43845			0.130768
$122$		0			0
$123$			4.44793i			0.401057i
$124$		0			0
$125$	11.8078			1.05612
$126$		0			0
$127$	−1.87285			−0.166189			−0.0830944	−	0.996542i	$-0.526480\pi$
$127$	−1.87285			−0.166189			−0.0830944	+	0.996542i	$0.526480\pi$
$128$		0			0
$129$		−	13.2084i		−	1.16294i
$130$		0			0
$131$		−	4.82860i		−	0.421876i	−0.977499	−	0.210938i	$-0.932348\pi$
$131$		−	4.82860i		−	0.421876i	0.977499	−	0.210938i	$-0.0676519\pi$
$132$		0			0
$133$	−2.68466	−	2.66732i	−0.232789	−	0.231286i
$134$		0			0
$135$	−8.54312			−0.735274
$136$		0			0
$137$	−3.87689			−0.331225			−0.165613	−	0.986191i	$-0.552960\pi$
$137$	−3.87689			−0.331225			−0.165613	+	0.986191i	$0.552960\pi$
$138$		0			0
$139$		−	0.380664i		−	0.0322875i	−0.999870	−	0.0161438i	$-0.994861\pi$
$139$		−	0.380664i		−	0.0322875i	0.999870	−	0.0161438i	$-0.00513894\pi$
$140$		0			0
$141$		−	11.1258i		−	0.936964i
$142$		0			0
$143$	14.6875			1.22823
$144$		0			0
$145$		−	13.2084i		−	1.09690i
$146$		0			0
$147$	−7.49141			−0.617881
$148$		0			0
$149$	17.8078			1.45887			0.729434	−	0.684051i	$-0.239783\pi$
$149$	17.8078			1.45887			0.729434	+	0.684051i	$0.239783\pi$
$150$		0			0
$151$	−24.2824			−1.97607			−0.988035	−	0.154231i	$-0.950710\pi$
$151$	−24.2824			−1.97607			−0.988035	+	0.154231i	$0.950710\pi$
$152$		0			0
$153$	1.56155			0.126244
$154$		0			0
$155$	−6.67026			−0.535769
$156$		0			0
$157$	−6.00000			−0.478852			−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000			−0.478852			−0.239426	+	0.970915i	$0.576959\pi$
$158$		0			0
$159$			1.24887i			0.0990422i
$160$		0			0
$161$	3.43845			0.270988
$162$		0			0
$163$		−	10.6323i		−	0.832785i	−0.909185	−	0.416392i	$-0.863294\pi$
$163$		−	10.6323i		−	0.832785i	0.909185	−	0.416392i	$-0.136706\pi$
$164$		0			0
$165$			5.79119i			0.450844i
$166$		0			0
$167$	−0.525853			−0.0406917			−0.0203459	−	0.999793i	$-0.506477\pi$
$167$	−0.525853			−0.0406917			−0.0203459	+	0.999793i	$0.506477\pi$
$168$		0			0
$169$	−9.56155			−0.735504
$170$		0			0
$171$	−4.79741	+	4.82860i	−0.366867	+	0.369252i
$172$		0			0
$173$		−	16.9170i		−	1.28618i	−0.765792	−	0.643089i	$-0.777653\pi$
$173$		−	16.9170i		−	1.28618i	0.765792	−	0.643089i	$-0.222347\pi$
$174$		0			0
$175$			2.22397i			0.168116i
$176$		0			0
$177$	−13.9309			−1.04711
$178$		0			0
$179$	9.06897			0.677847			0.338923	−	0.940814i	$-0.389937\pi$
$179$	9.06897			0.677847			0.338923	+	0.940814i	$0.389937\pi$
$180$		0			0
$181$		−	2.08258i		−	0.154797i	−0.997000	−	0.0773985i	$-0.975339\pi$
$181$		−	2.08258i		−	0.154797i	0.997000	−	0.0773985i	$-0.0246614\pi$
$182$		0			0
$183$	−0.821147			−0.0607009
$184$		0			0
$185$			5.79119i			0.425777i
$186$		0			0
$187$		−	3.09218i		−	0.226122i
$188$		0			0
$189$		−	4.74990i		−	0.345504i
$190$		0			0
$191$		−	8.78898i		−	0.635948i	−0.948099	−	0.317974i	$-0.896997\pi$
$191$		−	8.78898i		−	0.635948i	0.948099	−	0.317974i	$-0.103003\pi$
$192$		0			0
$193$		−	3.70861i		−	0.266952i	−0.991052	−	0.133476i	$-0.957386\pi$
$193$		−	3.70861i		−	0.266952i	0.991052	−	0.133476i	$-0.0426138\pi$
$194$		0			0
$195$		−	8.89586i		−	0.637046i
$196$		0			0
$197$	−10.4924			−0.747554			−0.373777	−	0.927519i	$-0.621937\pi$
$197$	−10.4924			−0.747554			−0.373777	+	0.927519i	$0.621937\pi$
$198$		0			0
$199$		−	13.2369i		−	0.938340i	−0.883108	−	0.469170i	$-0.844553\pi$
$199$		−	13.2369i		−	0.938340i	0.883108	−	0.469170i	$-0.155447\pi$
$200$		0			0
$201$	11.6847			0.824172
$202$		0			0
$203$	7.34376			0.515431
$204$		0			0
$205$			5.79119i			0.404474i
$206$		0			0
$207$		−	6.18435i		−	0.429842i
$208$		0			0
$209$	9.56155	+	9.49980i	0.661386	+	0.657115i
$210$		0			0
$211$	−2.02050			−0.139097			−0.0695485	−	0.997579i	$-0.522156\pi$
$211$	−2.02050			−0.139097			−0.0695485	+	0.997579i	$0.522156\pi$
$212$		0			0
$213$	−13.1231			−0.899180
$214$		0			0
$215$		−	17.1973i		−	1.17285i
$216$		0			0
$217$		−	3.70861i		−	0.251757i
$218$		0			0
$219$	−9.74247			−0.658335
$220$		0			0
$221$			4.74990i			0.319513i
$222$		0			0
$223$	16.2651			1.08919			0.544595	−	0.838699i	$-0.316683\pi$
$223$	16.2651			1.08919			0.544595	+	0.838699i	$0.316683\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$	−17.4644			−1.15916			−0.579578	−	0.814917i	$-0.696783\pi$
$227$	−17.4644			−1.15916			−0.579578	+	0.814917i	$0.696783\pi$
$228$		0			0
$229$	2.43845			0.161137			0.0805686	−	0.996749i	$-0.474326\pi$
$229$	2.43845			0.161137			0.0805686	+	0.996749i	$0.474326\pi$
$230$		0			0
$231$	−3.21985			−0.211851
$232$		0			0
$233$	−9.80776			−0.642528			−0.321264	−	0.946990i	$-0.604108\pi$
$233$	−9.80776			−0.642528			−0.321264	+	0.946990i	$0.604108\pi$
$234$		0			0
$235$		−	14.4858i		−	0.944949i
$236$		0			0
$237$	−9.61553			−0.624596
$238$		0			0
$239$			19.4213i			1.25626i	0.778110	+	0.628129i	$0.216179\pi$
$239$			19.4213i			1.25626i	−0.778110	+	0.628129i	$0.783821\pi$
$240$		0			0
$241$			15.2910i			0.984979i	0.870318	+	0.492490i	$0.163913\pi$
$241$			15.2910i			0.984979i	−0.870318	+	0.492490i	$0.836087\pi$
$242$		0			0
$243$	−14.1617			−0.908472
$244$		0			0
$245$	−9.75379			−0.623147
$246$		0			0
$247$	−14.6875	−	14.5927i	−0.934545	−	0.928509i
$248$		0			0
$249$		−	11.5824i		−	0.734004i
$250$		0			0
$251$			22.4066i			1.41429i	0.707069	+	0.707145i	$0.250017\pi$
$251$			22.4066i			1.41429i	−0.707069	+	0.707145i	$0.749983\pi$
$252$		0			0
$253$	−12.2462			−0.769913
$254$		0			0
$255$	−1.87285			−0.117283
$256$		0			0
$257$		−	18.9996i		−	1.18516i	−0.805511	−	0.592581i	$-0.798109\pi$
$257$		−	18.9996i		−	1.18516i	0.805511	−	0.592581i	$-0.201891\pi$
$258$		0			0
$259$	−3.21985			−0.200072
$260$		0			0
$261$		−	13.2084i		−	0.817580i
$262$		0			0
$263$		−	6.56502i		−	0.404816i	−0.979301	−	0.202408i	$-0.935123\pi$
$263$		−	6.56502i		−	0.404816i	0.979301	−	0.202408i	$-0.0648768\pi$
$264$		0			0
$265$			1.62603i			0.0998862i
$266$		0			0
$267$			6.94568i			0.425069i
$268$		0			0
$269$			3.70861i			0.226118i	0.993588	+	0.113059i	$0.0360649\pi$
$269$			3.70861i			0.226118i	−0.993588	+	0.113059i	$0.963935\pi$
$270$		0			0
$271$			15.3540i			0.932689i	0.884603	+	0.466345i	$0.154429\pi$
$271$			15.3540i			0.932689i	−0.884603	+	0.466345i	$0.845571\pi$
$272$		0			0
$273$	4.94602			0.299347
$274$		0			0
$275$		−	7.92077i		−	0.477641i
$276$		0			0
$277$	24.0540			1.44526			0.722632	−	0.691233i	$-0.242932\pi$
$277$	24.0540			1.44526			0.722632	+	0.691233i	$0.242932\pi$
$278$		0			0
$279$	−6.67026			−0.399338
$280$		0			0
$281$			11.5824i			0.690947i	0.938429	+	0.345473i	$0.112282\pi$
$281$			11.5824i			0.690947i	−0.938429	+	0.345473i	$0.887718\pi$
$282$		0			0
$283$		−	4.82860i		−	0.287030i	−0.989648	−	0.143515i	$-0.954159\pi$
$283$		−	4.82860i		−	0.287030i	0.989648	−	0.143515i	$-0.0458406\pi$
$284$		0			0
$285$	5.75379	−	5.79119i	0.340825	−	0.343041i
$286$		0			0
$287$	−3.21985			−0.190062
$288$		0			0
$289$	−16.0000			−0.941176
$290$		0			0
$291$		−	20.2895i		−	1.18939i
$292$		0			0
$293$			4.74990i			0.277492i	0.990328	+	0.138746i	$0.0443072\pi$
$293$			4.74990i			0.277492i	−0.990328	+	0.138746i	$0.955693\pi$
$294$		0			0
$295$	−18.1379			−1.05603
$296$		0			0
$297$			16.9170i			0.981625i
$298$		0			0
$299$	18.8114			1.08789
$300$		0			0
$301$	9.56155			0.551119
$302$		0			0
$303$	9.89012			0.568172
$304$		0			0
$305$	−1.06913			−0.0612182
$306$		0			0
$307$	−8.01726			−0.457569			−0.228785	−	0.973477i	$-0.573475\pi$
$307$	−8.01726			−0.457569			−0.228785	+	0.973477i	$0.573475\pi$
$308$		0			0
$309$	4.49242			0.255565
$310$		0			0
$311$		−	0.868210i		−	0.0492317i	−0.999697	−	0.0246158i	$-0.992164\pi$
$311$		−	0.868210i		−	0.0492317i	0.999697	−	0.0246158i	$-0.00783626\pi$
$312$		0			0
$313$	−4.56155			−0.257834			−0.128917	−	0.991655i	$-0.541150\pi$
$313$	−4.56155			−0.257834			−0.128917	+	0.991655i	$0.541150\pi$
$314$		0			0
$315$		−	2.11708i		−	0.119284i
$316$		0			0
$317$		−	4.74990i		−	0.266781i	−0.991064	−	0.133390i	$-0.957414\pi$
$317$		−	4.74990i		−	0.266781i	0.991064	−	0.133390i	$-0.0425864\pi$
$318$		0			0
$319$	−26.1552			−1.46441
$320$		0			0
$321$	−9.43845			−0.526803
$322$		0			0
$323$	−3.07221	+	3.09218i	−0.170942	+	0.172053i
$324$		0			0
$325$			12.1671i			0.674910i
$326$		0			0
$327$		−	19.0406i		−	1.05295i
$328$		0			0
$329$	8.05398			0.444030
$330$		0			0
$331$	2.25106			0.123729			0.0618647	−	0.998085i	$-0.480295\pi$
$331$	2.25106			0.123729			0.0618647	+	0.998085i	$0.480295\pi$
$332$		0			0
$333$			5.79119i			0.317355i
$334$		0			0
$335$	15.2134			0.831196
$336$		0			0
$337$		−	26.4168i		−	1.43902i	−0.694484	−	0.719508i	$-0.744367\pi$
$337$		−	26.4168i		−	1.43902i	0.694484	−	0.719508i	$-0.255633\pi$
$338$		0			0
$339$		−	4.44793i		−	0.241579i
$340$		0			0
$341$			13.2084i			0.715276i
$342$		0			0
$343$		−	11.5005i		−	0.620968i
$344$		0			0
$345$			7.41722i			0.399330i
$346$		0			0
$347$		−	24.1430i		−	1.29606i	−0.761613	−	0.648032i	$-0.775592\pi$
$347$		−	24.1430i		−	1.29606i	0.761613	−	0.648032i	$-0.224408\pi$
$348$		0			0
$349$	28.0540			1.50169			0.750847	−	0.660476i	$-0.229645\pi$
$349$	28.0540			1.50169			0.750847	+	0.660476i	$0.229645\pi$
$350$		0			0
$351$		−	25.9863i		−	1.38705i
$352$		0			0
$353$	6.31534			0.336132			0.168066	−	0.985776i	$-0.446248\pi$
$353$	6.31534			0.336132			0.168066	+	0.985776i	$0.446248\pi$
$354$		0			0
$355$	−17.0862			−0.906843
$356$		0			0
$357$		−	1.04129i		−	0.0551109i
$358$		0			0
$359$			27.3420i			1.44306i	0.692384	+	0.721529i	$0.256560\pi$
$359$			27.3420i			1.44306i	−0.692384	+	0.721529i	$0.743440\pi$
$360$		0			0
$361$	−0.123106	−	18.9996i	−0.00647924	−	0.999979i
$362$		0			0
$363$	−1.72521			−0.0905498
$364$		0			0
$365$	−12.6847			−0.663945
$366$		0			0
$367$			14.1051i			0.736282i	0.929770	+	0.368141i	$0.120006\pi$
$367$			14.1051i			0.736282i	−0.929770	+	0.368141i	$0.879994\pi$
$368$		0			0
$369$			5.79119i			0.301477i
$370$		0			0
$371$	−0.904059			−0.0469364
$372$		0			0
$373$			12.1671i			0.629990i	0.949093	+	0.314995i	$0.102003\pi$
$373$			12.1671i			0.629990i	−0.949093	+	0.314995i	$0.897997\pi$
$374$		0			0
$375$	−14.1617			−0.731306
$376$		0			0
$377$	40.1771			2.06922
$378$		0			0
$379$	6.81791			0.350213			0.175106	−	0.984550i	$-0.443973\pi$
$379$	6.81791			0.350213			0.175106	+	0.984550i	$0.443973\pi$
$380$		0			0
$381$	2.24621			0.115077
$382$		0			0
$383$	−16.2651			−0.831107			−0.415554	−	0.909569i	$-0.636412\pi$
$383$	−16.2651			−0.831107			−0.415554	+	0.909569i	$0.636412\pi$
$384$		0			0
$385$	−4.19224			−0.213656
$386$		0			0
$387$		−	17.1973i		−	0.874188i
$388$		0			0
$389$	5.80776			0.294465			0.147233	−	0.989102i	$-0.452963\pi$
$389$	5.80776			0.294465			0.147233	+	0.989102i	$0.452963\pi$
$390$		0			0
$391$		−	3.96039i		−	0.200285i
$392$		0			0
$393$			5.79119i			0.292127i
$394$		0			0
$395$	−12.5194			−0.629918
$396$		0			0
$397$	−24.9309			−1.25124			−0.625622	−	0.780126i	$-0.715155\pi$
$397$	−24.9309			−1.25124			−0.625622	+	0.780126i	$0.715155\pi$
$398$		0			0
$399$	3.21985	+	3.19906i	0.161194	+	0.160153i
$400$		0			0
$401$		−	20.6256i		−	1.02999i	−0.857192	−	0.514997i	$-0.827793\pi$
$401$		−	20.6256i		−	1.02999i	0.857192	−	0.514997i	$-0.172207\pi$
$402$		0			0
$403$		−	20.2895i		−	1.01069i
$404$		0			0
$405$	2.93087			0.145636
$406$		0			0
$407$	11.4677			0.568432
$408$		0			0
$409$			18.5431i			0.916895i	0.888722	+	0.458447i	$0.151594\pi$
$409$			18.5431i			0.916895i	−0.888722	+	0.458447i	$0.848406\pi$
$410$		0			0
$411$	4.64976			0.229356
$412$		0			0
$413$		−	10.0845i		−	0.496228i
$414$		0			0
$415$		−	15.0802i		−	0.740259i
$416$		0			0
$417$			0.456551i			0.0223574i
$418$		0			0
$419$			27.4489i			1.34097i	0.741924	+	0.670484i	$0.233913\pi$
$419$			27.4489i			1.34097i	−0.741924	+	0.670484i	$0.766087\pi$
$420$		0			0
$421$		−	23.2930i		−	1.13523i	−0.823294	−	0.567614i	$-0.807866\pi$
$421$		−	23.2930i		−	1.13523i	0.823294	−	0.567614i	$-0.192134\pi$
$422$		0			0
$423$		−	14.4858i		−	0.704323i
$424$		0			0
$425$	2.56155			0.124254
$426$		0			0
$427$		−	0.594427i		−	0.0287664i
$428$		0			0
$429$	−17.6155			−0.850486
$430$		0			0
$431$	17.0862			0.823015			0.411507	−	0.911406i	$-0.365002\pi$
$431$	17.0862			0.823015			0.411507	+	0.911406i	$0.365002\pi$
$432$		0			0
$433$			28.0429i			1.34765i	0.738889	+	0.673827i	$0.235351\pi$
$433$			28.0429i			1.34765i	−0.738889	+	0.673827i	$0.764649\pi$
$434$		0			0
$435$			15.8415i			0.759544i
$436$		0			0
$437$	12.2462	+	12.1671i	0.585816	+	0.582032i
$438$		0			0
$439$	24.8082			1.18403			0.592015	−	0.805927i	$-0.298332\pi$
$439$	24.8082			1.18403			0.592015	+	0.805927i	$0.298332\pi$
$440$		0			0
$441$	−9.75379			−0.464466
$442$		0			0
$443$			34.7753i			1.65222i	0.563507	+	0.826111i	$0.309452\pi$
$443$			34.7753i			1.65222i	−0.563507	+	0.826111i	$0.690548\pi$
$444$		0			0
$445$			9.04325i			0.428691i
$446$		0			0
$447$	−21.3578			−1.01019
$448$		0			0
$449$			35.9166i			1.69501i	0.530787	+	0.847505i	$0.321896\pi$
$449$			35.9166i			1.69501i	−0.530787	+	0.847505i	$0.678104\pi$
$450$		0			0
$451$	11.4677			0.539992
$452$		0			0
$453$	29.1231			1.36832
$454$		0			0
$455$	6.43971			0.301898
$456$		0			0
$457$	12.1231			0.567095			0.283547	−	0.958958i	$-0.408489\pi$
$457$	12.1231			0.567095			0.283547	+	0.958958i	$0.408489\pi$
$458$		0			0
$459$	−5.47091			−0.255360
$460$		0			0
$461$	−19.3153			−0.899605			−0.449803	−	0.893128i	$-0.648506\pi$
$461$	−19.3153			−0.899605			−0.449803	+	0.893128i	$0.648506\pi$
$462$		0			0
$463$			21.6452i			1.00594i	0.864304	+	0.502970i	$0.167759\pi$
$463$			21.6452i			1.00594i	−0.864304	+	0.502970i	$0.832241\pi$
$464$		0			0
$465$	8.00000			0.370991
$466$		0			0
$467$			3.09218i			0.143089i	0.997437	+	0.0715444i	$0.0227928\pi$
$467$			3.09218i			0.143089i	−0.997437	+	0.0715444i	$0.977207\pi$
$468$		0			0
$469$			8.45851i			0.390578i
$470$		0			0
$471$	7.19612			0.331580
$472$		0			0
$473$	−34.0540			−1.56580
$474$		0			0
$475$	−7.86962	+	7.92077i	−0.361083	+	0.363430i
$476$		0			0
$477$			1.62603i			0.0744508i
$478$		0			0
$479$		−	8.68210i		−	0.396695i	−0.980132	−	0.198348i	$-0.936442\pi$
$479$		−	8.68210i		−	0.396695i	0.980132	−	0.198348i	$-0.0635575\pi$
$480$		0			0
$481$	−17.6155			−0.803199
$482$		0			0
$483$	−4.12391			−0.187644
$484$		0			0
$485$		−	26.4168i		−	1.19953i
$486$		0			0
$487$	9.59482			0.434783			0.217391	−	0.976085i	$-0.430245\pi$
$487$	9.59482			0.434783			0.217391	+	0.976085i	$0.430245\pi$
$488$		0			0
$489$			12.7519i			0.576659i
$490$		0			0
$491$			2.71151i			0.122369i	0.998126	+	0.0611844i	$0.0194878\pi$
$491$			2.71151i			0.122369i	−0.998126	+	0.0611844i	$0.980512\pi$
$492$		0			0
$493$		−	8.45851i		−	0.380952i
$494$		0			0
$495$			7.54011i			0.338903i
$496$		0			0
$497$		−	9.49980i		−	0.426124i
$498$		0			0
$499$			11.0129i			0.493007i	0.969142	+	0.246504i	$0.0792817\pi$
$499$			11.0129i			0.493007i	−0.969142	+	0.246504i	$0.920718\pi$
$500$		0			0
$501$	0.630683			0.0281768
$502$		0			0
$503$			35.6435i			1.58926i	0.607091	+	0.794632i	$0.292336\pi$
$503$			35.6435i			1.58926i	−0.607091	+	0.794632i	$0.707664\pi$
$504$		0			0
$505$	12.8769			0.573014
$506$		0			0
$507$	11.4677			0.509297
$508$		0			0
$509$			9.49980i			0.421071i	0.977586	+	0.210536i	$0.0675208\pi$
$509$			9.49980i			0.421071i	−0.977586	+	0.210536i	$0.932479\pi$
$510$		0			0
$511$		−	7.05256i		−	0.311987i
$512$		0			0
$513$	16.8078	−	16.9170i	0.742081	−	0.746905i
$514$		0			0
$515$	5.84912			0.257743
$516$		0			0
$517$	−28.6847			−1.26155
$518$		0			0
$519$			20.2895i			0.890609i
$520$		0			0
$521$		−	15.2910i		−	0.669910i	−0.942234	−	0.334955i	$-0.891279\pi$
$521$		−	15.2910i		−	0.669910i	0.942234	−	0.334955i	$-0.108721\pi$
$522$		0			0
$523$	33.4990			1.46481			0.732404	−	0.680871i	$-0.238398\pi$
$523$	33.4990			1.46481			0.732404	+	0.680871i	$0.238398\pi$
$524$		0			0
$525$		−	2.66732i		−	0.116411i
$526$		0			0
$527$	−4.27156			−0.186072
$528$		0			0
$529$	7.31534			0.318058
$530$		0			0
$531$	−18.1379			−0.787120
$532$		0			0
$533$	−17.6155			−0.763013
$534$		0			0
$535$	−12.2888			−0.531292
$536$		0			0
$537$	−10.8769			−0.469373
$538$		0			0
$539$			19.3144i			0.831929i
$540$		0			0
$541$	12.1922			0.524185			0.262093	−	0.965043i	$-0.415587\pi$
$541$	12.1922			0.524185			0.262093	+	0.965043i	$0.415587\pi$
$542$		0			0
$543$			2.49775i			0.107189i
$544$		0			0
$545$		−	24.7908i		−	1.06192i
$546$		0			0
$547$	19.4849			0.833116			0.416558	−	0.909109i	$-0.363236\pi$
$547$	19.4849			0.833116			0.416558	+	0.909109i	$0.363236\pi$
$548$		0			0
$549$	−1.06913			−0.0456294
$550$		0			0
$551$	26.1552	+	25.9863i	1.11425	+	1.10705i
$552$		0			0
$553$		−	6.96067i		−	0.295998i
$554$		0			0
$555$		−	6.94568i		−	0.294828i
$556$		0			0
$557$	18.9309			0.802127			0.401063	−	0.916050i	$-0.368641\pi$
$557$	18.9309			0.802127			0.401063	+	0.916050i	$0.368641\pi$
$558$		0			0
$559$	52.3104			2.21249
$560$		0			0
$561$			3.70861i			0.156578i
$562$		0			0
$563$	33.6466			1.41804			0.709018	−	0.705191i	$-0.249139\pi$
$563$	33.6466			1.41804			0.709018	+	0.705191i	$0.249139\pi$
$564$		0			0
$565$		−	5.79119i		−	0.243637i
$566$		0			0
$567$			1.62954i			0.0684342i
$568$		0			0
$569$		−	12.7519i		−	0.534586i	−0.963615	−	0.267293i	$-0.913871\pi$
$569$		−	12.7519i		−	0.534586i	0.963615	−	0.267293i	$-0.0861291\pi$
$570$		0			0
$571$		−	17.5780i		−	0.735615i	−0.929902	−	0.367807i	$-0.880109\pi$
$571$		−	17.5780i		−	0.735615i	0.929902	−	0.367807i	$-0.119891\pi$
$572$		0			0
$573$			10.5411i			0.440360i
$574$		0			0
$575$		−	10.1447i		−	0.423065i
$576$		0			0
$577$	27.0000			1.12402			0.562012	−	0.827129i	$-0.310027\pi$
$577$	27.0000			1.12402			0.562012	+	0.827129i	$0.310027\pi$
$578$		0			0
$579$			4.44793i			0.184850i
$580$		0			0
$581$	8.38447			0.347847
$582$		0			0
$583$	3.21985			0.133353
$584$		0			0
$585$		−	11.5824i		−	0.478873i
$586$		0			0
$587$		−	4.82860i		−	0.199297i	−0.995023	−	0.0996487i	$-0.968228\pi$
$587$		−	4.82860i		−	0.199297i	0.995023	−	0.0996487i	$-0.0317719\pi$
$588$		0			0
$589$	13.1231	−	13.2084i	0.540728	−	0.544243i
$590$		0			0
$591$	12.5841			0.517641
$592$		0			0
$593$	−36.7386			−1.50867			−0.754337	−	0.656487i	$-0.772042\pi$
$593$	−36.7386			−1.50867			−0.754337	+	0.656487i	$0.772042\pi$
$594$		0			0
$595$		−	1.35576i		−	0.0555806i
$596$		0			0
$597$			15.8757i			0.649750i
$598$		0			0
$599$	−7.72197			−0.315511			−0.157756	−	0.987478i	$-0.550426\pi$
$599$	−7.72197			−0.315511			−0.157756	+	0.987478i	$0.550426\pi$
$600$		0			0
$601$		−	39.1687i		−	1.59772i	−0.601514	−	0.798862i	$-0.705436\pi$
$601$		−	39.1687i		−	1.59772i	0.601514	−	0.798862i	$-0.294564\pi$
$602$		0			0
$603$	15.2134			0.619537
$604$		0			0
$605$	−2.24621			−0.0913215
$606$		0			0
$607$	−13.6358			−0.553461			−0.276730	−	0.960948i	$-0.589251\pi$
$607$	−13.6358			−0.553461			−0.276730	+	0.960948i	$0.589251\pi$
$608$		0			0
$609$	−8.80776			−0.356909
$610$		0			0
$611$	44.0626			1.78258
$612$		0			0
$613$	−24.3002			−0.981475			−0.490738	−	0.871307i	$-0.663273\pi$
$613$	−24.3002			−0.981475			−0.490738	+	0.871307i	$0.663273\pi$
$614$		0			0
$615$		−	6.94568i		−	0.280077i
$616$		0			0
$617$	−28.5464			−1.14923			−0.574617	−	0.818422i	$-0.694849\pi$
$617$	−28.5464			−1.14923			−0.574617	+	0.818422i	$0.694849\pi$
$618$		0			0
$619$			34.3946i			1.38244i	0.722646	+	0.691218i	$0.242926\pi$
$619$			34.3946i			1.38244i	−0.722646	+	0.691218i	$0.757074\pi$
$620$		0			0
$621$			21.6669i			0.869464i
$622$		0			0
$623$	−5.02797			−0.201441
$624$		0			0
$625$	−5.63068			−0.225227
$626$		0			0
$627$	−11.4677	−	11.3936i	−0.457975	−	0.455017i
$628$		0			0
$629$			3.70861i			0.147872i
$630$		0			0
$631$			12.7494i			0.507544i	0.967264	+	0.253772i	$0.0816713\pi$
$631$			12.7494i			0.507544i	−0.967264	+	0.253772i	$0.918329\pi$
$632$		0			0
$633$	2.42329			0.0963172
$634$		0			0
$635$	2.92456			0.116058
$636$		0			0
$637$		−	29.6689i		−	1.17552i
$638$		0			0
$639$	−17.0862			−0.675921
$640$		0			0
$641$			48.6685i			1.92229i	0.276045	+	0.961145i	$0.410976\pi$
$641$			48.6685i			1.92229i	−0.276045	+	0.961145i	$0.589024\pi$
$642$		0			0
$643$			6.56502i			0.258899i	0.991586	+	0.129449i	$0.0413210\pi$
$643$			6.56502i			0.258899i	−0.991586	+	0.129449i	$0.958679\pi$
$644$		0			0
$645$			20.6256i			0.812133i
$646$		0			0
$647$			36.2379i			1.42466i	0.701845	+	0.712329i	$0.252360\pi$
$647$			36.2379i			1.42466i	−0.701845	+	0.712329i	$0.747640\pi$
$648$		0			0
$649$			35.9166i			1.40985i
$650$		0			0
$651$			4.44793i			0.174328i
$652$		0			0
$653$	45.8078			1.79260			0.896298	−	0.443452i	$-0.146246\pi$
$653$	45.8078			1.79260			0.896298	+	0.443452i	$0.146246\pi$
$654$		0			0
$655$			7.54011i			0.294616i
$656$		0			0
$657$	−12.6847			−0.494876
$658$		0			0
$659$	3.36750			0.131179			0.0655896	−	0.997847i	$-0.479107\pi$
$659$	3.36750			0.131179			0.0655896	+	0.997847i	$0.479107\pi$
$660$		0			0
$661$		−	17.5018i		−	0.680740i	−0.940292	−	0.340370i	$-0.889448\pi$
$661$		−	17.5018i		−	0.680740i	0.940292	−	0.340370i	$-0.110552\pi$
$662$		0			0
$663$		−	5.69681i		−	0.221246i
$664$		0			0
$665$	4.19224	+	4.16516i	0.162568	+	0.161518i
$666$		0			0
$667$	−33.4990			−1.29709
$668$		0			0
$669$	−19.5076			−0.754207
$670$		0			0
$671$			2.11708i			0.0817291i
$672$		0			0
$673$		−	31.7515i		−	1.22393i	−0.790885	−	0.611964i	$-0.790380\pi$
$673$		−	31.7515i		−	1.22393i	0.790885	−	0.611964i	$-0.209620\pi$
$674$		0			0
$675$	−14.0140			−0.539400
$676$		0			0
$677$		−	4.29335i		−	0.165007i	−0.996591	−	0.0825034i	$-0.973708\pi$
$677$		−	4.29335i		−	0.165007i	0.996591	−	0.0825034i	$-0.0262915\pi$
$678$		0			0
$679$	14.6875			0.563656
$680$		0			0
$681$	20.9460			0.802653
$682$		0			0
$683$	−13.6358			−0.521760			−0.260880	−	0.965371i	$-0.584013\pi$
$683$	−13.6358			−0.521760			−0.260880	+	0.965371i	$0.584013\pi$
$684$		0			0
$685$	6.05398			0.231311
$686$		0			0
$687$	−2.92456			−0.111579
$688$		0			0
$689$	−4.94602			−0.188429
$690$		0			0
$691$			23.3817i			0.889480i	0.895660	+	0.444740i	$0.146704\pi$
$691$			23.3817i			0.889480i	−0.895660	+	0.444740i	$0.853296\pi$
$692$		0			0
$693$	−4.19224			−0.159250
$694$		0			0
$695$			0.594427i			0.0225479i
$696$		0			0
$697$			3.70861i			0.140474i
$698$		0			0
$699$	11.7630			0.444916
$700$		0			0
$701$	−29.3693			−1.10926			−0.554632	−	0.832096i	$-0.687141\pi$
$701$	−29.3693			−1.10926			−0.554632	+	0.832096i	$0.687141\pi$
$702$		0			0
$703$	−11.4677	−	11.3936i	−0.432512	−	0.429718i
$704$		0			0
$705$			17.3736i			0.654327i
$706$		0			0
$707$			7.15944i			0.269259i
$708$		0			0
$709$	−37.3693			−1.40343			−0.701717	−	0.712456i	$-0.747583\pi$
$709$	−37.3693			−1.40343			−0.701717	+	0.712456i	$0.747583\pi$
$710$		0			0
$711$	−12.5194			−0.469513
$712$		0			0
$713$			16.9170i			0.633547i
$714$		0			0
$715$	−22.9354			−0.857733
$716$		0			0
$717$		−	23.2930i		−	0.869891i
$718$		0			0
$719$			15.9484i			0.594776i	0.954757	+	0.297388i	$0.0961155\pi$
$719$			15.9484i			0.594776i	−0.954757	+	0.297388i	$0.903885\pi$
$720$		0			0
$721$			3.25206i			0.121113i
$722$		0			0
$723$		−	18.3393i		−	0.682046i
$724$		0			0
$725$		−	21.6669i		−	0.804689i
$726$		0			0
$727$		−	23.6554i		−	0.877332i	−0.898650	−	0.438666i	$-0.855451\pi$
$727$		−	23.6554i		−	0.877332i	0.898650	−	0.438666i	$-0.144549\pi$
$728$		0			0
$729$	22.6155			0.837612
$730$		0			0
$731$		−	11.0129i		−	0.407329i
$732$		0			0
$733$	−3.12311			−0.115355			−0.0576773	−	0.998335i	$-0.518369\pi$
$733$	−3.12311			−0.115355			−0.0576773	+	0.998335i	$0.518369\pi$
$734$		0			0
$735$	11.6982			0.431496
$736$		0			0
$737$		−	30.1254i		−	1.10968i
$738$		0			0
$739$			30.3273i			1.11561i	0.829972	+	0.557804i	$0.188356\pi$
$739$			30.3273i			1.11561i	−0.829972	+	0.557804i	$0.811644\pi$
$740$		0			0
$741$	17.6155	+	17.5018i	0.647123	+	0.642943i
$742$		0			0
$743$	38.4440			1.41037			0.705187	−	0.709021i	$-0.250863\pi$
$743$	38.4440			1.41037			0.705187	+	0.709021i	$0.250863\pi$
$744$		0			0
$745$	−27.8078			−1.01880
$746$		0			0
$747$		−	15.0802i		−	0.551756i
$748$		0			0
$749$		−	6.83248i		−	0.249653i
$750$		0			0
$751$	8.24782			0.300967			0.150484	−	0.988612i	$-0.451917\pi$
$751$	8.24782			0.300967			0.150484	+	0.988612i	$0.451917\pi$
$752$		0			0
$753$		−	26.8734i		−	0.979320i
$754$		0			0
$755$	37.9182			1.37998
$756$		0			0
$757$	24.1922			0.879282			0.439641	−	0.898174i	$-0.355106\pi$
$757$	24.1922			0.879282			0.439641	+	0.898174i	$0.355106\pi$
$758$		0			0
$759$	14.6875			0.533123
$760$		0			0
$761$	−10.6155			−0.384813			−0.192406	−	0.981315i	$-0.561629\pi$
$761$	−10.6155			−0.384813			−0.192406	+	0.981315i	$0.561629\pi$
$762$		0			0
$763$	13.7835			0.498995
$764$		0			0
$765$	−2.43845			−0.0881622
$766$		0			0
$767$		−	55.1716i		−	1.99213i
$768$		0			0
$769$	54.2311			1.95562			0.977811	−	0.209489i	$-0.0671801\pi$
$769$	54.2311			1.95562			0.977811	+	0.209489i	$0.0671801\pi$
$770$		0			0
$771$			22.7872i			0.820662i
$772$		0			0
$773$		−	25.8321i		−	0.929115i	−0.885543	−	0.464558i	$-0.846213\pi$
$773$		−	25.8321i		−	0.929115i	0.885543	−	0.464558i	$-0.153787\pi$
$774$		0			0
$775$	−10.9418			−0.393042
$776$		0			0
$777$	3.86174			0.138539
$778$		0			0
$779$	−11.4677	−	11.3936i	−0.410872	−	0.408219i
$780$		0			0
$781$			33.8340i			1.21068i
$782$		0			0
$783$			46.2758i			1.65376i
$784$		0			0
$785$	9.36932			0.334405
$786$		0			0
$787$	−40.9904			−1.46115			−0.730575	−	0.682833i	$-0.760748\pi$
$787$	−40.9904			−1.46115			−0.730575	+	0.682833i	$0.760748\pi$
$788$		0			0
$789$			7.87377i			0.280314i
$790$		0			0
$791$	3.21985			0.114485
$792$		0			0
$793$		−	3.25206i		−	0.115484i
$794$		0			0
$795$		−	1.95018i		−	0.0691659i
$796$		0			0
$797$			23.2930i			0.825079i	0.910940	+	0.412539i	$0.135358\pi$
$797$			23.2930i			0.825079i	−0.910940	+	0.412539i	$0.864642\pi$
$798$		0			0
$799$		−	9.27653i		−	0.328180i
$800$		0			0
$801$			9.04325i			0.319528i
$802$		0			0
$803$			25.1181i			0.886398i
$804$		0			0
$805$	−5.36932			−0.189244
$806$		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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.19935 q^{3} -1.56155 q^{5} -0.868210i q^{7} -1.56155 q^{9} +O(q^{10})\)	\(q-1.19935 q^{3} -1.56155 q^{5} -0.868210i q^{7} -1.56155 q^{9} +3.09218i q^{11} -4.74990i q^{13} +1.87285 q^{15} -1.00000 q^{17} +(3.07221 - 3.09218i) q^{19} +1.04129i q^{21} +3.96039i q^{23} -2.56155 q^{25} +5.47091 q^{27} +8.45851i q^{29} +4.27156 q^{31} -3.70861i q^{33} +1.35576i q^{35} -3.70861i q^{37} +5.69681i q^{39} -3.70861i q^{41} +11.0129i q^{43} +2.43845 q^{45} +9.27653i q^{47} +6.24621 q^{49} +1.19935 q^{51} -1.04129i q^{53} -4.82860i q^{55} +(-3.68466 + 3.70861i) q^{57} +11.6153 q^{59} +0.684658 q^{61} +1.35576i q^{63} +7.41722i q^{65} -9.74247 q^{67} -4.74990i q^{69} +10.9418 q^{71} +8.12311 q^{73} +3.07221 q^{75} +2.68466 q^{77} +8.01726 q^{79} -1.87689 q^{81} +9.65719i q^{83} +1.56155 q^{85} -10.1447i q^{87} -5.79119i q^{89} -4.12391 q^{91} -5.12311 q^{93} +(-4.79741 + 4.82860i) q^{95} +16.9170i q^{97} -4.82860i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q + 4q^{5} + 4q^{9} + O(q^{10}) \)	\( 8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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