LMFDB - Embedded newform 272.10.a.f.1.1

Newspace parameters

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$140.089747437$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
Defining polynomial:	$x^{5} - 2 x^{4} - 1596 x^{3} + 5754 x^{2} + 488987 x - 2711704$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.77274$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

$f(q)$	$=$	$q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +O(q^{10})$	$q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +31779.3 q^{11} -132363. q^{13} +287481. q^{15} -83521.0 q^{17} +1603.79 q^{19} -325495. q^{21} -23254.0 q^{23} +671873. q^{25} +1.39856e6 q^{27} +3.73267e6 q^{29} -8.91479e6 q^{31} -5.63884e6 q^{33} -2.97211e6 q^{35} -1.20475e7 q^{37} +2.34861e7 q^{39} -1.26779e7 q^{41} -2.86352e7 q^{43} -1.91198e7 q^{45} +7.17761e6 q^{47} -3.69885e7 q^{49} +1.48197e7 q^{51} -5.96065e7 q^{53} -5.14884e7 q^{55} -284573. q^{57} -1.85990e8 q^{59} -2.00037e8 q^{61} +2.16480e7 q^{63} +2.14453e8 q^{65} +1.27030e8 q^{67} +4.12613e6 q^{69} +3.27860e8 q^{71} -1.48678e8 q^{73} -1.19215e8 q^{75} +5.82968e7 q^{77} +2.58778e8 q^{79} -4.80436e8 q^{81} -3.45060e8 q^{83} +1.35319e8 q^{85} -6.62314e8 q^{87} +4.03936e8 q^{89} -2.42810e8 q^{91} +1.58182e9 q^{93} -2.59844e6 q^{95} -9.89973e8 q^{97} +3.75028e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + O(q^{10}) $	$ 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + 68036q^{11} - 158862q^{13} + 687324q^{15} - 417605q^{17} + 370992q^{19} + 1783880q^{21} - 1645870q^{23} + 3270239q^{25} + 2998268q^{27} + 3668616q^{29} + 7262362q^{31} - 11334900q^{33} + 26503988q^{35} - 31420708q^{37} + 42449884q^{39} - 7996938q^{41} + 56908268q^{43} + 12799536q^{45} + 16903336q^{47} - 11784059q^{49} - 19710956q^{51} - 83362982q^{53} - 6363364q^{55} + 136615904q^{57} + 37946604q^{59} - 77685452q^{61} + 191945278q^{63} - 40321288q^{65} + 304503600q^{67} - 333409272q^{69} + 476602922q^{71} - 289980486q^{73} + 153685772q^{75} - 143385648q^{77} + 828240610q^{79} + 891328609q^{81} - 194681148q^{83} - 123611080q^{85} - 158149884q^{87} + 376848106q^{89} - 194543664q^{91} + 3494835920q^{93} - 1498679864q^{95} + 692035246q^{97} - 2027106408q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

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
$3$	−177.437	−1.26473	−0.632367	−	0.774669i	$-0.717917\pi$
$3$	−177.437	−1.26473	−0.632367	+	0.774669i	$0.717917\pi$
$4$	0	0
$5$	−1620.18	−1.15931	−0.579655	−	0.814862i	$-0.696813\pi$
$5$	−1620.18	−1.15931	−0.579655	+	0.814862i	$0.696813\pi$
$6$	0	0
$7$	1834.42	0.288774	0.144387	−	0.989521i	$-0.453879\pi$
$7$	1834.42	0.288774	0.144387	+	0.989521i	$0.453879\pi$
$8$	0	0
$9$	11801.0	0.599553
$10$	0	0
$11$	31779.3	0.654452	0.327226	−	0.944946i	$-0.393886\pi$
$11$	31779.3	0.654452	0.327226	+	0.944946i	$0.393886\pi$
$12$	0	0
$13$	−132363.	−1.28535	−0.642675	−	0.766139i	$-0.722176\pi$
$13$	−132363.	−1.28535	−0.642675	+	0.766139i	$0.722176\pi$
$14$	0	0
$15$	287481.	1.46622
$16$	0	0
$17$	−83521.0	−0.242536
$17$	−83521.0	−0.242536
$18$	0	0
$19$	1603.79	0.00282330	0.00141165	−	0.999999i	$-0.499551\pi$
$19$	1603.79	0.00282330	0.00141165	+	0.999999i	$0.499551\pi$
$20$	0	0
$21$	−325495.	−0.365223
$22$	0	0
$23$	−23254.0	−0.0173270	−0.00866348	−	0.999962i	$-0.502758\pi$
$23$	−23254.0	−0.0173270	−0.00866348	+	0.999962i	$0.502758\pi$
$24$	0	0
$25$	671873.	0.343999
$26$	0	0
$27$	1.39856e6	0.506460
$28$	0	0
$29$	3.73267e6	0.980005	0.490002	−	0.871721i	$-0.336996\pi$
$29$	3.73267e6	0.980005	0.490002	+	0.871721i	$0.336996\pi$
$30$	0	0
$31$	−8.91479e6	−1.73374	−0.866869	−	0.498536i	$-0.833871\pi$
$31$	−8.91479e6	−1.73374	−0.866869	+	0.498536i	$0.833871\pi$
$32$	0	0
$33$	−5.63884e6	−0.827707
$34$	0	0
$35$	−2.97211e6	−0.334779
$36$	0	0
$37$	−1.20475e7	−1.05679	−0.528395	−	0.848999i	$-0.677206\pi$
$37$	−1.20475e7	−1.05679	−0.528395	+	0.848999i	$0.677206\pi$
$38$	0	0
$39$	2.34861e7	1.62563
$40$	0	0
$41$	−1.26779e7	−0.700680	−0.350340	−	0.936623i	$-0.613934\pi$
$41$	−1.26779e7	−0.700680	−0.350340	+	0.936623i	$0.613934\pi$
$42$	0	0
$43$	−2.86352e7	−1.27730	−0.638650	−	0.769498i	$-0.720507\pi$
$43$	−2.86352e7	−1.27730	−0.638650	+	0.769498i	$0.720507\pi$
$44$	0	0
$45$	−1.91198e7	−0.695067
$46$	0	0
$47$	7.17761e6	0.214555	0.107278	−	0.994229i	$-0.465787\pi$
$47$	7.17761e6	0.214555	0.107278	+	0.994229i	$0.465787\pi$
$48$	0	0
$49$	−3.69885e7	−0.916609
$50$	0	0
$51$	1.48197e7	0.306743
$52$	0	0
$53$	−5.96065e7	−1.03765	−0.518826	−	0.854880i	$-0.673631\pi$
$53$	−5.96065e7	−1.03765	−0.518826	+	0.854880i	$0.673631\pi$
$54$	0	0
$55$	−5.14884e7	−0.758712
$56$	0	0
$57$	−284573.	−0.00357072
$58$	0	0
$59$	−1.85990e8	−1.99828	−0.999139	−	0.0414925i	$-0.986789\pi$
$59$	−1.85990e8	−1.99828	−0.999139	+	0.0414925i	$0.986789\pi$
$60$	0	0
$61$	−2.00037e8	−1.84981	−0.924905	−	0.380198i	$-0.875856\pi$
$61$	−2.00037e8	−1.84981	−0.924905	+	0.380198i	$0.875856\pi$
$62$	0	0
$63$	2.16480e7	0.173135
$64$	0	0
$65$	2.14453e8	1.49012
$66$	0	0
$67$	1.27030e8	0.770137	0.385069	−	0.922888i	$-0.374178\pi$
$67$	1.27030e8	0.770137	0.385069	+	0.922888i	$0.374178\pi$
$68$	0	0
$69$	4.12613e6	0.0219140
$70$	0	0
$71$	3.27860e8	1.53118	0.765590	−	0.643329i	$-0.222447\pi$
$71$	3.27860e8	1.53118	0.765590	+	0.643329i	$0.222447\pi$
$72$	0	0
$73$	−1.48678e8	−0.612766	−0.306383	−	0.951908i	$-0.599119\pi$
$73$	−1.48678e8	−0.612766	−0.306383	+	0.951908i	$0.599119\pi$
$74$	0	0
$75$	−1.19215e8	−0.435067
$76$	0	0
$77$	5.82968e7	0.188989
$78$	0	0
$79$	2.58778e8	0.747491	0.373746	−	0.927531i	$-0.378073\pi$
$79$	2.58778e8	0.747491	0.373746	+	0.927531i	$0.378073\pi$
$80$	0	0
$81$	−4.80436e8	−1.24009
$82$	0	0
$83$	−3.45060e8	−0.798073	−0.399037	−	0.916935i	$-0.630655\pi$
$83$	−3.45060e8	−0.798073	−0.399037	+	0.916935i	$0.630655\pi$
$84$	0	0
$85$	1.35319e8	0.281174
$86$	0	0
$87$	−6.62314e8	−1.23945
$88$	0	0
$89$	4.03936e8	0.682428	0.341214	−	0.939986i	$-0.389162\pi$
$89$	4.03936e8	0.682428	0.341214	+	0.939986i	$0.389162\pi$
$90$	0	0
$91$	−2.42810e8	−0.371176
$92$	0	0
$93$	1.58182e9	2.19272
$94$	0	0
$95$	−2.59844e6	−0.00327308
$96$	0	0
$97$	−9.89973e8	−1.13540	−0.567702	−	0.823234i	$-0.692167\pi$
$97$	−9.89973e8	−1.13540	−0.567702	+	0.823234i	$0.692167\pi$
$98$	0	0
$99$	3.75028e8	0.392378
$100$	0	0
$101$	−1.11917e9	−1.07017	−0.535084	−	0.844799i	$-0.679720\pi$
$101$	−1.11917e9	−1.07017	−0.535084	+	0.844799i	$0.679720\pi$
$102$	0	0
$103$	−6.58796e6	−0.00576744	−0.00288372	−	0.999996i	$-0.500918\pi$
$103$	−6.58796e6	−0.00576744	−0.00288372	+	0.999996i	$0.500918\pi$
$104$	0	0
$105$	5.27363e8	0.423407
$106$	0	0
$107$	−1.55590e9	−1.14751	−0.573754	−	0.819028i	$-0.694513\pi$
$107$	−1.55590e9	−1.14751	−0.573754	+	0.819028i	$0.694513\pi$
$108$	0	0
$109$	6.94766e8	0.471432	0.235716	−	0.971822i	$-0.424256\pi$
$109$	6.94766e8	0.471432	0.235716	+	0.971822i	$0.424256\pi$
$110$	0	0
$111$	2.13767e9	1.33656
$112$	0	0
$113$	2.09735e9	1.21009	0.605047	−	0.796190i	$-0.293154\pi$
$113$	2.09735e9	1.21009	0.605047	+	0.796190i	$0.293154\pi$
$114$	0	0
$115$	3.76758e7	0.0200873
$116$	0	0
$117$	−1.56202e9	−0.770635
$118$	0	0
$119$	−1.53213e8	−0.0700381
$120$	0	0
$121$	−1.34802e9	−0.571693
$122$	0	0
$123$	2.24953e9	0.886173
$124$	0	0
$125$	2.07586e9	0.760508
$126$	0	0
$127$	4.92229e9	1.67900	0.839499	−	0.543361i	$-0.182849\pi$
$127$	4.92229e9	1.67900	0.839499	+	0.543361i	$0.182849\pi$
$128$	0	0
$129$	5.08096e9	1.61544
$130$	0	0
$131$	−4.54604e9	−1.34869	−0.674345	−	0.738416i	$-0.735574\pi$
$131$	−4.54604e9	−1.34869	−0.674345	+	0.738416i	$0.735574\pi$
$132$	0	0
$133$	2.94204e6	0.000815297 0
$134$	0	0
$135$	−2.26593e9	−0.587143
$136$	0	0
$137$	5.25313e9	1.27402	0.637010	−	0.770856i	$-0.280171\pi$
$137$	5.25313e9	1.27402	0.637010	+	0.770856i	$0.280171\pi$
$138$	0	0
$139$	−1.55602e9	−0.353549	−0.176774	−	0.984251i	$-0.556566\pi$
$139$	−1.55602e9	−0.353549	−0.176774	+	0.984251i	$0.556566\pi$
$140$	0	0
$141$	−1.27358e9	−0.271356
$142$	0	0
$143$	−4.20641e9	−0.841200
$144$	0	0
$145$	−6.04761e9	−1.13613
$146$	0	0
$147$	6.56314e9	1.15927
$148$	0	0
$149$	2.13936e9	0.355587	0.177794	−	0.984068i	$-0.443104\pi$
$149$	2.13936e9	0.355587	0.177794	+	0.984068i	$0.443104\pi$
$150$	0	0
$151$	1.65162e9	0.258531	0.129266	−	0.991610i	$-0.458738\pi$
$151$	1.65162e9	0.258531	0.129266	+	0.991610i	$0.458738\pi$
$152$	0	0
$153$	−9.85631e8	−0.145413
$154$	0	0
$155$	1.44436e10	2.00994
$156$	0	0
$157$	−4.44203e9	−0.583490	−0.291745	−	0.956496i	$-0.594236\pi$
$157$	−4.44203e9	−0.583490	−0.291745	+	0.956496i	$0.594236\pi$
$158$	0	0
$159$	1.05764e10	1.31235
$160$	0	0
$161$	−4.26577e7	−0.00500358
$162$	0	0
$163$	−2.66196e9	−0.295363	−0.147682	−	0.989035i	$-0.547181\pi$
$163$	−2.66196e9	−0.295363	−0.147682	+	0.989035i	$0.547181\pi$
$164$	0	0
$165$	9.13596e9	0.959569
$166$	0	0
$167$	1.58326e10	1.57518	0.787588	−	0.616203i	$-0.211330\pi$
$167$	1.58326e10	1.57518	0.787588	+	0.616203i	$0.211330\pi$
$168$	0	0
$169$	6.91547e9	0.652126
$170$	0	0
$171$	1.89264e7	0.00169272
$172$	0	0
$173$	−1.27988e10	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$173$	−1.27988e10	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$174$	0	0
$175$	1.23250e9	0.0993382
$176$	0	0
$177$	3.30016e10	2.52729
$178$	0	0
$179$	8.33273e9	0.606665	0.303332	−	0.952885i	$-0.401901\pi$
$179$	8.33273e9	0.606665	0.303332	+	0.952885i	$0.401901\pi$
$180$	0	0
$181$	−8.70236e9	−0.602676	−0.301338	−	0.953517i	$-0.597433\pi$
$181$	−8.70236e9	−0.602676	−0.301338	+	0.953517i	$0.597433\pi$
$182$	0	0
$183$	3.54941e10	2.33952
$184$	0	0
$185$	1.95192e10	1.22515
$186$	0	0
$187$	−2.65424e9	−0.158728
$188$	0	0
$189$	2.56556e9	0.146253
$190$	0	0
$191$	−2.29558e10	−1.24808	−0.624040	−	0.781392i	$-0.714510\pi$
$191$	−2.29558e10	−1.24808	−0.624040	+	0.781392i	$0.714510\pi$
$192$	0	0
$193$	2.04685e10	1.06189	0.530943	−	0.847407i	$-0.321838\pi$
$193$	2.04685e10	1.06189	0.530943	+	0.847407i	$0.321838\pi$
$194$	0	0
$195$	−3.80519e10	−1.88460
$196$	0	0
$197$	5.81009e9	0.274843	0.137422	−	0.990513i	$-0.456118\pi$
$197$	5.81009e9	0.274843	0.137422	+	0.990513i	$0.456118\pi$
$198$	0	0
$199$	−4.49526e9	−0.203196	−0.101598	−	0.994826i	$-0.532396\pi$
$199$	−4.49526e9	−0.203196	−0.101598	+	0.994826i	$0.532396\pi$
$200$	0	0
$201$	−2.25398e10	−0.974019
$202$	0	0
$203$	6.84730e9	0.283000
$204$	0	0
$205$	2.05405e10	0.812305
$206$	0	0
$207$	−2.74420e8	−0.0103884
$208$	0	0
$209$	5.09674e7	0.00184771
$210$	0	0
$211$	−3.47865e10	−1.20820	−0.604101	−	0.796908i	$-0.706467\pi$
$211$	−3.47865e10	−1.20820	−0.604101	+	0.796908i	$0.706467\pi$
$212$	0	0
$213$	−5.81746e10	−1.93654
$214$	0	0
$215$	4.63944e10	1.48079
$216$	0	0
$217$	−1.63535e10	−0.500659
$218$	0	0
$219$	2.63811e10	0.774986
$220$	0	0
$221$	1.10551e10	0.311743
$222$	0	0
$223$	2.46348e10	0.667078	0.333539	−	0.942736i	$-0.391757\pi$
$223$	2.46348e10	0.667078	0.333539	+	0.942736i	$0.391757\pi$
$224$	0	0
$225$	7.92877e9	0.206246
$226$	0	0
$227$	−4.45657e10	−1.11400	−0.556999	−	0.830513i	$-0.688047\pi$
$227$	−4.45657e10	−1.11400	−0.556999	+	0.830513i	$0.688047\pi$
$228$	0	0
$229$	6.61817e10	1.59030	0.795148	−	0.606415i	$-0.207393\pi$
$229$	6.61817e10	1.59030	0.795148	+	0.606415i	$0.207393\pi$
$230$	0	0
$231$	−1.03440e10	−0.239021
$232$	0	0
$233$	−4.35721e10	−0.968516	−0.484258	−	0.874925i	$-0.660910\pi$
$233$	−4.35721e10	−0.968516	−0.484258	+	0.874925i	$0.660910\pi$
$234$	0	0
$235$	−1.16291e10	−0.248736
$236$	0	0
$237$	−4.59169e10	−0.945378
$238$	0	0
$239$	−3.67128e10	−0.727825	−0.363912	−	0.931433i	$-0.618559\pi$
$239$	−3.67128e10	−0.727825	−0.363912	+	0.931433i	$0.618559\pi$
$240$	0	0
$241$	3.62693e10	0.692568	0.346284	−	0.938130i	$-0.387443\pi$
$241$	3.62693e10	0.692568	0.346284	+	0.938130i	$0.387443\pi$
$242$	0	0
$243$	5.77194e10	1.06192
$244$	0	0
$245$	5.99282e10	1.06263
$246$	0	0
$247$	−2.12283e8	−0.00362893
$248$	0	0
$249$	6.12264e10	1.00935
$250$	0	0
$251$	9.34890e10	1.48672	0.743359	−	0.668892i	$-0.233231\pi$
$251$	9.34890e10	1.48672	0.743359	+	0.668892i	$0.233231\pi$
$252$	0	0
$253$	−7.38996e8	−0.0113397
$254$	0	0
$255$	−2.40107e10	−0.355610
$256$	0	0
$257$	1.91619e10	0.273993	0.136996	−	0.990572i	$-0.456255\pi$
$257$	1.91619e10	0.273993	0.136996	+	0.990572i	$0.456255\pi$
$258$	0	0
$259$	−2.21002e10	−0.305174
$260$	0	0
$261$	4.40492e10	0.587564
$262$	0	0
$263$	−3.75028e10	−0.483352	−0.241676	−	0.970357i	$-0.577697\pi$
$263$	−3.75028e10	−0.483352	−0.241676	+	0.970357i	$0.577697\pi$
$264$	0	0
$265$	9.65735e10	1.20296
$266$	0	0
$267$	−7.16732e10	−0.863090
$268$	0	0
$269$	−9.36804e10	−1.09085	−0.545423	−	0.838161i	$-0.683631\pi$
$269$	−9.36804e10	−1.09085	−0.545423	+	0.838161i	$0.683631\pi$
$270$	0	0
$271$	9.81985e10	1.10597	0.552985	−	0.833191i	$-0.313489\pi$
$271$	9.81985e10	1.10597	0.552985	+	0.833191i	$0.313489\pi$
$272$	0	0
$273$	4.30836e10	0.469439
$274$	0	0
$275$	2.13517e10	0.225131
$276$	0	0
$277$	9.27606e9	0.0946683	0.0473341	−	0.998879i	$-0.484927\pi$
$277$	9.27606e9	0.0946683	0.0473341	+	0.998879i	$0.484927\pi$
$278$	0	0
$279$	−1.05203e11	−1.03947
$280$	0	0
$281$	−1.04165e11	−0.996649	−0.498325	−	0.866991i	$-0.666051\pi$
$281$	−1.04165e11	−0.996649	−0.498325	+	0.866991i	$0.666051\pi$
$282$	0	0
$283$	9.57066e10	0.886958	0.443479	−	0.896285i	$-0.353744\pi$
$283$	9.57066e10	0.886958	0.443479	+	0.896285i	$0.353744\pi$
$284$	0	0
$285$	4.61060e8	0.00413958
$286$	0	0
$287$	−2.32566e10	−0.202338
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	1.75658e11	1.43598
$292$	0	0
$293$	−1.62664e11	−1.28940	−0.644698	−	0.764437i	$-0.723017\pi$
$293$	−1.62664e11	−1.28940	−0.644698	+	0.764437i	$0.723017\pi$
$294$	0	0
$295$	3.01338e11	2.31662
$296$	0	0
$297$	4.44453e10	0.331453
$298$	0	0
$299$	3.07797e9	0.0222712
$300$	0	0
$301$	−5.25292e10	−0.368851
$302$	0	0
$303$	1.98583e11	1.35348
$304$	0	0
$305$	3.24098e11	2.14450
$306$	0	0
$307$	2.26286e11	1.45390	0.726952	−	0.686688i	$-0.240936\pi$
$307$	2.26286e11	1.45390	0.726952	+	0.686688i	$0.240936\pi$
$308$	0	0
$309$	1.16895e9	0.00729428
$310$	0	0
$311$	−1.17225e11	−0.710556	−0.355278	−	0.934761i	$-0.615614\pi$
$311$	−1.17225e11	−0.710556	−0.355278	+	0.934761i	$0.615614\pi$
$312$	0	0
$313$	2.85060e11	1.67876	0.839378	−	0.543549i	$-0.182920\pi$
$313$	2.85060e11	1.67876	0.839378	+	0.543549i	$0.182920\pi$
$314$	0	0
$315$	−3.50738e10	−0.200718
$316$	0	0
$317$	−9.98016e9	−0.0555099	−0.0277550	−	0.999615i	$-0.508836\pi$
$317$	−9.98016e9	−0.0555099	−0.0277550	+	0.999615i	$0.508836\pi$
$318$	0	0
$319$	1.18622e11	0.641365
$320$	0	0
$321$	2.76075e11	1.45129
$322$	0	0
$323$	−1.33950e8	−0.000684751 0
$324$	0	0
$325$	−8.89312e10	−0.442159
$326$	0	0
$327$	−1.23277e11	−0.596236
$328$	0	0
$329$	1.31668e10	0.0619581
$330$	0	0
$331$	2.14841e11	0.983766	0.491883	−	0.870661i	$-0.336309\pi$
$331$	2.14841e11	0.983766	0.491883	+	0.870661i	$0.336309\pi$
$332$	0	0
$333$	−1.42172e11	−0.633601
$334$	0	0
$335$	−2.05811e11	−0.892828
$336$	0	0
$337$	−3.33228e11	−1.40737	−0.703683	−	0.710514i	$-0.748463\pi$
$337$	−3.33228e11	−1.40737	−0.703683	+	0.710514i	$0.748463\pi$
$338$	0	0
$339$	−3.72149e11	−1.53045
$340$	0	0
$341$	−2.83306e11	−1.13465
$342$	0	0
$343$	−1.41878e11	−0.553468
$344$	0	0
$345$	−6.68509e9	−0.0254051
$346$	0	0
$347$	1.63241e10	0.0604432	0.0302216	−	0.999543i	$-0.490379\pi$
$347$	1.63241e10	0.0604432	0.0302216	+	0.999543i	$0.490379\pi$
$348$	0	0
$349$	1.61528e11	0.582819	0.291410	−	0.956598i	$-0.405876\pi$
$349$	1.61528e11	0.582819	0.291410	+	0.956598i	$0.405876\pi$
$350$	0	0
$351$	−1.85118e11	−0.650978
$352$	0	0
$353$	2.89170e11	0.991212	0.495606	−	0.868547i	$-0.334946\pi$
$353$	2.89170e11	0.991212	0.495606	+	0.868547i	$0.334946\pi$
$354$	0	0
$355$	−5.31194e11	−1.77511
$356$	0	0
$357$	2.71857e10	0.0885796
$358$	0	0
$359$	−2.34120e11	−0.743898	−0.371949	−	0.928253i	$-0.621310\pi$
$359$	−2.34120e11	−0.743898	−0.371949	+	0.928253i	$0.621310\pi$
$360$	0	0
$361$	−3.22685e11	−0.999992
$362$	0	0
$363$	2.39190e11	0.723040
$364$	0	0
$365$	2.40886e11	0.710385
$366$	0	0
$367$	1.08932e11	0.313442	0.156721	−	0.987643i	$-0.449908\pi$
$367$	1.08932e11	0.313442	0.156721	+	0.987643i	$0.449908\pi$
$368$	0	0
$369$	−1.49612e11	−0.420094
$370$	0	0
$371$	−1.09344e11	−0.299648
$372$	0	0
$373$	−6.82746e11	−1.82629	−0.913145	−	0.407635i	$-0.866353\pi$
$373$	−6.82746e11	−1.82629	−0.913145	+	0.407635i	$0.866353\pi$
$374$	0	0
$375$	−3.68336e11	−0.961841
$376$	0	0
$377$	−4.94067e11	−1.25965
$378$	0	0
$379$	−1.87645e11	−0.467156	−0.233578	−	0.972338i	$-0.575043\pi$
$379$	−1.87645e11	−0.467156	−0.233578	+	0.972338i	$0.575043\pi$
$380$	0	0
$381$	−8.73397e11	−2.12349
$382$	0	0
$383$	−1.01456e11	−0.240925	−0.120462	−	0.992718i	$-0.538438\pi$
$383$	−1.01456e11	−0.240925	−0.120462	+	0.992718i	$0.538438\pi$
$384$	0	0
$385$	−9.44515e10	−0.219097
$386$	0	0
$387$	−3.37924e11	−0.765808
$388$	0	0
$389$	−4.62050e11	−1.02310	−0.511548	−	0.859255i	$-0.670928\pi$
$389$	−4.62050e11	−1.02310	−0.511548	+	0.859255i	$0.670928\pi$
$390$	0	0
$391$	1.94220e9	0.00420240
$392$	0	0
$393$	8.06637e11	1.70573
$394$	0	0
$395$	−4.19269e11	−0.866574
$396$	0	0
$397$	8.72806e11	1.76344	0.881720	−	0.471774i	$-0.156386\pi$
$397$	8.72806e11	1.76344	0.881720	+	0.471774i	$0.156386\pi$
$398$	0	0
$399$	−5.22027e8	−0.00103113
$400$	0	0
$401$	−1.68101e11	−0.324654	−0.162327	−	0.986737i	$-0.551900\pi$
$401$	−1.68101e11	−0.324654	−0.162327	+	0.986737i	$0.551900\pi$
$402$	0	0
$403$	1.17999e12	2.22846
$404$	0	0
$405$	7.78395e11	1.43765
$406$	0	0
$407$	−3.82861e11	−0.691618
$408$	0	0
$409$	−6.99127e11	−1.23538	−0.617691	−	0.786421i	$-0.711932\pi$
$409$	−6.99127e11	−1.23538	−0.617691	+	0.786421i	$0.711932\pi$
$410$	0	0
$411$	−9.32102e11	−1.61130
$412$	0	0
$413$	−3.41185e11	−0.577052
$414$	0	0
$415$	5.59060e11	0.925214
$416$	0	0
$417$	2.76097e11	0.447145
$418$	0	0
$419$	−9.68451e11	−1.53502	−0.767511	−	0.641036i	$-0.778505\pi$
$419$	−9.68451e11	−1.53502	−0.767511	+	0.641036i	$0.778505\pi$
$420$	0	0
$421$	−5.47606e11	−0.849569	−0.424784	−	0.905295i	$-0.639650\pi$
$421$	−5.47606e11	−0.849569	−0.424784	+	0.905295i	$0.639650\pi$
$422$	0	0
$423$	8.47029e10	0.128637
$424$	0	0
$425$	−5.61155e10	−0.0834320
$426$	0	0
$427$	−3.66954e11	−0.534178
$428$	0	0
$429$	7.46373e11	1.06389
$430$	0	0
$431$	−4.80670e11	−0.670964	−0.335482	−	0.942047i	$-0.608899\pi$
$431$	−4.80670e11	−0.670964	−0.335482	+	0.942047i	$0.608899\pi$
$432$	0	0
$433$	−7.09547e11	−0.970031	−0.485016	−	0.874506i	$-0.661186\pi$
$433$	−7.09547e11	−0.970031	−0.485016	+	0.874506i	$0.661186\pi$
$434$	0	0
$435$	1.07307e12	1.43690
$436$	0	0
$437$	−3.72946e7	−4.89192e−5 0
$438$	0	0
$439$	−1.17101e12	−1.50477	−0.752387	−	0.658721i	$-0.771098\pi$
$439$	−1.17101e12	−1.50477	−0.752387	+	0.658721i	$0.771098\pi$
$440$	0	0
$441$	−4.36501e11	−0.549555
$442$	0	0
$443$	4.75055e11	0.586040	0.293020	−	0.956106i	$-0.405340\pi$
$443$	4.75055e11	0.586040	0.293020	+	0.956106i	$0.405340\pi$
$444$	0	0
$445$	−6.54450e11	−0.791146
$446$	0	0
$447$	−3.79603e11	−0.449723
$448$	0	0
$449$	−9.33820e11	−1.08431	−0.542156	−	0.840278i	$-0.682392\pi$
$449$	−9.33820e11	−1.08431	−0.542156	+	0.840278i	$0.682392\pi$
$450$	0	0
$451$	−4.02895e11	−0.458561
$452$	0	0
$453$	−2.93058e11	−0.326973
$454$	0	0
$455$	3.93397e11	0.430308
$456$	0	0
$457$	1.28584e12	1.37900	0.689498	−	0.724288i	$-0.257831\pi$
$457$	1.28584e12	1.37900	0.689498	+	0.724288i	$0.257831\pi$
$458$	0	0
$459$	−1.16809e11	−0.122834
$460$	0	0
$461$	5.18444e11	0.534623	0.267311	−	0.963610i	$-0.413865\pi$
$461$	5.18444e11	0.534623	0.267311	+	0.963610i	$0.413865\pi$
$462$	0	0
$463$	7.76808e11	0.785596	0.392798	−	0.919625i	$-0.371507\pi$
$463$	7.76808e11	0.785596	0.392798	+	0.919625i	$0.371507\pi$
$464$	0	0
$465$	−2.56283e12	−2.54204
$466$	0	0
$467$	−8.28768e10	−0.0806319	−0.0403159	−	0.999187i	$-0.512836\pi$
$467$	−8.28768e10	−0.0806319	−0.0403159	+	0.999187i	$0.512836\pi$
$468$	0	0
$469$	2.33026e11	0.222396
$470$	0	0
$471$	7.88182e11	0.737959
$472$	0	0
$473$	−9.10008e11	−0.835930
$474$	0	0
$475$	1.07755e9	0.000971213 0
$476$	0	0
$477$	−7.03416e11	−0.622127
$478$	0	0
$479$	2.08230e12	1.80731	0.903657	−	0.428257i	$-0.140872\pi$
$479$	2.08230e12	1.80731	0.903657	+	0.428257i	$0.140872\pi$
$480$	0	0
$481$	1.59464e12	1.35835
$482$	0	0
$483$	7.56907e9	0.00632820
$484$	0	0
$485$	1.60394e12	1.31629
$486$	0	0
$487$	2.69427e11	0.217051	0.108525	−	0.994094i	$-0.465387\pi$
$487$	2.69427e11	0.217051	0.108525	+	0.994094i	$0.465387\pi$
$488$	0	0
$489$	4.72330e11	0.373556
$490$	0	0
$491$	5.87772e10	0.0456396	0.0228198	−	0.999740i	$-0.492736\pi$
$491$	5.87772e10	0.0456396	0.0228198	+	0.999740i	$0.492736\pi$
$492$	0	0
$493$	−3.11756e11	−0.237686
$494$	0	0
$495$	−6.07614e11	−0.454888
$496$	0	0
$497$	6.01435e11	0.442166
$498$	0	0
$499$	3.13142e11	0.226094	0.113047	−	0.993590i	$-0.463939\pi$
$499$	3.13142e11	0.226094	0.113047	+	0.993590i	$0.463939\pi$
$500$	0	0
$501$	−2.80930e12	−1.99218
$502$	0	0
$503$	−7.31761e11	−0.509699	−0.254849	−	0.966981i	$-0.582026\pi$
$503$	−7.31761e11	−0.509699	−0.254849	+	0.966981i	$0.582026\pi$
$504$	0	0
$505$	1.81327e12	1.24066
$506$	0	0
$507$	−1.22706e12	−0.824766
$508$	0	0
$509$	2.84601e12	1.87935	0.939673	−	0.342073i	$-0.111129\pi$
$509$	2.84601e12	1.87935	0.939673	+	0.342073i	$0.111129\pi$
$510$	0	0
$511$	−2.72739e11	−0.176951
$512$	0	0
$513$	2.24300e9	0.00142989
$514$	0	0
$515$	1.06737e10	0.00668625
$516$	0	0
$517$	2.28100e11	0.140416
$518$	0	0
$519$	2.27099e12	1.37392
$520$	0	0
$521$	−1.42270e10	−0.00845949	−0.00422974	−	0.999991i	$-0.501346\pi$
$521$	−1.42270e10	−0.00845949	−0.00422974	+	0.999991i	$0.501346\pi$
$522$	0	0
$523$	−1.20853e12	−0.706316	−0.353158	−	0.935564i	$-0.614892\pi$
$523$	−1.20853e12	−0.706316	−0.353158	+	0.935564i	$0.614892\pi$
$524$	0	0
$525$	−2.18692e11	−0.125636
$526$	0	0
$527$	7.44572e11	0.420493
$528$	0	0
$529$	−1.80061e12	−0.999700
$530$	0	0
$531$	−2.19487e12	−1.19807
$532$	0	0
$533$	1.67808e12	0.900619
$534$	0	0
$535$	2.52085e12	1.33032
$536$	0	0
$537$	−1.47854e12	−0.767270
$538$	0	0
$539$	−1.17547e12	−0.599876
$540$	0	0
$541$	2.35337e12	1.18115	0.590573	−	0.806985i	$-0.298902\pi$
$541$	2.35337e12	1.18115	0.590573	+	0.806985i	$0.298902\pi$
$542$	0	0
$543$	1.54412e12	0.762224
$544$	0	0
$545$	−1.12565e12	−0.546536
$546$	0	0
$547$	−1.30236e12	−0.621996	−0.310998	−	0.950411i	$-0.600663\pi$
$547$	−1.30236e12	−0.621996	−0.310998	+	0.950411i	$0.600663\pi$
$548$	0	0
$549$	−2.36064e12	−1.10906
$550$	0	0
$551$	5.98642e9	0.00276685
$552$	0	0
$553$	4.74710e11	0.215856
$554$	0	0
$555$	−3.46343e12	−1.54949
$556$	0	0
$557$	3.33519e12	1.46816	0.734078	−	0.679065i	$-0.237615\pi$
$557$	3.33519e12	1.46816	0.734078	+	0.679065i	$0.237615\pi$
$558$	0	0
$559$	3.79025e12	1.64178
$560$	0	0
$561$	4.70961e11	0.200748
$562$	0	0
$563$	9.51343e11	0.399070	0.199535	−	0.979891i	$-0.436057\pi$
$563$	9.51343e11	0.399070	0.199535	+	0.979891i	$0.436057\pi$
$564$	0	0
$565$	−3.39810e12	−1.40287
$566$	0	0
$567$	−8.81324e11	−0.358106
$568$	0	0
$569$	−8.14899e10	−0.0325911	−0.0162955	−	0.999867i	$-0.505187\pi$
$569$	−8.14899e10	−0.0325911	−0.0162955	+	0.999867i	$0.505187\pi$
$570$	0	0
$571$	−1.06368e12	−0.418744	−0.209372	−	0.977836i	$-0.567142\pi$
$571$	−1.06368e12	−0.418744	−0.209372	+	0.977836i	$0.567142\pi$
$572$	0	0
$573$	4.07322e12	1.57849
$574$	0	0
$575$	−1.56237e10	−0.00596046
$576$	0	0
$577$	−3.44925e12	−1.29549	−0.647743	−	0.761859i	$-0.724287\pi$
$577$	−3.44925e12	−1.29549	−0.647743	+	0.761859i	$0.724287\pi$
$578$	0	0
$579$	−3.63187e12	−1.34300
$580$	0	0
$581$	−6.32986e11	−0.230463
$582$	0	0
$583$	−1.89425e12	−0.679093
$584$	0	0
$585$	2.53075e12	0.893405
$586$	0	0
$587$	3.31180e12	1.15131	0.575655	−	0.817693i	$-0.304747\pi$
$587$	3.31180e12	1.15131	0.575655	+	0.817693i	$0.304747\pi$
$588$	0	0
$589$	−1.42975e10	−0.00489486
$590$	0	0
$591$	−1.03093e12	−0.347604
$592$	0	0
$593$	9.22763e10	0.0306439	0.0153219	−	0.999883i	$-0.495123\pi$
$593$	9.22763e10	0.0306439	0.0153219	+	0.999883i	$0.495123\pi$
$594$	0	0
$595$	2.48233e11	0.0811958
$596$	0	0
$597$	7.97627e11	0.256989
$598$	0	0
$599$	4.28034e11	0.135849	0.0679246	−	0.997690i	$-0.478362\pi$
$599$	4.28034e11	0.135849	0.0679246	+	0.997690i	$0.478362\pi$
$600$	0	0
$601$	2.92420e12	0.914266	0.457133	−	0.889398i	$-0.348876\pi$
$601$	2.92420e12	0.914266	0.457133	+	0.889398i	$0.348876\pi$
$602$	0	0
$603$	1.49908e12	0.461738
$604$	0	0
$605$	2.18405e12	0.662770
$606$	0	0
$607$	1.45237e12	0.434237	0.217118	−	0.976145i	$-0.430334\pi$
$607$	1.45237e12	0.434237	0.217118	+	0.976145i	$0.430334\pi$
$608$	0	0
$609$	−1.21497e12	−0.357920
$610$	0	0
$611$	−9.50050e11	−0.275779
$612$	0	0
$613$	5.13743e12	1.46951	0.734757	−	0.678331i	$-0.237296\pi$
$613$	5.13743e12	1.46951	0.734757	+	0.678331i	$0.237296\pi$
$614$	0	0
$615$	−3.64465e12	−1.02735
$616$	0	0
$617$	−5.31905e12	−1.47758	−0.738790	−	0.673936i	$-0.764603\pi$
$617$	−5.31905e12	−1.47758	−0.738790	+	0.673936i	$0.764603\pi$
$618$	0	0
$619$	3.23470e12	0.885577	0.442789	−	0.896626i	$-0.353989\pi$
$619$	3.23470e12	0.885577	0.442789	+	0.896626i	$0.353989\pi$
$620$	0	0
$621$	−3.25222e10	−0.00877540
$622$	0	0
$623$	7.40989e11	0.197068
$624$	0	0
$625$	−4.67554e12	−1.22566
$626$	0	0
$627$	−9.04353e9	−0.00233687
$628$	0	0
$629$	1.00622e12	0.256309
$630$	0	0
$631$	7.91021e12	1.98635	0.993175	−	0.116630i	$-0.0372091\pi$
$631$	7.91021e12	1.98635	0.993175	+	0.116630i	$0.0372091\pi$
$632$	0	0
$633$	6.17242e12	1.52805
$634$	0	0
$635$	−7.97501e12	−1.94648
$636$	0	0
$637$	4.89591e12	1.17816
$638$	0	0
$639$	3.86908e12	0.918023
$640$	0	0
$641$	−2.28438e12	−0.534451	−0.267225	−	0.963634i	$-0.586107\pi$
$641$	−2.28438e12	−0.534451	−0.267225	+	0.963634i	$0.586107\pi$
$642$	0	0
$643$	−7.52203e11	−0.173534	−0.0867672	−	0.996229i	$-0.527654\pi$
$643$	−7.52203e11	−0.173534	−0.0867672	+	0.996229i	$0.527654\pi$
$644$	0	0
$645$	−8.23209e12	−1.87280
$646$	0	0
$647$	4.59159e12	1.03013	0.515067	−	0.857150i	$-0.327767\pi$
$647$	4.59159e12	1.03013	0.515067	+	0.857150i	$0.327767\pi$
$648$	0	0
$649$	−5.91064e12	−1.30778
$650$	0	0
$651$	2.90172e12	0.633201
$652$	0	0
$653$	4.27664e12	0.920436	0.460218	−	0.887806i	$-0.347771\pi$
$653$	4.27664e12	0.920436	0.460218	+	0.887806i	$0.347771\pi$
$654$	0	0
$655$	7.36542e12	1.56355
$656$	0	0
$657$	−1.75455e12	−0.367385
$658$	0	0
$659$	8.26860e12	1.70784	0.853920	−	0.520404i	$-0.174218\pi$
$659$	8.26860e12	1.70784	0.853920	+	0.520404i	$0.174218\pi$
$660$	0	0
$661$	4.45784e12	0.908276	0.454138	−	0.890931i	$-0.349947\pi$
$661$	4.45784e12	0.908276	0.454138	+	0.890931i	$0.349947\pi$
$662$	0	0
$663$	−1.96159e12	−0.394272
$664$	0	0
$665$	−4.76664e9	−0.000945182 0
$666$	0	0
$667$	−8.67994e10	−0.0169805
$668$	0	0
$669$	−4.37112e12	−0.843676
$670$	0	0
$671$	−6.35705e12	−1.21061
$672$	0	0
$673$	3.90397e12	0.733566	0.366783	−	0.930307i	$-0.380459\pi$
$673$	3.90397e12	0.733566	0.366783	+	0.930307i	$0.380459\pi$
$674$	0	0
$675$	9.39656e11	0.174222
$676$	0	0
$677$	4.84414e12	0.886273	0.443137	−	0.896454i	$-0.353866\pi$
$677$	4.84414e12	0.886273	0.443137	+	0.896454i	$0.353866\pi$
$678$	0	0
$679$	−1.81603e12	−0.327876
$680$	0	0
$681$	7.90761e12	1.40891
$682$	0	0
$683$	3.67160e12	0.645598	0.322799	−	0.946468i	$-0.395376\pi$
$683$	3.67160e12	0.645598	0.322799	+	0.946468i	$0.395376\pi$
$684$	0	0
$685$	−8.51105e12	−1.47698
$686$	0	0
$687$	−1.17431e13	−2.01130
$688$	0	0
$689$	7.88969e12	1.33375
$690$	0	0
$691$	−3.94359e12	−0.658023	−0.329011	−	0.944326i	$-0.606715\pi$
$691$	−3.94359e12	−0.658023	−0.329011	+	0.944326i	$0.606715\pi$
$692$	0	0
$693$	6.87960e11	0.113309
$694$	0	0
$695$	2.52104e12	0.409873
$696$	0	0
$697$	1.05887e12	0.169940
$698$	0	0
$699$	7.73131e12	1.22492
$700$	0	0
$701$	−9.80834e12	−1.53414	−0.767069	−	0.641564i	$-0.778286\pi$
$701$	−9.80834e12	−1.53414	−0.767069	+	0.641564i	$0.778286\pi$
$702$	0	0
$703$	−1.93217e10	−0.00298364
$704$	0	0
$705$	2.06343e12	0.314585
$706$	0	0
$707$	−2.05304e12	−0.309037
$708$	0	0
$709$	4.04313e12	0.600910	0.300455	−	0.953796i	$-0.402861\pi$
$709$	4.04313e12	0.600910	0.300455	+	0.953796i	$0.402861\pi$
$710$	0	0
$711$	3.05384e12	0.448160
$712$	0	0
$713$	2.07305e11	0.0300404
$714$	0	0
$715$	6.81515e12	0.975211
$716$	0	0
$717$	6.51422e12	0.920505
$718$	0	0
$719$	4.48823e12	0.626319	0.313159	−	0.949701i	$-0.398613\pi$
$719$	4.48823e12	0.626319	0.313159	+	0.949701i	$0.398613\pi$
$720$	0	0
$721$	−1.20851e10	−0.00166549
$722$	0	0
$723$	−6.43553e12	−0.875915
$724$	0	0
$725$	2.50788e12	0.337121
$726$	0	0
$727$	1.02503e13	1.36092	0.680459	−	0.732786i	$-0.261780\pi$
$727$	1.02503e13	1.36092	0.680459	+	0.732786i	$0.261780\pi$
$728$	0	0
$729$	−7.85147e11	−0.102962
$730$	0	0
$731$	2.39164e12	0.309791
$732$	0	0
$733$	−1.77332e12	−0.226892	−0.113446	−	0.993544i	$-0.536189\pi$
$733$	−1.77332e12	−0.226892	−0.113446	+	0.993544i	$0.536189\pi$
$734$	0	0
$735$	−1.06335e13	−1.34395
$736$	0	0
$737$	4.03691e12	0.504018
$738$	0	0
$739$	−2.07204e12	−0.255563	−0.127781	−	0.991802i	$-0.540786\pi$
$739$	−2.07204e12	−0.255563	−0.127781	+	0.991802i	$0.540786\pi$
$740$	0	0
$741$	3.76669e10	0.00458963
$742$	0	0
$743$	−7.60966e12	−0.916042	−0.458021	−	0.888941i	$-0.651442\pi$
$743$	−7.60966e12	−0.916042	−0.458021	+	0.888941i	$0.651442\pi$
$744$	0	0
$745$	−3.46616e12	−0.412236
$746$	0	0
$747$	−4.07205e12	−0.478487
$748$	0	0
$749$	−2.85419e12	−0.331371
$750$	0	0
$751$	7.56877e12	0.868252	0.434126	−	0.900852i	$-0.357057\pi$
$751$	7.56877e12	0.868252	0.434126	+	0.900852i	$0.357057\pi$
$752$	0	0
$753$	−1.65884e13	−1.88030
$754$	0	0
$755$	−2.67592e12	−0.299718
$756$	0	0
$757$	−6.86633e12	−0.759965	−0.379982	−	0.924994i	$-0.624070\pi$
$757$	−6.86633e12	−0.759965	−0.379982	+	0.924994i	$0.624070\pi$
$758$	0	0
$759$	1.31125e11	0.0143416
$760$	0	0
$761$	9.51816e12	1.02878	0.514389	−	0.857557i	$-0.328019\pi$
$761$	9.51816e12	1.02878	0.514389	+	0.857557i	$0.328019\pi$
$762$	0	0
$763$	1.27450e12	0.136138
$764$	0	0
$765$	1.59690e12	0.168579
$766$	0	0
$767$	2.46182e13	2.56849
$768$	0	0
$769$	−6.30061e12	−0.649702	−0.324851	−	0.945765i	$-0.605314\pi$
$769$	−6.30061e12	−0.649702	−0.324851	+	0.945765i	$0.605314\pi$
$770$	0	0
$771$	−3.40003e12	−0.346528
$772$	0	0
$773$	−1.40318e13	−1.41354	−0.706768	−	0.707446i	$-0.749847\pi$
$773$	−1.40318e13	−1.41354	−0.706768	+	0.707446i	$0.749847\pi$
$774$	0	0
$775$	−5.98961e12	−0.596404
$776$	0	0
$777$	3.92140e12	0.385964
$778$	0	0
$779$	−2.03327e10	−0.00197823
$780$	0	0
$781$	1.04192e13	1.00208
$782$	0	0
$783$	5.22036e12	0.496333
$784$	0	0
$785$	7.19691e12	0.676445
$786$	0	0
$787$	−1.03038e13	−0.957440	−0.478720	−	0.877968i	$-0.658899\pi$
$787$	−1.03038e13	−0.957440	−0.478720	+	0.877968i	$0.658899\pi$
$788$	0	0
$789$	6.65440e12	0.611312
$790$	0	0
$791$	3.84744e12	0.349444
$792$	0	0
$793$	2.64776e13	2.37765
$794$	0	0
$795$	−1.71357e13	−1.52143
$796$	0	0
$797$	1.57641e13	1.38390	0.691952	−	0.721944i	$-0.256751\pi$
$797$	1.57641e13	1.38390	0.691952	+	0.721944i	$0.256751\pi$
$798$	0	0
$799$	−5.99481e11	−0.0520373
$800$	0	0
$801$	4.76684e12	0.409152
$802$	0	0
$803$	−4.72489e12	−0.401026
$804$	0	0
$805$	6.91134e10	0.00580070
$806$	0	0
$807$	1.66224e13	1.37963
$808$	0	0
$809$	1.37214e13	1.12623	0.563117	−	0.826377i	$-0.309602\pi$
$809$	1.37214e13	1.12623	0.563117	+	0.826377i	$0.309602\pi$
$810$	0	0
$811$	−1.55198e13	−1.25977	−0.629887	−	0.776687i	$-0.716899\pi$
$811$	−1.55198e13	−1.25977	−0.629887	+	0.776687i	$0.716899\pi$
$812$	0	0
$813$	−1.74241e13	−1.39876
$814$	0	0
$815$	4.31286e12	0.342418
$816$	0	0
$817$	−4.59250e10	−0.00360620
$818$	0	0
$819$	−2.86540e12	−0.222540
$820$	0	0
$821$	3.40134e12	0.261280	0.130640	−	0.991430i	$-0.458297\pi$
$821$	3.40134e12	0.261280	0.130640	+	0.991430i	$0.458297\pi$
$822$	0	0
$823$	−4.01848e12	−0.305325	−0.152663	−	0.988278i	$-0.548785\pi$
$823$	−4.01848e12	−0.305325	−0.152663	+	0.988278i	$0.548785\pi$
$824$	0	0
$825$	−3.78858e12	−0.284731
$826$	0	0
$827$	4.31147e12	0.320517	0.160258	−	0.987075i	$-0.448767\pi$
$827$	4.31147e12	0.320517	0.160258	+	0.987075i	$0.448767\pi$
$828$	0	0
$829$	4.42248e12	0.325215	0.162608	−	0.986691i	$-0.448010\pi$
$829$	4.42248e12	0.325215	0.162608	+	0.986691i	$0.448010\pi$
$830$	0	0
$831$	−1.64592e12	−0.119730
$832$	0	0
$833$	3.08932e12	0.222310
$834$	0	0
$835$	−2.56518e13	−1.82612
$836$	0	0
$837$	−1.24679e13	−0.878068
$838$	0	0
$839$	−1.12787e13	−0.785835	−0.392917	−	0.919574i	$-0.628534\pi$
$839$	−1.12787e13	−0.785835	−0.392917	+	0.919574i	$0.628534\pi$
$840$	0	0
$841$	−5.74353e11	−0.0395910
$842$	0	0
$843$	1.84827e13	1.26050
$844$	0	0
$845$	−1.12043e13	−0.756016
$846$	0	0
$847$	−2.47285e12	−0.165090
$848$	0	0
$849$	−1.69819e13	−1.12177
$850$	0	0
$851$	2.80152e11	0.0183110
$852$	0	0
$853$	−3.07683e12	−0.198991	−0.0994954	−	0.995038i	$-0.531723\pi$
$853$	−3.07683e12	−0.198991	−0.0994954	+	0.995038i	$0.531723\pi$
$854$	0	0
$855$	−3.06642e10	−0.00196238
$856$	0	0
$857$	−3.10840e13	−1.96845	−0.984223	−	0.176930i	$-0.943383\pi$
$857$	−3.10840e13	−1.96845	−0.984223	+	0.176930i	$0.943383\pi$
$858$	0	0
$859$	7.61893e12	0.477446	0.238723	−	0.971088i	$-0.423271\pi$
$859$	7.61893e12	0.477446	0.238723	+	0.971088i	$0.423271\pi$
$860$	0	0
$861$	4.12659e12	0.255904
$862$	0	0
$863$	2.61427e13	1.60436	0.802179	−	0.597083i	$-0.203674\pi$
$863$	2.61427e13	1.60436	0.802179	+	0.597083i	$0.203674\pi$
$864$	0	0
$865$	2.07364e13	1.25939
$866$	0	0
$867$	−1.23776e12	−0.0743961
$868$	0	0
$869$	8.22380e12	0.489197
$870$	0	0
$871$	−1.68140e13	−0.989896
$872$	0	0
$873$	−1.16827e13	−0.680735
$874$	0	0
$875$	3.80802e12	0.219615
$876$	0	0
$877$	−3.00641e13	−1.71613	−0.858065	−	0.513541i	$-0.828333\pi$
$877$	−3.00641e13	−1.71613	−0.858065	+	0.513541i	$0.828333\pi$
$878$	0	0
$879$	2.88626e13	1.63074
$880$	0	0
$881$	−2.06409e13	−1.15435	−0.577175	−	0.816620i	$-0.695845\pi$
$881$	−2.06409e13	−1.15435	−0.577175	+	0.816620i	$0.695845\pi$
$882$	0	0
$883$	−1.85711e13	−1.02805	−0.514026	−	0.857774i	$-0.671847\pi$
$883$	−1.85711e13	−1.02805	−0.514026	+	0.857774i	$0.671847\pi$
$884$	0	0
$885$	−5.34687e13	−2.92991
$886$	0	0
$887$	−1.35590e11	−0.00735482	−0.00367741	−	0.999993i	$-0.501171\pi$
$887$	−1.35590e11	−0.00735482	−0.00367741	+	0.999993i	$0.501171\pi$
$888$	0	0
$889$	9.02957e12	0.484852
$890$	0	0
$891$	−1.52679e13	−0.811578
$892$	0	0
$893$	1.15114e10	0.000605754 0
$894$	0	0
$895$	−1.35006e13	−0.703313
$896$	0	0
$897$	−5.46146e11	−0.0281672
$898$	0	0
$899$	−3.32759e13	−1.69907
$900$	0	0
$901$	4.97839e12	0.251668
$902$	0	0
$903$	9.32064e12	0.466499
$904$	0	0
$905$	1.40994e13	0.698688
$906$	0	0
$907$	1.36972e13	0.672045	0.336022	−	0.941854i	$-0.390918\pi$
$907$	1.36972e13	0.672045	0.336022	+	0.941854i	$0.390918\pi$
$908$	0	0
$909$	−1.32074e13	−0.641622
$910$	0	0
$911$	2.24844e13	1.08156	0.540778	−	0.841165i	$-0.318130\pi$
$911$	2.24844e13	1.08156	0.540778	+	0.841165i	$0.318130\pi$
$912$	0	0
$913$	−1.09658e13	−0.522300
$914$	0	0
$915$	−5.75070e13	−2.71223
$916$	0	0
$917$	−8.33936e12	−0.389467
$918$	0	0
$919$	−9.66637e12	−0.447037	−0.223519	−	0.974700i	$-0.571754\pi$
$919$	−9.66637e12	−0.447037	−0.223519	+	0.974700i	$0.571754\pi$
$920$	0	0
$921$	−4.01516e13	−1.83880
$922$	0	0
$923$	−4.33966e13	−1.96810
$924$	0	0
$925$	−8.09439e12	−0.363535
$926$	0	0
$927$	−7.77444e10	−0.00345788
$928$	0	0
$929$	8.19216e12	0.360851	0.180425	−	0.983589i	$-0.442253\pi$
$929$	8.19216e12	0.360851	0.180425	+	0.983589i	$0.442253\pi$
$930$	0	0
$931$	−5.93219e10	−0.00258786
$932$	0	0
$933$	2.08001e13	0.898665
$934$	0	0
$935$	4.30036e12	0.184015
$936$	0	0
$937$	3.49872e12	0.148279	0.0741396	−	0.997248i	$-0.476379\pi$
$937$	3.49872e12	0.148279	0.0741396	+	0.997248i	$0.476379\pi$
$938$	0	0
$939$	−5.05803e13	−2.12318
$940$	0	0
$941$	−1.51926e12	−0.0631655	−0.0315827	−	0.999501i	$-0.510055\pi$
$941$	−1.51926e12	−0.0631655	−0.0315827	+	0.999501i	$0.510055\pi$
$942$	0	0
$943$	2.94811e11	0.0121406
$944$	0	0
$945$	−4.15668e12	−0.169552
$946$	0	0
$947$	2.12785e13	0.859738	0.429869	−	0.902891i	$-0.358560\pi$
$947$	2.12785e13	0.859738	0.429869	+	0.902891i	$0.358560\pi$
$948$	0	0
$949$	1.96795e13	0.787619
$950$	0	0
$951$	1.77085e12	0.0702053
$952$	0	0
$953$	9.61359e12	0.377544	0.188772	−	0.982021i	$-0.439549\pi$
$953$	9.61359e12	0.377544	0.188772	+	0.982021i	$0.439549\pi$
$954$	0	0
$955$	3.71927e13	1.44691
$956$	0	0
$957$	−2.10479e13	−0.811157
$958$	0	0
$959$	9.63648e12	0.367904
$960$	0	0
$961$	5.30339e13	2.00585
$962$	0	0
$963$	−1.83612e13	−0.687992
$964$	0	0
$965$	−3.31627e13	−1.23106
$966$	0	0
$967$	1.08700e13	0.399772	0.199886	−	0.979819i	$-0.435943\pi$
$967$	1.08700e13	0.399772	0.199886	+	0.979819i	$0.435943\pi$
$968$	0	0
$969$	2.37678e10	0.000866028 0
$970$	0	0
$971$	−2.79438e13	−1.00879	−0.504393	−	0.863474i	$-0.668284\pi$
$971$	−2.79438e13	−1.00879	−0.504393	+	0.863474i	$0.668284\pi$
$972$	0	0
$973$	−2.85441e12	−0.102096
$974$	0	0
$975$	1.57797e13	0.559214
$976$	0	0
$977$	−1.49620e13	−0.525369	−0.262685	−	0.964882i	$-0.584608\pi$
$977$	−1.49620e13	−0.525369	−0.262685	+	0.964882i	$0.584608\pi$
$978$	0	0
$979$	1.28368e13	0.446616
$980$	0	0
$981$	8.19893e12	0.282648
$982$	0	0
$983$	−2.54966e13	−0.870946	−0.435473	−	0.900202i	$-0.643419\pi$
$983$	−2.54966e13	−0.870946	−0.435473	+	0.900202i	$0.643419\pi$
$984$	0	0
$985$	−9.41343e12	−0.318629
$986$	0	0
$987$	−2.33628e12	−0.0783605
$988$	0	0
$989$	6.65884e11	0.0221317
$990$	0	0
$991$	9.59893e12	0.316149	0.158074	−	0.987427i	$-0.449471\pi$
$991$	9.59893e12	0.316149	0.158074	+	0.987427i	$0.449471\pi$
$992$	0	0
$993$	−3.81209e13	−1.24420
$994$	0	0
$995$	7.28315e12	0.235568
$996$	0	0
$997$	−3.17533e13	−1.01780	−0.508898	−	0.860827i	$-0.669947\pi$
$997$	−3.17533e13	−1.01780	−0.508898	+	0.860827i	$0.669947\pi$
$998$	0	0
$999$	−1.68492e13	−0.535222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.f.1.1		5
4.3	odd	2		17.10.a.a.1.3	✓	5
12.11	even	2		153.10.a.c.1.3		5
68.67	odd	2		289.10.a.a.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.a.1.3	✓	5	4.3	odd	2
153.10.a.c.1.3		5	12.11	even	2
272.10.a.f.1.1		5	1.1	even	1	trivial
289.10.a.a.1.3		5	68.67	odd	2

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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +O(q^{10})\)	\(q-177.437 q^{3} -1620.18 q^{5} +1834.42 q^{7} +11801.0 q^{9} +31779.3 q^{11} -132363. q^{13} +287481. q^{15} -83521.0 q^{17} +1603.79 q^{19} -325495. q^{21} -23254.0 q^{23} +671873. q^{25} +1.39856e6 q^{27} +3.73267e6 q^{29} -8.91479e6 q^{31} -5.63884e6 q^{33} -2.97211e6 q^{35} -1.20475e7 q^{37} +2.34861e7 q^{39} -1.26779e7 q^{41} -2.86352e7 q^{43} -1.91198e7 q^{45} +7.17761e6 q^{47} -3.69885e7 q^{49} +1.48197e7 q^{51} -5.96065e7 q^{53} -5.14884e7 q^{55} -284573. q^{57} -1.85990e8 q^{59} -2.00037e8 q^{61} +2.16480e7 q^{63} +2.14453e8 q^{65} +1.27030e8 q^{67} +4.12613e6 q^{69} +3.27860e8 q^{71} -1.48678e8 q^{73} -1.19215e8 q^{75} +5.82968e7 q^{77} +2.58778e8 q^{79} -4.80436e8 q^{81} -3.45060e8 q^{83} +1.35319e8 q^{85} -6.62314e8 q^{87} +4.03936e8 q^{89} -2.42810e8 q^{91} +1.58182e9 q^{93} -2.59844e6 q^{95} -9.89973e8 q^{97} +3.75028e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + O(q^{10}) \)	\( 5q + 236q^{3} + 1480q^{5} + 13202q^{7} + 10981q^{9} + 68036q^{11} - 158862q^{13} + 687324q^{15} - 417605q^{17} + 370992q^{19} + 1783880q^{21} - 1645870q^{23} + 3270239q^{25} + 2998268q^{27} + 3668616q^{29} + 7262362q^{31} - 11334900q^{33} + 26503988q^{35} - 31420708q^{37} + 42449884q^{39} - 7996938q^{41} + 56908268q^{43} + 12799536q^{45} + 16903336q^{47} - 11784059q^{49} - 19710956q^{51} - 83362982q^{53} - 6363364q^{55} + 136615904q^{57} + 37946604q^{59} - 77685452q^{61} + 191945278q^{63} - 40321288q^{65} + 304503600q^{67} - 333409272q^{69} + 476602922q^{71} - 289980486q^{73} + 153685772q^{75} - 143385648q^{77} + 828240610q^{79} + 891328609q^{81} - 194681148q^{83} - 123611080q^{85} - 158149884q^{87} + 376848106q^{89} - 194543664q^{91} + 3494835920q^{93} - 1498679864q^{95} + 692035246q^{97} - 2027106408q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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