LMFDB - Embedded newform 48.22.a.e.1.1

Newspace parameters

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$134.149125258$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

$f(q)$

$=$

$q+59049.0 q^{3} +3.10995e6 q^{5} -3.63304e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$

$q+59049.0 q^{3} +3.10995e6 q^{5} -3.63304e8 q^{7} +3.48678e9 q^{9} -1.45818e10 q^{11} +1.13351e11 q^{13} +1.83639e11 q^{15} -8.58939e12 q^{17} +2.92029e13 q^{19} -2.14527e13 q^{21} +1.55899e14 q^{23} -4.67165e14 q^{25} +2.05891e14 q^{27} +2.40079e15 q^{29} -2.23982e15 q^{31} -8.61043e14 q^{33} -1.12986e15 q^{35} -3.07851e16 q^{37} +6.69325e15 q^{39} -1.03208e17 q^{41} +1.65557e17 q^{43} +1.08437e16 q^{45} +6.65872e16 q^{47} -4.26556e17 q^{49} -5.07195e17 q^{51} +4.35423e17 q^{53} -4.53488e16 q^{55} +1.72440e18 q^{57} -5.53437e18 q^{59} -7.17621e18 q^{61} -1.26676e18 q^{63} +3.52515e17 q^{65} +1.57554e19 q^{67} +9.20569e18 q^{69} -2.64579e19 q^{71} +1.34712e19 q^{73} -2.75856e19 q^{75} +5.29764e18 q^{77} +1.68861e19 q^{79} +1.21577e19 q^{81} +1.70688e20 q^{83} -2.67126e19 q^{85} +1.41764e20 q^{87} -3.12592e20 q^{89} -4.11808e19 q^{91} -1.32259e20 q^{93} +9.08197e19 q^{95} +9.49015e20 q^{97} -5.08437e19 q^{99} +O(q^{100})$

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
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	0	0
$5$	3.10995e6	0.142419	0.0712096	−	0.997461i	$-0.477314\pi$
$5$	3.10995e6	0.142419	0.0712096	+	0.997461i	$0.477314\pi$
$6$	0	0
$7$	−3.63304e8	−0.486117	−0.243058	−	0.970012i	$-0.578151\pi$
$7$	−3.63304e8	−0.486117	−0.243058	+	0.970012i	$0.578151\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−1.45818e10	−0.169508	−0.0847538	−	0.996402i	$-0.527010\pi$
$11$	−1.45818e10	−0.169508	−0.0847538	+	0.996402i	$0.527010\pi$
$12$	0	0
$13$	1.13351e11	0.228044	0.114022	−	0.993478i	$-0.463627\pi$
$13$	1.13351e11	0.228044	0.114022	+	0.993478i	$0.463627\pi$
$14$	0	0
$15$	1.83639e11	0.0822257
$16$	0	0
$17$	−8.58939e12	−1.03335	−0.516676	−	0.856181i	$-0.672831\pi$
$17$	−8.58939e12	−1.03335	−0.516676	+	0.856181i	$0.672831\pi$
$18$	0	0
$19$	2.92029e13	1.09273	0.546366	−	0.837546i	$-0.316011\pi$
$19$	2.92029e13	1.09273	0.546366	+	0.837546i	$0.316011\pi$
$20$	0	0
$21$	−2.14527e13	−0.280660
$22$	0	0
$23$	1.55899e14	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$23$	1.55899e14	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$24$	0	0
$25$	−4.67165e14	−0.979717
$26$	0	0
$27$	2.05891e14	0.192450
$28$	0	0
$29$	2.40079e15	1.05968	0.529840	−	0.848097i	$-0.322252\pi$
$29$	2.40079e15	1.05968	0.529840	+	0.848097i	$0.322252\pi$
$30$	0	0
$31$	−2.23982e15	−0.490812	−0.245406	−	0.969420i	$-0.578921\pi$
$31$	−2.23982e15	−0.490812	−0.245406	+	0.969420i	$0.578921\pi$
$32$	0	0
$33$	−8.61043e14	−0.0978652
$34$	0	0
$35$	−1.12986e15	−0.0692323
$36$	0	0
$37$	−3.07851e16	−1.05250	−0.526250	−	0.850330i	$-0.676402\pi$
$37$	−3.07851e16	−1.05250	−0.526250	+	0.850330i	$0.676402\pi$
$38$	0	0
$39$	6.69325e15	0.131661
$40$	0	0
$41$	−1.03208e17	−1.20083	−0.600414	−	0.799689i	$-0.704998\pi$
$41$	−1.03208e17	−1.20083	−0.600414	+	0.799689i	$0.704998\pi$
$42$	0	0
$43$	1.65557e17	1.16823	0.584117	−	0.811670i	$-0.301441\pi$
$43$	1.65557e17	1.16823	0.584117	+	0.811670i	$0.301441\pi$
$44$	0	0
$45$	1.08437e16	0.0474730
$46$	0	0
$47$	6.65872e16	0.184656	0.0923280	−	0.995729i	$-0.470569\pi$
$47$	6.65872e16	0.184656	0.0923280	+	0.995729i	$0.470569\pi$
$48$	0	0
$49$	−4.26556e17	−0.763690
$50$	0	0
$51$	−5.07195e17	−0.596607
$52$	0	0
$53$	4.35423e17	0.341991	0.170995	−	0.985272i	$-0.445302\pi$
$53$	4.35423e17	0.341991	0.170995	+	0.985272i	$0.445302\pi$
$54$	0	0
$55$	−4.53488e16	−0.0241411
$56$	0	0
$57$	1.72440e18	0.630889
$58$	0	0
$59$	−5.53437e18	−1.40968	−0.704842	−	0.709364i	$-0.748982\pi$
$59$	−5.53437e18	−1.40968	−0.704842	+	0.709364i	$0.748982\pi$
$60$	0	0
$61$	−7.17621e18	−1.28805	−0.644023	−	0.765006i	$-0.722736\pi$
$61$	−7.17621e18	−1.28805	−0.644023	+	0.765006i	$0.722736\pi$
$62$	0	0
$63$	−1.26676e18	−0.162039
$64$	0	0
$65$	3.52515e17	0.0324779
$66$	0	0
$67$	1.57554e19	1.05595	0.527977	−	0.849258i	$-0.322951\pi$
$67$	1.57554e19	1.05595	0.527977	+	0.849258i	$0.322951\pi$
$68$	0	0
$69$	9.20569e18	0.453045
$70$	0	0
$71$	−2.64579e19	−0.964588	−0.482294	−	0.876009i	$-0.660196\pi$
$71$	−2.64579e19	−0.964588	−0.482294	+	0.876009i	$0.660196\pi$
$72$	0	0
$73$	1.34712e19	0.366875	0.183437	−	0.983031i	$-0.441278\pi$
$73$	1.34712e19	0.366875	0.183437	+	0.983031i	$0.441278\pi$
$74$	0	0
$75$	−2.75856e19	−0.565640
$76$	0	0
$77$	5.29764e18	0.0824005
$78$	0	0
$79$	1.68861e19	0.200653	0.100326	−	0.994955i	$-0.468011\pi$
$79$	1.68861e19	0.200653	0.100326	+	0.994955i	$0.468011\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	1.70688e20	1.20749	0.603744	−	0.797178i	$-0.293675\pi$
$83$	1.70688e20	1.20749	0.603744	+	0.797178i	$0.293675\pi$
$84$	0	0
$85$	−2.67126e19	−0.147169
$86$	0	0
$87$	1.41764e20	0.611807
$88$	0	0
$89$	−3.12592e20	−1.06263	−0.531317	−	0.847173i	$-0.678303\pi$
$89$	−3.12592e20	−1.06263	−0.531317	+	0.847173i	$0.678303\pi$
$90$	0	0
$91$	−4.11808e19	−0.110856
$92$	0	0
$93$	−1.32259e20	−0.283371
$94$	0	0
$95$	9.08197e19	0.155626
$96$	0	0
$97$	9.49015e20	1.30668	0.653341	−	0.757064i	$-0.273367\pi$
$97$	9.49015e20	1.30668	0.653341	+	0.757064i	$0.273367\pi$
$98$	0	0
$99$	−5.08437e19	−0.0565025
$100$	0	0
$101$	1.44798e20	0.130433	0.0652166	−	0.997871i	$-0.479226\pi$
$101$	1.44798e20	0.130433	0.0652166	+	0.997871i	$0.479226\pi$
$102$	0	0
$103$	−2.19627e21	−1.61025	−0.805127	−	0.593102i	$-0.797903\pi$
$103$	−2.19627e21	−1.61025	−0.805127	+	0.593102i	$0.797903\pi$
$104$	0	0
$105$	−6.67169e19	−0.0399713
$106$	0	0
$107$	1.63087e20	0.0801473	0.0400737	−	0.999197i	$-0.487241\pi$
$107$	1.63087e20	0.0801473	0.0400737	+	0.999197i	$0.487241\pi$
$108$	0	0
$109$	2.24852e20	0.0909743	0.0454871	−	0.998965i	$-0.485516\pi$
$109$	2.24852e20	0.0909743	0.0454871	+	0.998965i	$0.485516\pi$
$110$	0	0
$111$	−1.81783e21	−0.607661
$112$	0	0
$113$	−4.24118e21	−1.17534	−0.587670	−	0.809101i	$-0.699955\pi$
$113$	−4.24118e21	−1.17534	−0.587670	+	0.809101i	$0.699955\pi$
$114$	0	0
$115$	4.84839e20	0.111756
$116$	0	0
$117$	3.95230e20	0.0760148
$118$	0	0
$119$	3.12056e21	0.502330
$120$	0	0
$121$	−7.18762e21	−0.971267
$122$	0	0
$123$	−6.09430e21	−0.693299
$124$	0	0
$125$	−2.93580e21	−0.281950
$126$	0	0
$127$	−1.66312e21	−0.135202	−0.0676012	−	0.997712i	$-0.521535\pi$
$127$	−1.66312e21	−0.135202	−0.0676012	+	0.997712i	$0.521535\pi$
$128$	0	0
$129$	9.77599e21	0.674480
$130$	0	0
$131$	−6.40663e21	−0.376081	−0.188040	−	0.982161i	$-0.560214\pi$
$131$	−6.40663e21	−0.376081	−0.188040	+	0.982161i	$0.560214\pi$
$132$	0	0
$133$	−1.06095e22	−0.531196
$134$	0	0
$135$	6.40311e20	0.0274086
$136$	0	0
$137$	−1.98314e22	−0.727423	−0.363711	−	0.931512i	$-0.618491\pi$
$137$	−1.98314e22	−0.727423	−0.363711	+	0.931512i	$0.618491\pi$
$138$	0	0
$139$	5.20143e21	0.163858	0.0819290	−	0.996638i	$-0.473892\pi$
$139$	5.20143e21	0.163858	0.0819290	+	0.996638i	$0.473892\pi$
$140$	0	0
$141$	3.93191e21	0.106611
$142$	0	0
$143$	−1.65286e21	−0.0386552
$144$	0	0
$145$	7.46633e21	0.150919
$146$	0	0
$147$	−2.51877e22	−0.440917
$148$	0	0
$149$	−7.47631e22	−1.13562	−0.567808	−	0.823161i	$-0.692208\pi$
$149$	−7.47631e22	−1.13562	−0.567808	+	0.823161i	$0.692208\pi$
$150$	0	0
$151$	−1.11044e23	−1.46635	−0.733174	−	0.680042i	$-0.761962\pi$
$151$	−1.11044e23	−1.46635	−0.733174	+	0.680042i	$0.761962\pi$
$152$	0	0
$153$	−2.99493e22	−0.344451
$154$	0	0
$155$	−6.96573e21	−0.0699011
$156$	0	0
$157$	4.36563e22	0.382913	0.191457	−	0.981501i	$-0.438679\pi$
$157$	4.36563e22	0.382913	0.191457	+	0.981501i	$0.438679\pi$
$158$	0	0
$159$	2.57113e22	0.197449
$160$	0	0
$161$	−5.66388e22	−0.381454
$162$	0	0
$163$	−2.85661e23	−1.68998	−0.844989	−	0.534783i	$-0.820393\pi$
$163$	−2.85661e23	−1.68998	−0.844989	+	0.534783i	$0.820393\pi$
$164$	0	0
$165$	−2.67780e21	−0.0139379
$166$	0	0
$167$	−2.66950e23	−1.22435	−0.612177	−	0.790720i	$-0.709706\pi$
$167$	−2.66950e23	−1.22435	−0.612177	+	0.790720i	$0.709706\pi$
$168$	0	0
$169$	−2.34216e23	−0.947996
$170$	0	0
$171$	1.01824e23	0.364244
$172$	0	0
$173$	−2.28496e23	−0.723425	−0.361713	−	0.932290i	$-0.617808\pi$
$173$	−2.28496e23	−0.723425	−0.361713	+	0.932290i	$0.617808\pi$
$174$	0	0
$175$	1.69723e23	0.476257
$176$	0	0
$177$	−3.26799e23	−0.813881
$178$	0	0
$179$	1.29151e22	0.0285852	0.0142926	−	0.999898i	$-0.495450\pi$
$179$	1.29151e22	0.0285852	0.0142926	+	0.999898i	$0.495450\pi$
$180$	0	0
$181$	−8.75338e23	−1.72405	−0.862026	−	0.506863i	$-0.830805\pi$
$181$	−8.75338e23	−1.72405	−0.862026	+	0.506863i	$0.830805\pi$
$182$	0	0
$183$	−4.23748e23	−0.743654
$184$	0	0
$185$	−9.57400e22	−0.149896
$186$	0	0
$187$	1.25249e23	0.175161
$188$	0	0
$189$	−7.48011e22	−0.0935532
$190$	0	0
$191$	−1.33961e24	−1.50013	−0.750066	−	0.661363i	$-0.769978\pi$
$191$	−1.33961e24	−1.50013	−0.750066	+	0.661363i	$0.769978\pi$
$192$	0	0
$193$	−2.13970e23	−0.214783	−0.107392	−	0.994217i	$-0.534250\pi$
$193$	−2.13970e23	−0.214783	−0.107392	+	0.994217i	$0.534250\pi$
$194$	0	0
$195$	2.08157e22	0.0187511
$196$	0	0
$197$	1.42895e24	1.15644	0.578220	−	0.815881i	$-0.303748\pi$
$197$	1.42895e24	1.15644	0.578220	+	0.815881i	$0.303748\pi$
$198$	0	0
$199$	−7.45856e23	−0.542872	−0.271436	−	0.962456i	$-0.587499\pi$
$199$	−7.45856e23	−0.542872	−0.271436	+	0.962456i	$0.587499\pi$
$200$	0	0
$201$	9.30344e23	0.609656
$202$	0	0
$203$	−8.72216e23	−0.515129
$204$	0	0
$205$	−3.20970e23	−0.171021
$206$	0	0
$207$	5.43587e23	0.261565
$208$	0	0
$209$	−4.25832e23	−0.185226
$210$	0	0
$211$	2.65090e24	1.04334	0.521671	−	0.853147i	$-0.325309\pi$
$211$	2.65090e24	1.04334	0.521671	+	0.853147i	$0.325309\pi$
$212$	0	0
$213$	−1.56231e24	−0.556905
$214$	0	0
$215$	5.14875e23	0.166379
$216$	0	0
$217$	8.13736e23	0.238592
$218$	0	0
$219$	7.95464e23	0.211815
$220$	0	0
$221$	−9.73614e23	−0.235650
$222$	0	0
$223$	−3.97174e24	−0.874539	−0.437270	−	0.899330i	$-0.644054\pi$
$223$	−3.97174e24	−0.874539	−0.437270	+	0.899330i	$0.644054\pi$
$224$	0	0
$225$	−1.62890e24	−0.326572
$226$	0	0
$227$	2.14690e24	0.392229	0.196114	−	0.980581i	$-0.437168\pi$
$227$	2.14690e24	0.392229	0.196114	+	0.980581i	$0.437168\pi$
$228$	0	0
$229$	3.66024e24	0.609869	0.304934	−	0.952373i	$-0.401365\pi$
$229$	3.66024e24	0.609869	0.304934	+	0.952373i	$0.401365\pi$
$230$	0	0
$231$	3.12820e23	0.0475739
$232$	0	0
$233$	−2.90921e24	−0.404145	−0.202073	−	0.979371i	$-0.564768\pi$
$233$	−2.90921e24	−0.404145	−0.202073	+	0.979371i	$0.564768\pi$
$234$	0	0
$235$	2.07083e23	0.0262985
$236$	0	0
$237$	9.97109e23	0.115847
$238$	0	0
$239$	4.80224e24	0.510818	0.255409	−	0.966833i	$-0.417790\pi$
$239$	4.80224e24	0.510818	0.255409	+	0.966833i	$0.417790\pi$
$240$	0	0
$241$	7.86263e24	0.766282	0.383141	−	0.923690i	$-0.374842\pi$
$241$	7.86263e24	0.766282	0.383141	+	0.923690i	$0.374842\pi$
$242$	0	0
$243$	7.17898e23	0.0641500
$244$	0	0
$245$	−1.32657e24	−0.108764
$246$	0	0
$247$	3.31018e24	0.249191
$248$	0	0
$249$	1.00790e25	0.697144
$250$	0	0
$251$	1.99525e25	1.26889	0.634443	−	0.772969i	$-0.281229\pi$
$251$	1.99525e25	1.26889	0.634443	+	0.772969i	$0.281229\pi$
$252$	0	0
$253$	−2.27330e24	−0.133012
$254$	0	0
$255$	−1.57735e24	−0.0849682
$256$	0	0
$257$	1.33580e25	0.662894	0.331447	−	0.943474i	$-0.392463\pi$
$257$	1.33580e25	0.662894	0.331447	+	0.943474i	$0.392463\pi$
$258$	0	0
$259$	1.11843e25	0.511638
$260$	0	0
$261$	8.37103e24	0.353227
$262$	0	0
$263$	5.83637e24	0.227304	0.113652	−	0.993521i	$-0.463745\pi$
$263$	5.83637e24	0.227304	0.113652	+	0.993521i	$0.463745\pi$
$264$	0	0
$265$	1.35414e24	0.0487060
$266$	0	0
$267$	−1.84583e25	−0.613512
$268$	0	0
$269$	5.35297e25	1.64511	0.822557	−	0.568683i	$-0.192547\pi$
$269$	5.35297e25	1.64511	0.822557	+	0.568683i	$0.192547\pi$
$270$	0	0
$271$	1.04403e25	0.296849	0.148425	−	0.988924i	$-0.452580\pi$
$271$	1.04403e25	0.296849	0.148425	+	0.988924i	$0.452580\pi$
$272$	0	0
$273$	−2.43168e24	−0.0640029
$274$	0	0
$275$	6.81213e24	0.166069
$276$	0	0
$277$	3.14884e25	0.711399	0.355699	−	0.934600i	$-0.384243\pi$
$277$	3.14884e25	0.711399	0.355699	+	0.934600i	$0.384243\pi$
$278$	0	0
$279$	−7.80977e24	−0.163604
$280$	0	0
$281$	−1.15887e25	−0.225225	−0.112613	−	0.993639i	$-0.535922\pi$
$281$	−1.15887e25	−0.225225	−0.112613	+	0.993639i	$0.535922\pi$
$282$	0	0
$283$	4.80399e25	0.866652	0.433326	−	0.901237i	$-0.357340\pi$
$283$	4.80399e25	0.866652	0.433326	+	0.901237i	$0.357340\pi$
$284$	0	0
$285$	5.36281e24	0.0898507
$286$	0	0
$287$	3.74957e25	0.583743
$288$	0	0
$289$	4.68568e24	0.0678180
$290$	0	0
$291$	5.60384e25	0.754413
$292$	0	0
$293$	−7.96714e25	−0.998142	−0.499071	−	0.866561i	$-0.666325\pi$
$293$	−7.96714e25	−0.998142	−0.499071	+	0.866561i	$0.666325\pi$
$294$	0	0
$295$	−1.72116e25	−0.200766
$296$	0	0
$297$	−3.00227e24	−0.0326217
$298$	0	0
$299$	1.76713e25	0.178946
$300$	0	0
$301$	−6.01476e25	−0.567898
$302$	0	0
$303$	8.55018e24	0.0753056
$304$	0	0
$305$	−2.23176e25	−0.183442
$306$	0	0
$307$	−1.51498e26	−1.16266	−0.581332	−	0.813666i	$-0.697468\pi$
$307$	−1.51498e26	−1.16266	−0.581332	+	0.813666i	$0.697468\pi$
$308$	0	0
$309$	−1.29687e26	−0.929681
$310$	0	0
$311$	−1.41406e26	−0.947292	−0.473646	−	0.880715i	$-0.657062\pi$
$311$	−1.41406e26	−0.947292	−0.473646	+	0.880715i	$0.657062\pi$
$312$	0	0
$313$	4.99598e25	0.312899	0.156450	−	0.987686i	$-0.449995\pi$
$313$	4.99598e25	0.312899	0.156450	+	0.987686i	$0.449995\pi$
$314$	0	0
$315$	−3.93957e24	−0.0230774
$316$	0	0
$317$	1.81535e26	0.995033	0.497517	−	0.867454i	$-0.334245\pi$
$317$	1.81535e26	0.995033	0.497517	+	0.867454i	$0.334245\pi$
$318$	0	0
$319$	−3.50079e25	−0.179624
$320$	0	0
$321$	9.63011e24	0.0462731
$322$	0	0
$323$	−2.50835e26	−1.12918
$324$	0	0
$325$	−5.29536e25	−0.223419
$326$	0	0
$327$	1.32773e25	0.0525240
$328$	0	0
$329$	−2.41914e25	−0.0897644
$330$	0	0
$331$	−1.44090e26	−0.501695	−0.250848	−	0.968027i	$-0.580709\pi$
$331$	−1.44090e26	−0.501695	−0.250848	+	0.968027i	$0.580709\pi$
$332$	0	0
$333$	−1.07341e26	−0.350833
$334$	0	0
$335$	4.89987e25	0.150388
$336$	0	0
$337$	−3.63051e26	−1.04678	−0.523388	−	0.852095i	$-0.675332\pi$
$337$	−3.63051e26	−1.04678	−0.523388	+	0.852095i	$0.675332\pi$
$338$	0	0
$339$	−2.50438e26	−0.678583
$340$	0	0
$341$	3.26607e25	0.0831964
$342$	0	0
$343$	3.57891e26	0.857360
$344$	0	0
$345$	2.86292e25	0.0645222
$346$	0	0
$347$	7.09622e25	0.150511	0.0752554	−	0.997164i	$-0.476023\pi$
$347$	7.09622e25	0.150511	0.0752554	+	0.997164i	$0.476023\pi$
$348$	0	0
$349$	−7.03939e26	−1.40562	−0.702810	−	0.711378i	$-0.748072\pi$
$349$	−7.03939e26	−1.40562	−0.702810	+	0.711378i	$0.748072\pi$
$350$	0	0
$351$	2.33379e25	0.0438872
$352$	0	0
$353$	−1.08085e26	−0.191483	−0.0957414	−	0.995406i	$-0.530522\pi$
$353$	−1.08085e26	−0.191483	−0.0957414	+	0.995406i	$0.530522\pi$
$354$	0	0
$355$	−8.22826e25	−0.137376
$356$	0	0
$357$	1.84266e26	0.290020
$358$	0	0
$359$	1.67492e26	0.248601	0.124301	−	0.992245i	$-0.460331\pi$
$359$	1.67492e26	0.248601	0.124301	+	0.992245i	$0.460331\pi$
$360$	0	0
$361$	1.38602e26	0.194064
$362$	0	0
$363$	−4.24422e26	−0.560761
$364$	0	0
$365$	4.18949e25	0.0522500
$366$	0	0
$367$	−9.83667e26	−1.15839	−0.579194	−	0.815190i	$-0.696633\pi$
$367$	−9.83667e26	−1.15839	−0.579194	+	0.815190i	$0.696633\pi$
$368$	0	0
$369$	−3.59863e26	−0.400276
$370$	0	0
$371$	−1.58191e26	−0.166248
$372$	0	0
$373$	1.00058e26	0.0993824	0.0496912	−	0.998765i	$-0.484176\pi$
$373$	1.00058e26	0.0993824	0.0496912	+	0.998765i	$0.484176\pi$
$374$	0	0
$375$	−1.73356e26	−0.162784
$376$	0	0
$377$	2.72131e26	0.241654
$378$	0	0
$379$	−9.23905e25	−0.0776096	−0.0388048	−	0.999247i	$-0.512355\pi$
$379$	−9.23905e25	−0.0776096	−0.0388048	+	0.999247i	$0.512355\pi$
$380$	0	0
$381$	−9.82055e25	−0.0780591
$382$	0	0
$383$	2.13677e27	1.60757	0.803786	−	0.594919i	$-0.202816\pi$
$383$	2.13677e27	1.60757	0.803786	+	0.594919i	$0.202816\pi$
$384$	0	0
$385$	1.64754e25	0.0117354
$386$	0	0
$387$	5.77263e26	0.389411
$388$	0	0
$389$	−1.25581e27	−0.802516	−0.401258	−	0.915965i	$-0.631427\pi$
$389$	−1.25581e27	−0.802516	−0.401258	+	0.915965i	$0.631427\pi$
$390$	0	0
$391$	−1.33908e27	−0.810868
$392$	0	0
$393$	−3.78305e26	−0.217130
$394$	0	0
$395$	5.25150e25	0.0285768
$396$	0	0
$397$	1.78165e27	0.919439	0.459719	−	0.888064i	$-0.347950\pi$
$397$	1.78165e27	0.919439	0.459719	+	0.888064i	$0.347950\pi$
$398$	0	0
$399$	−6.26483e26	−0.306686
$400$	0	0
$401$	1.40181e27	0.651141	0.325570	−	0.945518i	$-0.394444\pi$
$401$	1.40181e27	0.651141	0.325570	+	0.945518i	$0.394444\pi$
$402$	0	0
$403$	−2.53885e26	−0.111927
$404$	0	0
$405$	3.78097e25	0.0158243
$406$	0	0
$407$	4.48903e26	0.178407
$408$	0	0
$409$	2.17493e27	0.821015	0.410507	−	0.911857i	$-0.365352\pi$
$409$	2.17493e27	0.821015	0.410507	+	0.911857i	$0.365352\pi$
$410$	0	0
$411$	−1.17102e27	−0.419978
$412$	0	0
$413$	2.01066e27	0.685271
$414$	0	0
$415$	5.30831e26	0.171969
$416$	0	0
$417$	3.07139e26	0.0946034
$418$	0	0
$419$	6.07636e27	1.77990	0.889952	−	0.456055i	$-0.150738\pi$
$419$	6.07636e27	1.77990	0.889952	+	0.456055i	$0.150738\pi$
$420$	0	0
$421$	−1.89993e27	−0.529389	−0.264695	−	0.964332i	$-0.585271\pi$
$421$	−1.89993e27	−0.529389	−0.264695	+	0.964332i	$0.585271\pi$
$422$	0	0
$423$	2.32175e26	0.0615520
$424$	0	0
$425$	4.01267e27	1.01239
$426$	0	0
$427$	2.60714e27	0.626141
$428$	0	0
$429$	−9.75999e25	−0.0223176
$430$	0	0
$431$	8.08572e24	0.00176079	0.000880395	−	1.00000i	$-0.499720\pi$
$431$	8.08572e24	0.00176079	0.000880395		1.00000i	$0.499720\pi$
$432$	0	0
$433$	−5.60439e27	−1.16253	−0.581267	−	0.813713i	$-0.697443\pi$
$433$	−5.60439e27	−1.16253	−0.581267	+	0.813713i	$0.697443\pi$
$434$	0	0
$435$	4.40879e26	0.0871330
$436$	0	0
$437$	4.55272e27	0.857463
$438$	0	0
$439$	8.51110e27	1.52795	0.763973	−	0.645248i	$-0.223246\pi$
$439$	8.51110e27	1.52795	0.763973	+	0.645248i	$0.223246\pi$
$440$	0	0
$441$	−1.48731e27	−0.254563
$442$	0	0
$443$	6.63134e27	1.08234	0.541168	−	0.840915i	$-0.317982\pi$
$443$	6.63134e27	1.08234	0.541168	+	0.840915i	$0.317982\pi$
$444$	0	0
$445$	−9.72147e26	−0.151339
$446$	0	0
$447$	−4.41468e27	−0.655648
$448$	0	0
$449$	1.30394e28	1.84787	0.923933	−	0.382555i	$-0.124956\pi$
$449$	1.30394e28	1.84787	0.923933	+	0.382555i	$0.124956\pi$
$450$	0	0
$451$	1.50496e27	0.203549
$452$	0	0
$453$	−6.55703e27	−0.846596
$454$	0	0
$455$	−1.28070e26	−0.0157880
$456$	0	0
$457$	4.72949e27	0.556793	0.278397	−	0.960466i	$-0.410197\pi$
$457$	4.72949e27	0.556793	0.278397	+	0.960466i	$0.410197\pi$
$458$	0	0
$459$	−1.76848e27	−0.198869
$460$	0	0
$461$	4.92722e27	0.529349	0.264675	−	0.964338i	$-0.414735\pi$
$461$	4.92722e27	0.529349	0.264675	+	0.964338i	$0.414735\pi$
$462$	0	0
$463$	−1.20207e28	−1.23404	−0.617021	−	0.786947i	$-0.711661\pi$
$463$	−1.20207e28	−1.23404	−0.617021	+	0.786947i	$0.711661\pi$
$464$	0	0
$465$	−4.11319e26	−0.0403574
$466$	0	0
$467$	−1.09969e28	−1.03144	−0.515719	−	0.856758i	$-0.672475\pi$
$467$	−1.09969e28	−1.03144	−0.515719	+	0.856758i	$0.672475\pi$
$468$	0	0
$469$	−5.72402e27	−0.513317
$470$	0	0
$471$	2.57786e27	0.221075
$472$	0	0
$473$	−2.41413e27	−0.198024
$474$	0	0
$475$	−1.36426e28	−1.07057
$476$	0	0
$477$	1.51823e27	0.113997
$478$	0	0
$479$	1.32717e28	0.953684	0.476842	−	0.878989i	$-0.341781\pi$
$479$	1.32717e28	0.953684	0.476842	+	0.878989i	$0.341781\pi$
$480$	0	0
$481$	−3.48951e27	−0.240017
$482$	0	0
$483$	−3.34446e27	−0.220233
$484$	0	0
$485$	2.95139e27	0.186096
$486$	0	0
$487$	−2.62576e28	−1.58563	−0.792813	−	0.609466i	$-0.791384\pi$
$487$	−2.62576e28	−1.58563	−0.792813	+	0.609466i	$0.791384\pi$
$488$	0	0
$489$	−1.68680e28	−0.975710
$490$	0	0
$491$	2.54066e28	1.40796	0.703982	−	0.710218i	$-0.251404\pi$
$491$	2.54066e28	1.40796	0.703982	+	0.710218i	$0.251404\pi$
$492$	0	0
$493$	−2.06213e28	−1.09502
$494$	0	0
$495$	−1.58121e26	−0.00804704
$496$	0	0
$497$	9.61224e27	0.468903
$498$	0	0
$499$	1.30048e28	0.608204	0.304102	−	0.952640i	$-0.401644\pi$
$499$	1.30048e28	0.608204	0.304102	+	0.952640i	$0.401644\pi$
$500$	0	0
$501$	−1.57631e28	−0.706882
$502$	0	0
$503$	−1.34993e27	−0.0580559	−0.0290280	−	0.999579i	$-0.509241\pi$
$503$	−1.34993e27	−0.0580559	−0.0290280	+	0.999579i	$0.509241\pi$
$504$	0	0
$505$	4.50314e26	0.0185762
$506$	0	0
$507$	−1.38302e28	−0.547326
$508$	0	0
$509$	−4.04902e28	−1.53749	−0.768746	−	0.639554i	$-0.779119\pi$
$509$	−4.04902e28	−1.53749	−0.768746	+	0.639554i	$0.779119\pi$
$510$	0	0
$511$	−4.89416e27	−0.178344
$512$	0	0
$513$	6.01263e27	0.210296
$514$	0	0
$515$	−6.83028e27	−0.229331
$516$	0	0
$517$	−9.70964e26	−0.0313006
$518$	0	0
$519$	−1.34924e28	−0.417670
$520$	0	0
$521$	5.40378e28	1.60658	0.803289	−	0.595590i	$-0.203082\pi$
$521$	5.40378e28	1.60658	0.803289	+	0.595590i	$0.203082\pi$
$522$	0	0
$523$	−1.54066e28	−0.439988	−0.219994	−	0.975501i	$-0.570604\pi$
$523$	−1.54066e28	−0.439988	−0.219994	+	0.975501i	$0.570604\pi$
$524$	0	0
$525$	1.00220e28	0.274967
$526$	0	0
$527$	1.92387e28	0.507182
$528$	0	0
$529$	−1.51670e28	−0.384252
$530$	0	0
$531$	−1.92971e28	−0.469895
$532$	0	0
$533$	−1.16987e28	−0.273842
$534$	0	0
$535$	5.07192e26	0.0114145
$536$	0	0
$537$	7.62623e26	0.0165037
$538$	0	0
$539$	6.21997e27	0.129451
$540$	0	0
$541$	−7.54478e28	−1.51034	−0.755171	−	0.655528i	$-0.772446\pi$
$541$	−7.54478e28	−1.51034	−0.755171	+	0.655528i	$0.772446\pi$
$542$	0	0
$543$	−5.16878e28	−0.995382
$544$	0	0
$545$	6.99279e26	0.0129565
$546$	0	0
$547$	−7.90524e28	−1.40944	−0.704722	−	0.709483i	$-0.748928\pi$
$547$	−7.90524e28	−1.40944	−0.704722	+	0.709483i	$0.748928\pi$
$548$	0	0
$549$	−2.50219e28	−0.429349
$550$	0	0
$551$	7.01101e28	1.15795
$552$	0	0
$553$	−6.13480e27	−0.0975408
$554$	0	0
$555$	−5.65335e27	−0.0865426
$556$	0	0
$557$	1.17729e29	1.73541	0.867707	−	0.497075i	$-0.165593\pi$
$557$	1.17729e29	1.73541	0.867707	+	0.497075i	$0.165593\pi$
$558$	0	0
$559$	1.87660e28	0.266409
$560$	0	0
$561$	7.39583e27	0.101129
$562$	0	0
$563$	9.20807e28	1.21291	0.606457	−	0.795116i	$-0.292590\pi$
$563$	9.20807e28	1.21291	0.606457	+	0.795116i	$0.292590\pi$
$564$	0	0
$565$	−1.31899e28	−0.167391
$566$	0	0
$567$	−4.41693e27	−0.0540130
$568$	0	0
$569$	−2.71795e27	−0.0320304	−0.0160152	−	0.999872i	$-0.505098\pi$
$569$	−2.71795e27	−0.0320304	−0.0160152	+	0.999872i	$0.505098\pi$
$570$	0	0
$571$	−1.28086e28	−0.145487	−0.0727434	−	0.997351i	$-0.523175\pi$
$571$	−1.28086e28	−0.145487	−0.0727434	+	0.997351i	$0.523175\pi$
$572$	0	0
$573$	−7.91029e28	−0.866102
$574$	0	0
$575$	−7.28307e28	−0.768780
$576$	0	0
$577$	1.49329e29	1.51984	0.759922	−	0.650015i	$-0.225237\pi$
$577$	1.49329e29	1.51984	0.759922	+	0.650015i	$0.225237\pi$
$578$	0	0
$579$	−1.26347e28	−0.124005
$580$	0	0
$581$	−6.20116e28	−0.586980
$582$	0	0
$583$	−6.34926e27	−0.0579700
$584$	0	0
$585$	1.22914e27	0.0108260
$586$	0	0
$587$	−2.62975e28	−0.223468	−0.111734	−	0.993738i	$-0.535640\pi$
$587$	−2.62975e28	−0.223468	−0.111734	+	0.993738i	$0.535640\pi$
$588$	0	0
$589$	−6.54093e28	−0.536326
$590$	0	0
$591$	8.43783e28	0.667671
$592$	0	0
$593$	1.92294e29	1.46856	0.734278	−	0.678849i	$-0.237521\pi$
$593$	1.92294e29	1.46856	0.734278	+	0.678849i	$0.237521\pi$
$594$	0	0
$595$	9.70478e27	0.0715414
$596$	0	0
$597$	−4.40421e28	−0.313428
$598$	0	0
$599$	2.45874e29	1.68940	0.844699	−	0.535242i	$-0.179780\pi$
$599$	2.45874e29	1.68940	0.844699	+	0.535242i	$0.179780\pi$
$600$	0	0
$601$	7.47252e28	0.495776	0.247888	−	0.968789i	$-0.420264\pi$
$601$	7.47252e28	0.495776	0.247888	+	0.968789i	$0.420264\pi$
$602$	0	0
$603$	5.49359e28	0.351985
$604$	0	0
$605$	−2.23531e28	−0.138327
$606$	0	0
$607$	1.23466e29	0.738016	0.369008	−	0.929426i	$-0.379698\pi$
$607$	1.23466e29	0.738016	0.369008	+	0.929426i	$0.379698\pi$
$608$	0	0
$609$	−5.15035e28	−0.297410
$610$	0	0
$611$	7.54771e27	0.0421098
$612$	0	0
$613$	−1.59244e29	−0.858476	−0.429238	−	0.903191i	$-0.641218\pi$
$613$	−1.59244e29	−0.858476	−0.429238	+	0.903191i	$0.641218\pi$
$614$	0	0
$615$	−1.89530e28	−0.0987390
$616$	0	0
$617$	1.23256e29	0.620602	0.310301	−	0.950638i	$-0.399570\pi$
$617$	1.23256e29	0.620602	0.310301	+	0.950638i	$0.399570\pi$
$618$	0	0
$619$	5.18990e28	0.252585	0.126292	−	0.991993i	$-0.459692\pi$
$619$	5.18990e28	0.252585	0.126292	+	0.991993i	$0.459692\pi$
$620$	0	0
$621$	3.20983e28	0.151015
$622$	0	0
$623$	1.13566e29	0.516564
$624$	0	0
$625$	2.13632e29	0.939562
$626$	0	0
$627$	−2.51450e28	−0.106940
$628$	0	0
$629$	2.64425e29	1.08760
$630$	0	0
$631$	2.05208e28	0.0816366	0.0408183	−	0.999167i	$-0.487004\pi$
$631$	2.05208e28	0.0816366	0.0408183	+	0.999167i	$0.487004\pi$
$632$	0	0
$633$	1.56533e29	0.602374
$634$	0	0
$635$	−5.17222e27	−0.0192554
$636$	0	0
$637$	−4.83505e28	−0.174155
$638$	0	0
$639$	−9.22528e28	−0.321529
$640$	0	0
$641$	5.14342e29	1.73477	0.867386	−	0.497636i	$-0.165798\pi$
$641$	5.14342e29	1.73477	0.867386	+	0.497636i	$0.165798\pi$
$642$	0	0
$643$	8.85766e28	0.289137	0.144569	−	0.989495i	$-0.453821\pi$
$643$	8.85766e28	0.289137	0.144569	+	0.989495i	$0.453821\pi$
$644$	0	0
$645$	3.04028e28	0.0960588
$646$	0	0
$647$	−5.46916e29	−1.67273	−0.836364	−	0.548174i	$-0.815323\pi$
$647$	−5.46916e29	−1.67273	−0.836364	+	0.548174i	$0.815323\pi$
$648$	0	0
$649$	8.07012e28	0.238952
$650$	0	0
$651$	4.80503e28	0.137751
$652$	0	0
$653$	−5.66153e29	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$653$	−5.66153e29	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$654$	0	0
$655$	−1.99243e28	−0.0535611
$656$	0	0
$657$	4.69713e28	0.122292
$658$	0	0
$659$	1.48653e29	0.374867	0.187434	−	0.982277i	$-0.439983\pi$
$659$	1.48653e29	0.374867	0.187434	+	0.982277i	$0.439983\pi$
$660$	0	0
$661$	−4.04669e29	−0.988517	−0.494259	−	0.869315i	$-0.664560\pi$
$661$	−4.04669e29	−0.988517	−0.494259	+	0.869315i	$0.664560\pi$
$662$	0	0
$663$	−5.74909e28	−0.136053
$664$	0	0
$665$	−3.29951e28	−0.0756524
$666$	0	0
$667$	3.74281e29	0.831527
$668$	0	0
$669$	−2.34527e29	−0.504916
$670$	0	0
$671$	1.04642e29	0.218334
$672$	0	0
$673$	−1.54590e29	−0.312625	−0.156313	−	0.987708i	$-0.549961\pi$
$673$	−1.54590e29	−0.312625	−0.156313	+	0.987708i	$0.549961\pi$
$674$	0	0
$675$	−9.61852e28	−0.188547
$676$	0	0
$677$	−9.88421e29	−1.87828	−0.939142	−	0.343530i	$-0.888377\pi$
$677$	−9.88421e29	−1.87828	−0.939142	+	0.343530i	$0.888377\pi$
$678$	0	0
$679$	−3.44781e29	−0.635200
$680$	0	0
$681$	1.26772e29	0.226453
$682$	0	0
$683$	1.41169e29	0.244524	0.122262	−	0.992498i	$-0.460985\pi$
$683$	1.41169e29	0.244524	0.122262	+	0.992498i	$0.460985\pi$
$684$	0	0
$685$	−6.16746e28	−0.103599
$686$	0	0
$687$	2.16133e29	0.352108
$688$	0	0
$689$	4.93555e28	0.0779891
$690$	0	0
$691$	−7.11585e29	−1.09070	−0.545352	−	0.838207i	$-0.683604\pi$
$691$	−7.11585e29	−1.09070	−0.545352	+	0.838207i	$0.683604\pi$
$692$	0	0
$693$	1.84717e28	0.0274668
$694$	0	0
$695$	1.61762e28	0.0233365
$696$	0	0
$697$	8.86490e29	1.24088
$698$	0	0
$699$	−1.71786e29	−0.233334
$700$	0	0
$701$	−8.80754e29	−1.16096	−0.580478	−	0.814276i	$-0.697134\pi$
$701$	−8.80754e29	−1.16096	−0.580478	+	0.814276i	$0.697134\pi$
$702$	0	0
$703$	−8.99015e29	−1.15010
$704$	0	0
$705$	1.22280e28	0.0151835
$706$	0	0
$707$	−5.26057e28	−0.0634058
$708$	0	0
$709$	−2.79000e29	−0.326452	−0.163226	−	0.986589i	$-0.552190\pi$
$709$	−2.79000e29	−0.326452	−0.163226	+	0.986589i	$0.552190\pi$
$710$	0	0
$711$	5.88783e28	0.0668843
$712$	0	0
$713$	−3.49186e29	−0.385139
$714$	0	0
$715$	−5.14032e27	−0.00550524
$716$	0	0
$717$	2.83568e29	0.294921
$718$	0	0
$719$	−1.21404e30	−1.22625	−0.613124	−	0.789987i	$-0.710087\pi$
$719$	−1.21404e30	−1.22625	−0.613124	+	0.789987i	$0.710087\pi$
$720$	0	0
$721$	7.97913e29	0.782772
$722$	0	0
$723$	4.64280e29	0.442413
$724$	0	0
$725$	−1.12157e30	−1.03819
$726$	0	0
$727$	6.54831e29	0.588868	0.294434	−	0.955672i	$-0.404869\pi$
$727$	6.54831e29	0.588868	0.294434	+	0.955672i	$0.404869\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−1.42204e30	−1.20720
$732$	0	0
$733$	2.20665e29	0.182029	0.0910147	−	0.995850i	$-0.470989\pi$
$733$	2.20665e29	0.182029	0.0910147	+	0.995850i	$0.470989\pi$
$734$	0	0
$735$	−7.83325e28	−0.0627950
$736$	0	0
$737$	−2.29743e29	−0.178992
$738$	0	0
$739$	4.07297e29	0.308422	0.154211	−	0.988038i	$-0.450717\pi$
$739$	4.07297e29	0.308422	0.154211	+	0.988038i	$0.450717\pi$
$740$	0	0
$741$	1.95463e29	0.143871
$742$	0	0
$743$	−3.97218e29	−0.284214	−0.142107	−	0.989851i	$-0.545388\pi$
$743$	−3.97218e29	−0.284214	−0.142107	+	0.989851i	$0.545388\pi$
$744$	0	0
$745$	−2.32509e29	−0.161733
$746$	0	0
$747$	5.95152e29	0.402496
$748$	0	0
$749$	−5.92500e28	−0.0389610
$750$	0	0
$751$	1.86890e30	1.19500	0.597498	−	0.801871i	$-0.296162\pi$
$751$	1.86890e30	1.19500	0.597498	+	0.801871i	$0.296162\pi$
$752$	0	0
$753$	1.17817e30	0.732592
$754$	0	0
$755$	−3.45341e29	−0.208836
$756$	0	0
$757$	−2.99103e30	−1.75919	−0.879597	−	0.475720i	$-0.842188\pi$
$757$	−2.99103e30	−1.75919	−0.879597	+	0.475720i	$0.842188\pi$
$758$	0	0
$759$	−1.34236e29	−0.0767945
$760$	0	0
$761$	1.51341e30	0.842206	0.421103	−	0.907013i	$-0.361643\pi$
$761$	1.51341e30	0.842206	0.421103	+	0.907013i	$0.361643\pi$
$762$	0	0
$763$	−8.16896e28	−0.0442241
$764$	0	0
$765$	−9.31410e28	−0.0490564
$766$	0	0
$767$	−6.27325e29	−0.321470
$768$	0	0
$769$	2.53401e30	1.26352	0.631759	−	0.775165i	$-0.282333\pi$
$769$	2.53401e30	1.26352	0.631759	+	0.775165i	$0.282333\pi$
$770$	0	0
$771$	7.88777e29	0.382722
$772$	0	0
$773$	−1.69545e30	−0.800571	−0.400285	−	0.916390i	$-0.631089\pi$
$773$	−1.69545e30	−0.800571	−0.400285	+	0.916390i	$0.631089\pi$
$774$	0	0
$775$	1.04637e30	0.480857
$776$	0	0
$777$	6.60424e29	0.295394
$778$	0	0
$779$	−3.01396e30	−1.31218
$780$	0	0
$781$	3.85804e29	0.163505
$782$	0	0
$783$	4.94301e29	0.203936
$784$	0	0
$785$	1.35769e29	0.0545342
$786$	0	0
$787$	1.87332e30	0.732616	0.366308	−	0.930494i	$-0.380622\pi$
$787$	1.87332e30	0.732616	0.366308	+	0.930494i	$0.380622\pi$
$788$	0	0
$789$	3.44632e29	0.131234
$790$	0	0
$791$	1.54084e30	0.571353
$792$	0	0
$793$	−8.13429e29	−0.293732
$794$	0	0
$795$	7.99608e28	0.0281204
$796$	0	0
$797$	−7.79664e29	−0.267051	−0.133526	−	0.991045i	$-0.542630\pi$
$797$	−7.79664e29	−0.267051	−0.133526	+	0.991045i	$0.542630\pi$
$798$	0	0
$799$	−5.71944e29	−0.190815
$800$	0	0
$801$	−1.08994e30	−0.354211
$802$	0	0
$803$	−1.96436e29	−0.0621880
$804$	0	0
$805$	−1.76144e29	−0.0543264
$806$	0	0
$807$	3.16088e30	0.949807
$808$	0	0
$809$	8.82262e29	0.258308	0.129154	−	0.991625i	$-0.458774\pi$
$809$	8.82262e29	0.258308	0.129154	+	0.991625i	$0.458774\pi$
$810$	0	0
$811$	2.06044e30	0.587815	0.293907	−	0.955834i	$-0.405044\pi$
$811$	2.06044e30	0.587815	0.293907	+	0.955834i	$0.405044\pi$
$812$	0	0
$813$	6.16490e29	0.171386
$814$	0	0
$815$	−8.88392e29	−0.240685
$816$	0	0
$817$	4.83476e30	1.27657
$818$	0	0
$819$	−1.43589e29	−0.0369521
$820$	0	0
$821$	−1.83846e30	−0.461160	−0.230580	−	0.973053i	$-0.574062\pi$
$821$	−1.83846e30	−0.461160	−0.230580	+	0.973053i	$0.574062\pi$
$822$	0	0
$823$	−7.73766e29	−0.189196	−0.0945979	−	0.995516i	$-0.530157\pi$
$823$	−7.73766e29	−0.189196	−0.0945979	+	0.995516i	$0.530157\pi$
$824$	0	0
$825$	4.02249e29	0.0958802
$826$	0	0
$827$	4.59989e30	1.06891	0.534453	−	0.845198i	$-0.320518\pi$
$827$	4.59989e30	1.06891	0.534453	+	0.845198i	$0.320518\pi$
$828$	0	0
$829$	7.93000e30	1.79660	0.898298	−	0.439386i	$-0.144804\pi$
$829$	7.93000e30	1.79660	0.898298	+	0.439386i	$0.144804\pi$
$830$	0	0
$831$	1.85936e30	0.410726
$832$	0	0
$833$	3.66386e30	0.789162
$834$	0	0
$835$	−8.30202e29	−0.174372
$836$	0	0
$837$	−4.61159e29	−0.0944569
$838$	0	0
$839$	−4.84033e30	−0.966884	−0.483442	−	0.875376i	$-0.660614\pi$
$839$	−4.84033e30	−0.966884	−0.483442	+	0.875376i	$0.660614\pi$
$840$	0	0
$841$	6.30944e29	0.122923
$842$	0	0
$843$	−6.84300e29	−0.130034
$844$	0	0
$845$	−7.28400e29	−0.135013
$846$	0	0
$847$	2.61129e30	0.472149
$848$	0	0
$849$	2.83671e30	0.500362
$850$	0	0
$851$	−4.79937e30	−0.825893
$852$	0	0
$853$	2.96903e30	0.498483	0.249242	−	0.968441i	$-0.419819\pi$
$853$	2.96903e30	0.498483	0.249242	+	0.968441i	$0.419819\pi$
$854$	0	0
$855$	3.16669e29	0.0518753
$856$	0	0
$857$	−3.70009e30	−0.591444	−0.295722	−	0.955274i	$-0.595560\pi$
$857$	−3.70009e30	−0.591444	−0.295722	+	0.955274i	$0.595560\pi$
$858$	0	0
$859$	−7.75385e29	−0.120945	−0.0604726	−	0.998170i	$-0.519261\pi$
$859$	−7.75385e29	−0.120945	−0.0604726	+	0.998170i	$0.519261\pi$
$860$	0	0
$861$	2.21408e30	0.337024
$862$	0	0
$863$	1.29544e31	1.92443	0.962217	−	0.272283i	$-0.0877786\pi$
$863$	1.29544e31	1.92443	0.962217	+	0.272283i	$0.0877786\pi$
$864$	0	0
$865$	−7.10610e29	−0.103030
$866$	0	0
$867$	2.76685e29	0.0391548
$868$	0	0
$869$	−2.46231e29	−0.0340122
$870$	0	0
$871$	1.78589e30	0.240805
$872$	0	0
$873$	3.30901e30	0.435560
$874$	0	0
$875$	1.06659e30	0.137060
$876$	0	0
$877$	−1.57355e31	−1.97417	−0.987083	−	0.160207i	$-0.948784\pi$
$877$	−1.57355e31	−1.97417	−0.987083	+	0.160207i	$0.948784\pi$
$878$	0	0
$879$	−4.70452e30	−0.576278
$880$	0	0
$881$	−1.47526e31	−1.76450	−0.882252	−	0.470778i	$-0.843973\pi$
$881$	−1.47526e31	−1.76450	−0.882252	+	0.470778i	$0.843973\pi$
$882$	0	0
$883$	5.64453e30	0.659235	0.329617	−	0.944115i	$-0.393080\pi$
$883$	5.64453e30	0.659235	0.329617	+	0.944115i	$0.393080\pi$
$884$	0	0
$885$	−1.01633e30	−0.115912
$886$	0	0
$887$	−5.89300e30	−0.656354	−0.328177	−	0.944616i	$-0.606434\pi$
$887$	−5.89300e30	−0.656354	−0.328177	+	0.944616i	$0.606434\pi$
$888$	0	0
$889$	6.04218e29	0.0657242
$890$	0	0
$891$	−1.77281e29	−0.0188342
$892$	0	0
$893$	1.94454e30	0.201780
$894$	0	0
$895$	4.01653e28	0.00407107
$896$	0	0
$897$	1.04347e30	0.103314
$898$	0	0
$899$	−5.37734e30	−0.520104
$900$	0	0
$901$	−3.74002e30	−0.353397
$902$	0	0
$903$	−3.55166e30	−0.327876
$904$	0	0
$905$	−2.72226e30	−0.245538
$906$	0	0
$907$	−8.32782e30	−0.733931	−0.366965	−	0.930235i	$-0.619603\pi$
$907$	−8.32782e30	−0.733931	−0.366965	+	0.930235i	$0.619603\pi$
$908$	0	0
$909$	5.04879e29	0.0434777
$910$	0	0
$911$	−1.08086e31	−0.909548	−0.454774	−	0.890607i	$-0.650280\pi$
$911$	−1.08086e31	−0.909548	−0.454774	+	0.890607i	$0.650280\pi$
$912$	0	0
$913$	−2.48894e30	−0.204678
$914$	0	0
$915$	−1.31783e30	−0.105911
$916$	0	0
$917$	2.32755e30	0.182819
$918$	0	0
$919$	−6.55205e30	−0.502996	−0.251498	−	0.967858i	$-0.580923\pi$
$919$	−6.55205e30	−0.502996	−0.251498	+	0.967858i	$0.580923\pi$
$920$	0	0
$921$	−8.94582e30	−0.671265
$922$	0	0
$923$	−2.99902e30	−0.219969
$924$	0	0
$925$	1.43817e31	1.03115
$926$	0	0
$927$	−7.65791e30	−0.536752
$928$	0	0
$929$	1.04036e31	0.712883	0.356442	−	0.934318i	$-0.383990\pi$
$929$	1.04036e31	0.712883	0.356442	+	0.934318i	$0.383990\pi$
$930$	0	0
$931$	−1.24567e31	−0.834509
$932$	0	0
$933$	−8.34989e30	−0.546919
$934$	0	0
$935$	3.89518e29	0.0249463
$936$	0	0
$937$	−2.09831e31	−1.31402	−0.657012	−	0.753880i	$-0.728180\pi$
$937$	−2.09831e31	−1.31402	−0.657012	+	0.753880i	$0.728180\pi$
$938$	0	0
$939$	2.95007e30	0.180653
$940$	0	0
$941$	2.20967e31	1.32323	0.661617	−	0.749842i	$-0.269870\pi$
$941$	2.20967e31	1.32323	0.661617	+	0.749842i	$0.269870\pi$
$942$	0	0
$943$	−1.60900e31	−0.942286
$944$	0	0
$945$	−2.32628e29	−0.0133238
$946$	0	0
$947$	2.59604e31	1.45424	0.727121	−	0.686509i	$-0.240858\pi$
$947$	2.59604e31	1.45424	0.727121	+	0.686509i	$0.240858\pi$
$948$	0	0
$949$	1.52698e30	0.0836637
$950$	0	0
$951$	1.07194e31	0.574483
$952$	0	0
$953$	−2.94729e31	−1.54507	−0.772535	−	0.634973i	$-0.781011\pi$
$953$	−2.94729e31	−1.54507	−0.772535	+	0.634973i	$0.781011\pi$
$954$	0	0
$955$	−4.16613e30	−0.213648
$956$	0	0
$957$	−2.06718e30	−0.103706
$958$	0	0
$959$	7.20482e30	0.353612
$960$	0	0
$961$	−1.58087e31	−0.759103
$962$	0	0
$963$	5.68648e29	0.0267158
$964$	0	0
$965$	−6.65436e29	−0.0305893
$966$	0	0
$967$	−6.98077e30	−0.313997	−0.156998	−	0.987599i	$-0.550182\pi$
$967$	−6.98077e30	−0.313997	−0.156998	+	0.987599i	$0.550182\pi$
$968$	0	0
$969$	−1.48116e31	−0.651931
$970$	0	0
$971$	1.30234e30	0.0560946	0.0280473	−	0.999607i	$-0.491071\pi$
$971$	1.30234e30	0.0560946	0.0280473	+	0.999607i	$0.491071\pi$
$972$	0	0
$973$	−1.88970e30	−0.0796541
$974$	0	0
$975$	−3.12685e30	−0.128991
$976$	0	0
$977$	3.06599e31	1.23788	0.618939	−	0.785439i	$-0.287563\pi$
$977$	3.06599e31	1.23788	0.618939	+	0.785439i	$0.287563\pi$
$978$	0	0
$979$	4.55817e30	0.180124
$980$	0	0
$981$	7.84011e29	0.0303248
$982$	0	0
$983$	−8.31325e30	−0.314745	−0.157373	−	0.987539i	$-0.550302\pi$
$983$	−8.31325e30	−0.314745	−0.157373	+	0.987539i	$0.550302\pi$
$984$	0	0
$985$	4.44398e30	0.164699
$986$	0	0
$987$	−1.42848e30	−0.0518255
$988$	0	0
$989$	2.58102e31	0.916708
$990$	0	0
$991$	1.32568e31	0.460962	0.230481	−	0.973077i	$-0.425970\pi$
$991$	1.32568e31	0.460962	0.230481	+	0.973077i	$0.425970\pi$
$992$	0	0
$993$	−8.50837e30	−0.289654
$994$	0	0
$995$	−2.31958e30	−0.0773154
$996$	0	0
$997$	3.42048e31	1.11632	0.558159	−	0.829734i	$-0.311508\pi$
$997$	3.42048e31	1.11632	0.558159	+	0.829734i	$0.311508\pi$
$998$	0	0
$999$	−6.33837e30	−0.202554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.e.1.1		1
4.3	odd	2		3.22.a.a.1.1	✓	1
12.11	even	2		9.22.a.d.1.1		1
20.3	even	4		75.22.b.a.49.2		2
20.7	even	4		75.22.b.a.49.1		2
20.19	odd	2		75.22.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.a.1.1	✓	1	4.3	odd	2
9.22.a.d.1.1		1	12.11	even	2
48.22.a.e.1.1		1	1.1	even	1	trivial
75.22.a.c.1.1		1	20.19	odd	2
75.22.b.a.49.1		2	20.7	even	4
75.22.b.a.49.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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