LMFDB - Embedded newform 4140.2.a.q.1.2

Newspace parameters

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$33.0580664368$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

$f(q)$	$=$	$q+1.00000 q^{5} +1.44949 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.44949 q^{7} -2.44949 q^{11} -0.449490 q^{13} -0.550510 q^{17} -6.44949 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.89898 q^{29} -7.00000 q^{31} +1.44949 q^{35} -8.34847 q^{37} +1.89898 q^{41} +0.898979 q^{43} -2.44949 q^{47} -4.89898 q^{49} -10.3485 q^{53} -2.44949 q^{55} -7.89898 q^{59} +9.34847 q^{61} -0.449490 q^{65} -3.44949 q^{67} -3.00000 q^{71} -5.34847 q^{73} -3.55051 q^{77} -4.00000 q^{79} -4.34847 q^{83} -0.550510 q^{85} +7.10102 q^{89} -0.651531 q^{91} -6.44949 q^{95} -5.10102 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 8 q^{19} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 14 q^{31} - 2 q^{35} - 2 q^{37} - 6 q^{41} - 8 q^{43} - 6 q^{53} - 6 q^{59} + 4 q^{61} + 4 q^{65} - 2 q^{67} - 6 q^{71} + 4 q^{73} - 12 q^{77} - 8 q^{79} + 6 q^{83} - 6 q^{85} + 24 q^{89} - 16 q^{91} - 8 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 8 * q^19 - 2 * q^23 + 2 * q^25 + 6 * q^29 - 14 * q^31 - 2 * q^35 - 2 * q^37 - 6 * q^41 - 8 * q^43 - 6 * q^53 - 6 * q^59 + 4 * q^61 + 4 * q^65 - 2 * q^67 - 6 * q^71 + 4 * q^73 - 12 * q^77 - 8 * q^79 + 6 * q^83 - 6 * q^85 + 24 * q^89 - 16 * q^91 - 8 * q^95 - 20 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.44949	0.547856	0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949	0.547856	0.273928	+	0.961750i	$0.411677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	−0.449490	−0.124666	−0.0623330	−	0.998055i	$-0.519854\pi$
$13$	−0.449490	−0.124666	−0.0623330	+	0.998055i	$0.519854\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.550510	−0.133518	−0.0667592	−	0.997769i	$-0.521266\pi$
$17$	−0.550510	−0.133518	−0.0667592	+	0.997769i	$0.521266\pi$
$18$	0	0
$19$	−6.44949	−1.47961	−0.739807	−	0.672819i	$-0.765083\pi$
$19$	−6.44949	−1.47961	−0.739807	+	0.672819i	$0.765083\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.89898	1.46680	0.733402	−	0.679795i	$-0.237931\pi$
$29$	7.89898	1.46680	0.733402	+	0.679795i	$0.237931\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.44949	0.245008
$36$	0	0
$37$	−8.34847	−1.37248	−0.686240	−	0.727375i	$-0.740740\pi$
$37$	−8.34847	−1.37248	−0.686240	+	0.727375i	$0.740740\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.89898	0.296571	0.148285	−	0.988945i	$-0.452625\pi$
$41$	1.89898	0.296571	0.148285	+	0.988945i	$0.452625\pi$
$42$	0	0
$43$	0.898979	0.137093	0.0685465	−	0.997648i	$-0.478164\pi$
$43$	0.898979	0.137093	0.0685465	+	0.997648i	$0.478164\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.44949	−0.357295	−0.178647	−	0.983913i	$-0.557172\pi$
$47$	−2.44949	−0.357295	−0.178647	+	0.983913i	$0.557172\pi$
$48$	0	0
$49$	−4.89898	−0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.3485	−1.42147	−0.710736	−	0.703459i	$-0.751638\pi$
$53$	−10.3485	−1.42147	−0.710736	+	0.703459i	$0.751638\pi$
$54$	0	0
$55$	−2.44949	−0.330289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.89898	−1.02836	−0.514180	−	0.857682i	$-0.671904\pi$
$59$	−7.89898	−1.02836	−0.514180	+	0.857682i	$0.671904\pi$
$60$	0	0
$61$	9.34847	1.19695	0.598474	−	0.801142i	$-0.295774\pi$
$61$	9.34847	1.19695	0.598474	+	0.801142i	$0.295774\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.449490	−0.0557523
$66$	0	0
$67$	−3.44949	−0.421422	−0.210711	−	0.977548i	$-0.567578\pi$
$67$	−3.44949	−0.421422	−0.210711	+	0.977548i	$0.567578\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−5.34847	−0.625991	−0.312995	−	0.949755i	$-0.601332\pi$
$73$	−5.34847	−0.625991	−0.312995	+	0.949755i	$0.601332\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.55051	−0.404618
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−4.34847	−0.477307	−0.238653	−	0.971105i	$-0.576706\pi$
$83$	−4.34847	−0.477307	−0.238653	+	0.971105i	$0.576706\pi$
$84$	0	0
$85$	−0.550510	−0.0597112
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.10102	0.752707	0.376353	−	0.926476i	$-0.377178\pi$
$89$	7.10102	0.752707	0.376353	+	0.926476i	$0.377178\pi$
$90$	0	0
$91$	−0.651531	−0.0682990
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.44949	−0.661704
$96$	0	0
$97$	−5.10102	−0.517930	−0.258965	−	0.965887i	$-0.583381\pi$
$97$	−5.10102	−0.517930	−0.258965	+	0.965887i	$0.583381\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.797959	0.0793999	0.0396999	−	0.999212i	$-0.487360\pi$
$101$	0.797959	0.0793999	0.0396999	+	0.999212i	$0.487360\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.1464	1.94763	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	20.1464	1.94763	0.973814	+	0.227345i	$0.0730044\pi$
$108$	0	0
$109$	3.34847	0.320725	0.160363	−	0.987058i	$-0.448734\pi$
$109$	3.34847	0.320725	0.160363	+	0.987058i	$0.448734\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.65153	0.155363	0.0776815	−	0.996978i	$-0.475248\pi$
$113$	1.65153	0.155363	0.0776815	+	0.996978i	$0.475248\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.797959	−0.0731488
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.3485	−1.00701	−0.503507	−	0.863991i	$-0.667957\pi$
$127$	−11.3485	−1.00701	−0.503507	+	0.863991i	$0.667957\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.10102	−0.0961966	−0.0480983	−	0.998843i	$-0.515316\pi$
$131$	−1.10102	−0.0961966	−0.0480983	+	0.998843i	$0.515316\pi$
$132$	0	0
$133$	−9.34847	−0.810615
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.10102	0.606681	0.303341	−	0.952882i	$-0.401898\pi$
$137$	7.10102	0.606681	0.303341	+	0.952882i	$0.401898\pi$
$138$	0	0
$139$	7.69694	0.652846	0.326423	−	0.945224i	$-0.394157\pi$
$139$	7.69694	0.652846	0.326423	+	0.945224i	$0.394157\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.10102	0.0920720
$144$	0	0
$145$	7.89898	0.655975
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.24745	0.511811	0.255905	−	0.966702i	$-0.417626\pi$
$149$	6.24745	0.511811	0.255905	+	0.966702i	$0.417626\pi$
$150$	0	0
$151$	−0.202041	−0.0164419	−0.00822093	−	0.999966i	$-0.502617\pi$
$151$	−0.202041	−0.0164419	−0.00822093	+	0.999966i	$0.502617\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.00000	−0.562254
$156$	0	0
$157$	0.348469	0.0278109	0.0139054	−	0.999903i	$-0.495574\pi$
$157$	0.348469	0.0278109	0.0139054	+	0.999903i	$0.495574\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.44949	−0.114236
$162$	0	0
$163$	4.69694	0.367893	0.183946	−	0.982936i	$-0.441113\pi$
$163$	4.69694	0.367893	0.183946	+	0.982936i	$0.441113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.2474	1.87632	0.938162	−	0.346197i	$-0.112527\pi$
$167$	24.2474	1.87632	0.938162	+	0.346197i	$0.112527\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	1.44949	0.109571
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−8.89898	−0.661456	−0.330728	−	0.943726i	$-0.607294\pi$
$181$	−8.89898	−0.661456	−0.330728	+	0.943726i	$0.607294\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.34847	−0.613792
$186$	0	0
$187$	1.34847	0.0986098
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.4495	−1.47967	−0.739837	−	0.672787i	$-0.765097\pi$
$191$	−20.4495	−1.47967	−0.739837	+	0.672787i	$0.765097\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.79796	−0.270593	−0.135297	−	0.990805i	$-0.543199\pi$
$197$	−3.79796	−0.270593	−0.135297	+	0.990805i	$0.543199\pi$
$198$	0	0
$199$	−7.79796	−0.552783	−0.276391	−	0.961045i	$-0.589139\pi$
$199$	−7.79796	−0.552783	−0.276391	+	0.961045i	$0.589139\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	11.4495	0.803597
$204$	0	0
$205$	1.89898	0.132630
$206$	0	0
$207$	0	0
$208$	0	0
$209$	15.7980	1.09277
$210$	0	0
$211$	−15.6969	−1.08062	−0.540311	−	0.841465i	$-0.681693\pi$
$211$	−15.6969	−1.08062	−0.540311	+	0.841465i	$0.681693\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.898979	0.0613099
$216$	0	0
$217$	−10.1464	−0.688784
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.247449	0.0166452
$222$	0	0
$223$	−26.8990	−1.80129	−0.900644	−	0.434557i	$-0.856905\pi$
$223$	−26.8990	−1.80129	−0.900644	+	0.434557i	$0.856905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	17.7980	1.17612	0.588061	−	0.808816i	$-0.299891\pi$
$229$	17.7980	1.17612	0.588061	+	0.808816i	$0.299891\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.4949	1.60471	0.802357	−	0.596844i	$-0.203579\pi$
$233$	24.4949	1.60471	0.802357	+	0.596844i	$0.203579\pi$
$234$	0	0
$235$	−2.44949	−0.159787
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.1010	−0.653381	−0.326690	−	0.945131i	$-0.605933\pi$
$239$	−10.1010	−0.653381	−0.326690	+	0.945131i	$0.605933\pi$
$240$	0	0
$241$	−11.3485	−0.731019	−0.365510	−	0.930808i	$-0.619105\pi$
$241$	−11.3485	−0.731019	−0.365510	+	0.930808i	$0.619105\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.89898	−0.312984
$246$	0	0
$247$	2.89898	0.184458
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.8990	0.687937	0.343969	−	0.938981i	$-0.388229\pi$
$251$	10.8990	0.687937	0.343969	+	0.938981i	$0.388229\pi$
$252$	0	0
$253$	2.44949	0.153998
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.65153	0.290155	0.145077	−	0.989420i	$-0.453657\pi$
$257$	4.65153	0.290155	0.145077	+	0.989420i	$0.453657\pi$
$258$	0	0
$259$	−12.1010	−0.751921
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.24745	0.200246	0.100123	−	0.994975i	$-0.468076\pi$
$263$	3.24745	0.200246	0.100123	+	0.994975i	$0.468076\pi$
$264$	0	0
$265$	−10.3485	−0.635701
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.6969	−1.44483	−0.722414	−	0.691461i	$-0.756967\pi$
$269$	−23.6969	−1.44483	−0.722414	+	0.691461i	$0.756967\pi$
$270$	0	0
$271$	−3.69694	−0.224573	−0.112287	−	0.993676i	$-0.535817\pi$
$271$	−3.69694	−0.224573	−0.112287	+	0.993676i	$0.535817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.44949	−0.147710
$276$	0	0
$277$	−5.10102	−0.306491	−0.153245	−	0.988188i	$-0.548972\pi$
$277$	−5.10102	−0.306491	−0.153245	+	0.988188i	$0.548972\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.34847	−0.438373	−0.219186	−	0.975683i	$-0.570340\pi$
$281$	−7.34847	−0.438373	−0.219186	+	0.975683i	$0.570340\pi$
$282$	0	0
$283$	13.4495	0.799489	0.399745	−	0.916627i	$-0.369099\pi$
$283$	13.4495	0.799489	0.399745	+	0.916627i	$0.369099\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.75255	0.162478
$288$	0	0
$289$	−16.6969	−0.982173
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.65153	0.447007	0.223504	−	0.974703i	$-0.428251\pi$
$293$	7.65153	0.447007	0.223504	+	0.974703i	$0.428251\pi$
$294$	0	0
$295$	−7.89898	−0.459896
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.449490	0.0259947
$300$	0	0
$301$	1.30306	0.0751072
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.34847	0.535292
$306$	0	0
$307$	−7.55051	−0.430930	−0.215465	−	0.976512i	$-0.569127\pi$
$307$	−7.55051	−0.430930	−0.215465	+	0.976512i	$0.569127\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.6969	−1.17362	−0.586808	−	0.809726i	$-0.699616\pi$
$311$	−20.6969	−1.17362	−0.586808	+	0.809726i	$0.699616\pi$
$312$	0	0
$313$	18.3485	1.03712	0.518558	−	0.855042i	$-0.326469\pi$
$313$	18.3485	1.03712	0.518558	+	0.855042i	$0.326469\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.247449	−0.0138981	−0.00694905	−	0.999976i	$-0.502212\pi$
$317$	−0.247449	−0.0138981	−0.00694905	+	0.999976i	$0.502212\pi$
$318$	0	0
$319$	−19.3485	−1.08331
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.55051	0.197556
$324$	0	0
$325$	−0.449490	−0.0249332
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.55051	−0.195746
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.44949	−0.188466
$336$	0	0
$337$	24.8990	1.35633	0.678167	−	0.734908i	$-0.262775\pi$
$337$	24.8990	1.35633	0.678167	+	0.734908i	$0.262775\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.1464	0.928531
$342$	0	0
$343$	−17.2474	−0.931275
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.7980	−1.17018	−0.585088	−	0.810970i	$-0.698940\pi$
$347$	−21.7980	−1.17018	−0.585088	+	0.810970i	$0.698940\pi$
$348$	0	0
$349$	25.6969	1.37553	0.687763	−	0.725935i	$-0.258593\pi$
$349$	25.6969	1.37553	0.687763	+	0.725935i	$0.258593\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.55051	−0.508322	−0.254161	−	0.967162i	$-0.581799\pi$
$353$	−9.55051	−0.508322	−0.254161	+	0.967162i	$0.581799\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.1464	1.22162	0.610811	−	0.791777i	$-0.290844\pi$
$359$	23.1464	1.22162	0.610811	+	0.791777i	$0.290844\pi$
$360$	0	0
$361$	22.5959	1.18926
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.34847	−0.279952
$366$	0	0
$367$	−19.2474	−1.00471	−0.502354	−	0.864662i	$-0.667533\pi$
$367$	−19.2474	−1.00471	−0.502354	+	0.864662i	$0.667533\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.0000	−0.778761
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.55051	−0.182861
$378$	0	0
$379$	−33.3939	−1.71533	−0.857664	−	0.514210i	$-0.828085\pi$
$379$	−33.3939	−1.71533	−0.857664	+	0.514210i	$0.828085\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.34847	−0.222196	−0.111098	−	0.993809i	$-0.535437\pi$
$383$	−4.34847	−0.222196	−0.111098	+	0.993809i	$0.535437\pi$
$384$	0	0
$385$	−3.55051	−0.180951
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.1010	0.664248	0.332124	−	0.943236i	$-0.392235\pi$
$389$	13.1010	0.664248	0.332124	+	0.943236i	$0.392235\pi$
$390$	0	0
$391$	0.550510	0.0278405
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	17.7980	0.893254	0.446627	−	0.894720i	$-0.352625\pi$
$397$	17.7980	0.893254	0.446627	+	0.894720i	$0.352625\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.6969	−1.03356	−0.516778	−	0.856120i	$-0.672869\pi$
$401$	−20.6969	−1.03356	−0.516778	+	0.856120i	$0.672869\pi$
$402$	0	0
$403$	3.14643	0.156735
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.4495	1.01364
$408$	0	0
$409$	18.5959	0.919509	0.459754	−	0.888046i	$-0.347937\pi$
$409$	18.5959	0.919509	0.459754	+	0.888046i	$0.347937\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.4495	−0.563393
$414$	0	0
$415$	−4.34847	−0.213458
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.1464	−0.837658	−0.418829	−	0.908065i	$-0.637559\pi$
$419$	−17.1464	−0.837658	−0.418829	+	0.908065i	$0.637559\pi$
$420$	0	0
$421$	−35.3485	−1.72278	−0.861389	−	0.507945i	$-0.830405\pi$
$421$	−35.3485	−1.72278	−0.861389	+	0.507945i	$0.830405\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.550510	−0.0267037
$426$	0	0
$427$	13.5505	0.655755
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.5959	−0.943902	−0.471951	−	0.881625i	$-0.656450\pi$
$431$	−19.5959	−0.943902	−0.471951	+	0.881625i	$0.656450\pi$
$432$	0	0
$433$	36.8434	1.77058	0.885290	−	0.465040i	$-0.153960\pi$
$433$	36.8434	1.77058	0.885290	+	0.465040i	$0.153960\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.44949	0.308521
$438$	0	0
$439$	−2.89898	−0.138361	−0.0691804	−	0.997604i	$-0.522038\pi$
$439$	−2.89898	−0.138361	−0.0691804	+	0.997604i	$0.522038\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.9444	0.710029	0.355015	−	0.934861i	$-0.384476\pi$
$443$	14.9444	0.710029	0.355015	+	0.934861i	$0.384476\pi$
$444$	0	0
$445$	7.10102	0.336621
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.3939	−1.52876	−0.764381	−	0.644765i	$-0.776955\pi$
$449$	−32.3939	−1.52876	−0.764381	+	0.644765i	$0.776955\pi$
$450$	0	0
$451$	−4.65153	−0.219032
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.651531	−0.0305442
$456$	0	0
$457$	−11.6515	−0.545036	−0.272518	−	0.962151i	$-0.587856\pi$
$457$	−11.6515	−0.545036	−0.272518	+	0.962151i	$0.587856\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−38.6969	−1.80230	−0.901148	−	0.433511i	$-0.857274\pi$
$461$	−38.6969	−1.80230	−0.901148	+	0.433511i	$0.857274\pi$
$462$	0	0
$463$	−20.0454	−0.931589	−0.465795	−	0.884893i	$-0.654231\pi$
$463$	−20.0454	−0.931589	−0.465795	+	0.884893i	$0.654231\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.9444	1.66331	0.831654	−	0.555294i	$-0.187394\pi$
$467$	35.9444	1.66331	0.831654	+	0.555294i	$0.187394\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.20204	−0.101250
$474$	0	0
$475$	−6.44949	−0.295923
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.3485	1.15820	0.579101	−	0.815256i	$-0.303404\pi$
$479$	25.3485	1.15820	0.579101	+	0.815256i	$0.303404\pi$
$480$	0	0
$481$	3.75255	0.171102
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.10102	−0.231625
$486$	0	0
$487$	−33.1464	−1.50201	−0.751004	−	0.660298i	$-0.770430\pi$
$487$	−33.1464	−1.50201	−0.751004	+	0.660298i	$0.770430\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.3939	0.920363	0.460181	−	0.887825i	$-0.347784\pi$
$491$	20.3939	0.920363	0.460181	+	0.887825i	$0.347784\pi$
$492$	0	0
$493$	−4.34847	−0.195845
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.34847	−0.195056
$498$	0	0
$499$	14.7980	0.662448	0.331224	−	0.943552i	$-0.392538\pi$
$499$	14.7980	0.662448	0.331224	+	0.943552i	$0.392538\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.1464	−0.898285	−0.449142	−	0.893460i	$-0.648270\pi$
$503$	−20.1464	−0.898285	−0.449142	+	0.893460i	$0.648270\pi$
$504$	0	0
$505$	0.797959	0.0355087
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.3939	1.56881	0.784403	−	0.620252i	$-0.212969\pi$
$509$	35.3939	1.56881	0.784403	+	0.620252i	$0.212969\pi$
$510$	0	0
$511$	−7.75255	−0.342953
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16.6515	−0.729517	−0.364758	−	0.931102i	$-0.618848\pi$
$521$	−16.6515	−0.729517	−0.364758	+	0.931102i	$0.618848\pi$
$522$	0	0
$523$	−31.7980	−1.39043	−0.695214	−	0.718803i	$-0.744690\pi$
$523$	−31.7980	−1.39043	−0.695214	+	0.718803i	$0.744690\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.85357	0.167864
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.853572	−0.0369723
$534$	0	0
$535$	20.1464	0.871006
$536$	0	0
$537$	0	0
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	24.8990	1.07049	0.535245	−	0.844697i	$-0.320219\pi$
$541$	24.8990	1.07049	0.535245	+	0.844697i	$0.320219\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.34847	0.143433
$546$	0	0
$547$	25.3939	1.08576	0.542882	−	0.839809i	$-0.317333\pi$
$547$	25.3939	1.08576	0.542882	+	0.839809i	$0.317333\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−50.9444	−2.17030
$552$	0	0
$553$	−5.79796	−0.246554
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.4495	1.24781	0.623907	−	0.781498i	$-0.285544\pi$
$557$	29.4495	1.24781	0.623907	+	0.781498i	$0.285544\pi$
$558$	0	0
$559$	−0.404082	−0.0170909
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.85357	0.162409	0.0812043	−	0.996697i	$-0.474123\pi$
$563$	3.85357	0.162409	0.0812043	+	0.996697i	$0.474123\pi$
$564$	0	0
$565$	1.65153	0.0694804
$566$	0	0
$567$	0	0
$568$	0	0
$569$	47.3939	1.98685	0.993427	−	0.114465i	$-0.0365152\pi$
$569$	47.3939	1.98685	0.993427	+	0.114465i	$0.0365152\pi$
$570$	0	0
$571$	12.0454	0.504085	0.252042	−	0.967716i	$-0.418898\pi$
$571$	12.0454	0.504085	0.252042	+	0.967716i	$0.418898\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	42.2929	1.76067	0.880337	−	0.474348i	$-0.157316\pi$
$577$	42.2929	1.76067	0.880337	+	0.474348i	$0.157316\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.30306	−0.261495
$582$	0	0
$583$	25.3485	1.04983
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.5959	1.55175	0.775875	−	0.630887i	$-0.217309\pi$
$587$	37.5959	1.55175	0.775875	+	0.630887i	$0.217309\pi$
$588$	0	0
$589$	45.1464	1.86023
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.85357	0.281442	0.140721	−	0.990049i	$-0.455058\pi$
$593$	6.85357	0.281442	0.140721	+	0.990049i	$0.455058\pi$
$594$	0	0
$595$	−0.797959	−0.0327131
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−4.79796	−0.195713	−0.0978564	−	0.995201i	$-0.531199\pi$
$601$	−4.79796	−0.195713	−0.0978564	+	0.995201i	$0.531199\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	39.3485	1.59711	0.798553	−	0.601925i	$-0.205599\pi$
$607$	39.3485	1.59711	0.798553	+	0.601925i	$0.205599\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.10102	0.0445425
$612$	0	0
$613$	−6.69694	−0.270487	−0.135243	−	0.990812i	$-0.543182\pi$
$613$	−6.69694	−0.270487	−0.135243	+	0.990812i	$0.543182\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.65153	0.0664881	0.0332441	−	0.999447i	$-0.489416\pi$
$617$	1.65153	0.0664881	0.0332441	+	0.999447i	$0.489416\pi$
$618$	0	0
$619$	17.7980	0.715360	0.357680	−	0.933844i	$-0.383568\pi$
$619$	17.7980	0.715360	0.357680	+	0.933844i	$0.383568\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.2929	0.412375
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.59592	0.183251
$630$	0	0
$631$	2.24745	0.0894695	0.0447348	−	0.998999i	$-0.485756\pi$
$631$	2.24745	0.0894695	0.0447348	+	0.998999i	$0.485756\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.3485	−0.450350
$636$	0	0
$637$	2.20204	0.0872480
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.5505	−0.851194	−0.425597	−	0.904913i	$-0.639936\pi$
$641$	−21.5505	−0.851194	−0.425597	+	0.904913i	$0.639936\pi$
$642$	0	0
$643$	9.65153	0.380619	0.190310	−	0.981724i	$-0.439051\pi$
$643$	9.65153	0.380619	0.190310	+	0.981724i	$0.439051\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.5505	−0.611354	−0.305677	−	0.952135i	$-0.598883\pi$
$647$	−15.5505	−0.611354	−0.305677	+	0.952135i	$0.598883\pi$
$648$	0	0
$649$	19.3485	0.759494
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.4495	−1.26985	−0.634923	−	0.772575i	$-0.718968\pi$
$653$	−32.4495	−1.26985	−0.634923	+	0.772575i	$0.718968\pi$
$654$	0	0
$655$	−1.10102	−0.0430204
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.55051	−0.372035	−0.186018	−	0.982546i	$-0.559558\pi$
$659$	−9.55051	−0.372035	−0.186018	+	0.982546i	$0.559558\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.34847	−0.362518
$666$	0	0
$667$	−7.89898	−0.305850
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.8990	−0.884005
$672$	0	0
$673$	−26.0454	−1.00398	−0.501988	−	0.864874i	$-0.667398\pi$
$673$	−26.0454	−1.00398	−0.501988	+	0.864874i	$0.667398\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.7526	0.566987	0.283493	−	0.958974i	$-0.408507\pi$
$677$	14.7526	0.566987	0.283493	+	0.958974i	$0.408507\pi$
$678$	0	0
$679$	−7.39388	−0.283751
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.65153	0.177986	0.0889929	−	0.996032i	$-0.471635\pi$
$683$	4.65153	0.177986	0.0889929	+	0.996032i	$0.471635\pi$
$684$	0	0
$685$	7.10102	0.271316
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.65153	0.177209
$690$	0	0
$691$	−15.3939	−0.585611	−0.292805	−	0.956172i	$-0.594589\pi$
$691$	−15.3939	−0.585611	−0.292805	+	0.956172i	$0.594589\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.69694	0.291962
$696$	0	0
$697$	−1.04541	−0.0395976
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−28.0454	−1.05926	−0.529630	−	0.848229i	$-0.677669\pi$
$701$	−28.0454	−1.05926	−0.529630	+	0.848229i	$0.677669\pi$
$702$	0	0
$703$	53.8434	2.03074
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.15663	0.0434997
$708$	0	0
$709$	−0.944387	−0.0354672	−0.0177336	−	0.999843i	$-0.505645\pi$
$709$	−0.944387	−0.0354672	−0.0177336	+	0.999843i	$0.505645\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.00000	0.262152
$714$	0	0
$715$	1.10102	0.0411758
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19.2929	−0.719502	−0.359751	−	0.933048i	$-0.617138\pi$
$719$	−19.2929	−0.719502	−0.359751	+	0.933048i	$0.617138\pi$
$720$	0	0
$721$	−5.79796	−0.215927
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.89898	0.293361
$726$	0	0
$727$	−32.3485	−1.19974	−0.599869	−	0.800098i	$-0.704781\pi$
$727$	−32.3485	−1.19974	−0.599869	+	0.800098i	$0.704781\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.494897	−0.0183044
$732$	0	0
$733$	36.3485	1.34256	0.671281	−	0.741203i	$-0.265745\pi$
$733$	36.3485	1.34256	0.671281	+	0.741203i	$0.265745\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.44949	0.311241
$738$	0	0
$739$	−16.3031	−0.599718	−0.299859	−	0.953984i	$-0.596940\pi$
$739$	−16.3031	−0.599718	−0.299859	+	0.953984i	$0.596940\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.7980	1.23993	0.619963	−	0.784631i	$-0.287147\pi$
$743$	33.7980	1.23993	0.619963	+	0.784631i	$0.287147\pi$
$744$	0	0
$745$	6.24745	0.228889
$746$	0	0
$747$	0	0
$748$	0	0
$749$	29.2020	1.06702
$750$	0	0
$751$	12.6515	0.461661	0.230830	−	0.972994i	$-0.425856\pi$
$751$	12.6515	0.461661	0.230830	+	0.972994i	$0.425856\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−0.202041	−0.00735303
$756$	0	0
$757$	35.2474	1.28109	0.640545	−	0.767921i	$-0.278708\pi$
$757$	35.2474	1.28109	0.640545	+	0.767921i	$0.278708\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.4949	−1.64919	−0.824594	−	0.565724i	$-0.808597\pi$
$761$	−45.4949	−1.64919	−0.824594	+	0.565724i	$0.808597\pi$
$762$	0	0
$763$	4.85357	0.175711
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.55051	0.128201
$768$	0	0
$769$	−3.14643	−0.113463	−0.0567316	−	0.998389i	$-0.518068\pi$
$769$	−3.14643	−0.113463	−0.0567316	+	0.998389i	$0.518068\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.69694	0.312807	0.156404	−	0.987693i	$-0.450010\pi$
$773$	8.69694	0.312807	0.156404	+	0.987693i	$0.450010\pi$
$774$	0	0
$775$	−7.00000	−0.251447
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.2474	−0.438810
$780$	0	0
$781$	7.34847	0.262949
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.348469	0.0124374
$786$	0	0
$787$	−48.1464	−1.71623	−0.858117	−	0.513454i	$-0.828366\pi$
$787$	−48.1464	−1.71623	−0.858117	+	0.513454i	$0.828366\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.39388	0.0851165
$792$	0	0
$793$	−4.20204	−0.149219
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.1464	1.35122	0.675608	−	0.737261i	$-0.263881\pi$
$797$	38.1464	1.35122	0.675608	+	0.737261i	$0.263881\pi$
$798$	0	0
$799$	1.34847	0.0477054
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.1010	0.462325
$804$	0	0
$805$	−1.44949	−0.0510878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.797959	−0.0280547	−0.0140274	−	0.999902i	$-0.504465\pi$
$809$	−0.797959	−0.0280547	−0.0140274	+	0.999902i	$0.504465\pi$
$810$	0	0
$811$	−16.7980	−0.589856	−0.294928	−	0.955519i	$-0.595296\pi$
$811$	−16.7980	−0.589856	−0.294928	+	0.955519i	$0.595296\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.69694	0.164527
$816$	0	0
$817$	−5.79796	−0.202845
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.5959	−0.893304	−0.446652	−	0.894708i	$-0.647384\pi$
$821$	−25.5959	−0.893304	−0.446652	+	0.894708i	$0.647384\pi$
$822$	0	0
$823$	3.10102	0.108095	0.0540474	−	0.998538i	$-0.482788\pi$
$823$	3.10102	0.108095	0.0540474	+	0.998538i	$0.482788\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.2474	−1.57341	−0.786704	−	0.617330i	$-0.788214\pi$
$827$	−45.2474	−1.57341	−0.786704	+	0.617330i	$0.788214\pi$
$828$	0	0
$829$	34.3939	1.19455	0.597274	−	0.802037i	$-0.296250\pi$
$829$	34.3939	1.19455	0.597274	+	0.802037i	$0.296250\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.69694	0.0934434
$834$	0	0
$835$	24.2474	0.839118
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13.5959	−0.469383	−0.234692	−	0.972070i	$-0.575408\pi$
$839$	−13.5959	−0.469383	−0.234692	+	0.972070i	$0.575408\pi$
$840$	0	0
$841$	33.3939	1.15151
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.7980	−0.440263
$846$	0	0
$847$	−7.24745	−0.249025
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.34847	0.286182
$852$	0	0
$853$	4.20204	0.143875	0.0719376	−	0.997409i	$-0.477082\pi$
$853$	4.20204	0.143875	0.0719376	+	0.997409i	$0.477082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	52.2929	1.78629	0.893145	−	0.449769i	$-0.148494\pi$
$857$	52.2929	1.78629	0.893145	+	0.449769i	$0.148494\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.5959	0.462810	0.231405	−	0.972857i	$-0.425668\pi$
$863$	13.5959	0.462810	0.231405	+	0.972857i	$0.425668\pi$
$864$	0	0
$865$	−9.79796	−0.333141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.79796	0.332373
$870$	0	0
$871$	1.55051	0.0525370
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.44949	0.0490017
$876$	0	0
$877$	4.20204	0.141893	0.0709464	−	0.997480i	$-0.477398\pi$
$877$	4.20204	0.141893	0.0709464	+	0.997480i	$0.477398\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.853572	0.0287576	0.0143788	−	0.999897i	$-0.495423\pi$
$881$	0.853572	0.0287576	0.0143788	+	0.999897i	$0.495423\pi$
$882$	0	0
$883$	−42.4495	−1.42854	−0.714270	−	0.699871i	$-0.753241\pi$
$883$	−42.4495	−1.42854	−0.714270	+	0.699871i	$0.753241\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.59592	−0.0535857	−0.0267928	−	0.999641i	$-0.508529\pi$
$887$	−1.59592	−0.0535857	−0.0267928	+	0.999641i	$0.508529\pi$
$888$	0	0
$889$	−16.4495	−0.551698
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.7980	0.528659
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−55.2929	−1.84412
$900$	0	0
$901$	5.69694	0.189793
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.89898	−0.295812
$906$	0	0
$907$	12.3485	0.410024	0.205012	−	0.978759i	$-0.434277\pi$
$907$	12.3485	0.410024	0.205012	+	0.978759i	$0.434277\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.4949	0.811552	0.405776	−	0.913973i	$-0.367001\pi$
$911$	24.4949	0.811552	0.405776	+	0.913973i	$0.367001\pi$
$912$	0	0
$913$	10.6515	0.352514
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.59592	−0.0527019
$918$	0	0
$919$	27.1010	0.893980	0.446990	−	0.894539i	$-0.352496\pi$
$919$	27.1010	0.893980	0.446990	+	0.894539i	$0.352496\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.34847	0.0443854
$924$	0	0
$925$	−8.34847	−0.274496
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	31.5959	1.03551
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.34847	0.0440997
$936$	0	0
$937$	−0.696938	−0.0227680	−0.0113840	−	0.999935i	$-0.503624\pi$
$937$	−0.696938	−0.0227680	−0.0113840	+	0.999935i	$0.503624\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.5505	−0.898121	−0.449060	−	0.893501i	$-0.648241\pi$
$941$	−27.5505	−0.898121	−0.449060	+	0.893501i	$0.648241\pi$
$942$	0	0
$943$	−1.89898	−0.0618393
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.4949	0.406030	0.203015	−	0.979176i	$-0.434926\pi$
$947$	12.4949	0.406030	0.203015	+	0.979176i	$0.434926\pi$
$948$	0	0
$949$	2.40408	0.0780398
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51.7980	1.67790	0.838950	−	0.544208i	$-0.183170\pi$
$953$	51.7980	1.67790	0.838950	+	0.544208i	$0.183170\pi$
$954$	0	0
$955$	−20.4495	−0.661730
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.2929	0.332374
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.0000	−0.515058
$966$	0	0
$967$	−23.3485	−0.750836	−0.375418	−	0.926856i	$-0.622501\pi$
$967$	−23.3485	−0.750836	−0.375418	+	0.926856i	$0.622501\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−36.4949	−1.17118	−0.585588	−	0.810608i	$-0.699137\pi$
$971$	−36.4949	−1.17118	−0.585588	+	0.810608i	$0.699137\pi$
$972$	0	0
$973$	11.1566	0.357665
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.6515	−0.820665	−0.410333	−	0.911936i	$-0.634587\pi$
$977$	−25.6515	−0.820665	−0.410333	+	0.911936i	$0.634587\pi$
$978$	0	0
$979$	−17.3939	−0.555911
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.65153	0.0526757	0.0263378	−	0.999653i	$-0.491615\pi$
$983$	1.65153	0.0526757	0.0263378	+	0.999653i	$0.491615\pi$
$984$	0	0
$985$	−3.79796	−0.121013
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.898979	−0.0285859
$990$	0	0
$991$	39.8990	1.26743	0.633716	−	0.773565i	$-0.281529\pi$
$991$	39.8990	1.26743	0.633716	+	0.773565i	$0.281529\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.79796	−0.247212
$996$	0	0
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.q.1.2	yes	2
3.2	odd	2		4140.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.l.1.2	✓	2	3.2	odd	2
4140.2.a.q.1.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.44949 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.44949 q^{7} -2.44949 q^{11} -0.449490 q^{13} -0.550510 q^{17} -6.44949 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.89898 q^{29} -7.00000 q^{31} +1.44949 q^{35} -8.34847 q^{37} +1.89898 q^{41} +0.898979 q^{43} -2.44949 q^{47} -4.89898 q^{49} -10.3485 q^{53} -2.44949 q^{55} -7.89898 q^{59} +9.34847 q^{61} -0.449490 q^{65} -3.44949 q^{67} -3.00000 q^{71} -5.34847 q^{73} -3.55051 q^{77} -4.00000 q^{79} -4.34847 q^{83} -0.550510 q^{85} +7.10102 q^{89} -0.651531 q^{91} -6.44949 q^{95} -5.10102 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} + 4 q^{13} - 6 q^{17} - 8 q^{19} - 2 q^{23} + 2 q^{25} + 6 q^{29} - 14 q^{31} - 2 q^{35} - 2 q^{37} - 6 q^{41} - 8 q^{43} - 6 q^{53} - 6 q^{59} + 4 q^{61} + 4 q^{65} - 2 q^{67} - 6 q^{71} + 4 q^{73} - 12 q^{77} - 8 q^{79} + 6 q^{83} - 6 q^{85} + 24 q^{89} - 16 q^{91} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + 4 * q^13 - 6 * q^17 - 8 * q^19 - 2 * q^23 + 2 * q^25 + 6 * q^29 - 14 * q^31 - 2 * q^35 - 2 * q^37 - 6 * q^41 - 8 * q^43 - 6 * q^53 - 6 * q^59 + 4 * q^61 + 4 * q^65 - 2 * q^67 - 6 * q^71 + 4 * q^73 - 12 * q^77 - 8 * q^79 + 6 * q^83 - 6 * q^85 + 24 * q^89 - 16 * q^91 - 8 * q^95 - 20 * q^97

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