LMFDB - Embedded newform 3696.2.a.bl.1.2

Newspace parameters

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

$f(q)$	$=$	$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} +7.58258 q^{17} +6.58258 q^{19} -1.00000 q^{21} +5.58258 q^{23} +4.00000 q^{25} +1.00000 q^{27} -8.16515 q^{29} -3.58258 q^{31} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} -11.1652 q^{41} -1.58258 q^{43} +3.00000 q^{45} -1.41742 q^{47} +1.00000 q^{49} +7.58258 q^{51} -9.58258 q^{53} +3.00000 q^{55} +6.58258 q^{57} -4.58258 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -8.58258 q^{67} +5.58258 q^{69} -11.1652 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} -7.16515 q^{79} +1.00000 q^{81} +11.5826 q^{83} +22.7477 q^{85} -8.16515 q^{87} +9.16515 q^{89} -1.00000 q^{91} -3.58258 q^{93} +19.7477 q^{95} -2.41742 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 + 6 * q^43 + 6 * q^45 - 12 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 6 * q^55 + 4 * q^57 + 20 * q^61 - 2 * q^63 + 6 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 + 14 * q^73 + 8 * q^75 - 2 * q^77 + 4 * q^79 + 2 * q^81 + 14 * q^83 + 18 * q^85 + 2 * q^87 - 2 * q^91 + 2 * q^93 + 12 * q^95 - 14 * q^97 + 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	7.58258	1.83904	0.919522	−	0.393038i	$-0.128576\pi$
$17$	7.58258	1.83904	0.919522	+	0.393038i	$0.128576\pi$
$18$	0	0
$19$	6.58258	1.51015	0.755073	−	0.655640i	$-0.227601\pi$
$19$	6.58258	1.51015	0.755073	+	0.655640i	$0.227601\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.58258	1.16405	0.582024	−	0.813172i	$-0.302261\pi$
$23$	5.58258	1.16405	0.582024	+	0.813172i	$0.302261\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.16515	−1.51623	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	−8.16515	−1.51623	−0.758115	+	0.652121i	$0.773880\pi$
$30$	0	0
$31$	−3.58258	−0.643450	−0.321725	−	0.946833i	$-0.604263\pi$
$31$	−3.58258	−0.643450	−0.321725	+	0.946833i	$0.604263\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−11.1652	−1.74370	−0.871852	−	0.489770i	$-0.837081\pi$
$41$	−11.1652	−1.74370	−0.871852	+	0.489770i	$0.837081\pi$
$42$	0	0
$43$	−1.58258	−0.241341	−0.120670	−	0.992693i	$-0.538504\pi$
$43$	−1.58258	−0.241341	−0.120670	+	0.992693i	$0.538504\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.58258	1.06177
$52$	0	0
$53$	−9.58258	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$53$	−9.58258	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	6.58258	0.871883
$58$	0	0
$59$	−4.58258	−0.596601	−0.298300	−	0.954472i	$-0.596420\pi$
$59$	−4.58258	−0.596601	−0.298300	+	0.954472i	$0.596420\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−8.58258	−1.04853	−0.524264	−	0.851556i	$-0.675660\pi$
$67$	−8.58258	−1.04853	−0.524264	+	0.851556i	$0.675660\pi$
$68$	0	0
$69$	5.58258	0.672063
$70$	0	0
$71$	−11.1652	−1.32506	−0.662530	−	0.749036i	$-0.730517\pi$
$71$	−11.1652	−1.32506	−0.662530	+	0.749036i	$0.730517\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−7.16515	−0.806143	−0.403071	−	0.915169i	$-0.632057\pi$
$79$	−7.16515	−0.806143	−0.403071	+	0.915169i	$0.632057\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.5826	1.27135	0.635676	−	0.771956i	$-0.280721\pi$
$83$	11.5826	1.27135	0.635676	+	0.771956i	$0.280721\pi$
$84$	0	0
$85$	22.7477	2.46734
$86$	0	0
$87$	−8.16515	−0.875396
$88$	0	0
$89$	9.16515	0.971504	0.485752	−	0.874097i	$-0.338546\pi$
$89$	9.16515	0.971504	0.485752	+	0.874097i	$0.338546\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.58258	−0.371496
$94$	0	0
$95$	19.7477	2.02607
$96$	0	0
$97$	−2.41742	−0.245452	−0.122726	−	0.992441i	$-0.539164\pi$
$97$	−2.41742	−0.245452	−0.122726	+	0.992441i	$0.539164\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	11.5826	1.15251	0.576255	−	0.817270i	$-0.304514\pi$
$101$	11.5826	1.15251	0.576255	+	0.817270i	$0.304514\pi$
$102$	0	0
$103$	1.16515	0.114806	0.0574029	−	0.998351i	$-0.481718\pi$
$103$	1.16515	0.114806	0.0574029	+	0.998351i	$0.481718\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	12.5826	1.21640	0.608202	−	0.793782i	$-0.291891\pi$
$107$	12.5826	1.21640	0.608202	+	0.793782i	$0.291891\pi$
$108$	0	0
$109$	3.58258	0.343149	0.171574	−	0.985171i	$-0.445115\pi$
$109$	3.58258	0.343149	0.171574	+	0.985171i	$0.445115\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	9.16515	0.862185	0.431092	−	0.902308i	$-0.358128\pi$
$113$	9.16515	0.862185	0.431092	+	0.902308i	$0.358128\pi$
$114$	0	0
$115$	16.7477	1.56173
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−7.58258	−0.695094
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.1652	−1.00673
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	11.5826	1.02779	0.513894	−	0.857854i	$-0.328203\pi$
$127$	11.5826	1.02779	0.513894	+	0.857854i	$0.328203\pi$
$128$	0	0
$129$	−1.58258	−0.139338
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−6.58258	−0.570782
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−11.5826	−0.989566	−0.494783	−	0.869016i	$-0.664752\pi$
$137$	−11.5826	−0.989566	−0.494783	+	0.869016i	$0.664752\pi$
$138$	0	0
$139$	−11.1652	−0.947016	−0.473508	−	0.880790i	$-0.657012\pi$
$139$	−11.1652	−0.947016	−0.473508	+	0.880790i	$0.657012\pi$
$140$	0	0
$141$	−1.41742	−0.119369
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−24.4955	−2.03424
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	6.16515	0.505069	0.252534	−	0.967588i	$-0.418736\pi$
$149$	6.16515	0.505069	0.252534	+	0.967588i	$0.418736\pi$
$150$	0	0
$151$	−3.58258	−0.291546	−0.145773	−	0.989318i	$-0.546567\pi$
$151$	−3.58258	−0.291546	−0.145773	+	0.989318i	$0.546567\pi$
$152$	0	0
$153$	7.58258	0.613015
$154$	0	0
$155$	−10.7477	−0.863278
$156$	0	0
$157$	19.1652	1.52955	0.764773	−	0.644300i	$-0.222851\pi$
$157$	19.1652	1.52955	0.764773	+	0.644300i	$0.222851\pi$
$158$	0	0
$159$	−9.58258	−0.759948
$160$	0	0
$161$	−5.58258	−0.439969
$162$	0	0
$163$	−8.58258	−0.672239	−0.336120	−	0.941819i	$-0.609115\pi$
$163$	−8.58258	−0.672239	−0.336120	+	0.941819i	$0.609115\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	4.74773	0.367390	0.183695	−	0.982983i	$-0.441194\pi$
$167$	4.74773	0.367390	0.183695	+	0.982983i	$0.441194\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	6.58258	0.503382
$172$	0	0
$173$	−7.16515	−0.544756	−0.272378	−	0.962190i	$-0.587810\pi$
$173$	−7.16515	−0.544756	−0.272378	+	0.962190i	$0.587810\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−4.58258	−0.344447
$178$	0	0
$179$	−14.3303	−1.07110	−0.535549	−	0.844504i	$-0.679895\pi$
$179$	−14.3303	−1.07110	−0.535549	+	0.844504i	$0.679895\pi$
$180$	0	0
$181$	−5.58258	−0.414950	−0.207475	−	0.978240i	$-0.566524\pi$
$181$	−5.58258	−0.414950	−0.207475	+	0.978240i	$0.566524\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	7.58258	0.554493
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−11.5826	−0.838086	−0.419043	−	0.907966i	$-0.637634\pi$
$191$	−11.5826	−0.838086	−0.419043	+	0.907966i	$0.637634\pi$
$192$	0	0
$193$	−2.41742	−0.174010	−0.0870050	−	0.996208i	$-0.527730\pi$
$193$	−2.41742	−0.174010	−0.0870050	+	0.996208i	$0.527730\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	5.16515	0.368002	0.184001	−	0.982926i	$-0.441095\pi$
$197$	5.16515	0.368002	0.184001	+	0.982926i	$0.441095\pi$
$198$	0	0
$199$	9.58258	0.679291	0.339645	−	0.940554i	$-0.389693\pi$
$199$	9.58258	0.679291	0.339645	+	0.940554i	$0.389693\pi$
$200$	0	0
$201$	−8.58258	−0.605368
$202$	0	0
$203$	8.16515	0.573081
$204$	0	0
$205$	−33.4955	−2.33942
$206$	0	0
$207$	5.58258	0.388016
$208$	0	0
$209$	6.58258	0.455326
$210$	0	0
$211$	13.1652	0.906326	0.453163	−	0.891428i	$-0.350295\pi$
$211$	13.1652	0.906326	0.453163	+	0.891428i	$0.350295\pi$
$212$	0	0
$213$	−11.1652	−0.765024
$214$	0	0
$215$	−4.74773	−0.323792
$216$	0	0
$217$	3.58258	0.243201
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	7.58258	0.510059
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	0.747727	0.0494112	0.0247056	−	0.999695i	$-0.492135\pi$
$229$	0.747727	0.0494112	0.0247056	+	0.999695i	$0.492135\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−4.25227	−0.277388
$236$	0	0
$237$	−7.16515	−0.465427
$238$	0	0
$239$	−16.5826	−1.07264	−0.536319	−	0.844015i	$-0.680186\pi$
$239$	−16.5826	−1.07264	−0.536319	+	0.844015i	$0.680186\pi$
$240$	0	0
$241$	−10.1652	−0.654795	−0.327397	−	0.944887i	$-0.606172\pi$
$241$	−10.1652	−0.654795	−0.327397	+	0.944887i	$0.606172\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	6.58258	0.418839
$248$	0	0
$249$	11.5826	0.734016
$250$	0	0
$251$	−7.41742	−0.468184	−0.234092	−	0.972214i	$-0.575212\pi$
$251$	−7.41742	−0.468184	−0.234092	+	0.972214i	$0.575212\pi$
$252$	0	0
$253$	5.58258	0.350974
$254$	0	0
$255$	22.7477	1.42452
$256$	0	0
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−8.16515	−0.505410
$262$	0	0
$263$	22.9129	1.41287	0.706434	−	0.707779i	$-0.250303\pi$
$263$	22.9129	1.41287	0.706434	+	0.707779i	$0.250303\pi$
$264$	0	0
$265$	−28.7477	−1.76596
$266$	0	0
$267$	9.16515	0.560898
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−5.41742	−0.329085	−0.164543	−	0.986370i	$-0.552615\pi$
$271$	−5.41742	−0.329085	−0.164543	+	0.986370i	$0.552615\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−19.1652	−1.15152	−0.575761	−	0.817618i	$-0.695294\pi$
$277$	−19.1652	−1.15152	−0.575761	+	0.817618i	$0.695294\pi$
$278$	0	0
$279$	−3.58258	−0.214483
$280$	0	0
$281$	−27.3303	−1.63039	−0.815195	−	0.579187i	$-0.803370\pi$
$281$	−27.3303	−1.63039	−0.815195	+	0.579187i	$0.803370\pi$
$282$	0	0
$283$	−27.7477	−1.64943	−0.824716	−	0.565548i	$-0.808665\pi$
$283$	−27.7477	−1.64943	−0.824716	+	0.565548i	$0.808665\pi$
$284$	0	0
$285$	19.7477	1.16975
$286$	0	0
$287$	11.1652	0.659058
$288$	0	0
$289$	40.4955	2.38209
$290$	0	0
$291$	−2.41742	−0.141712
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−13.7477	−0.800424
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	5.58258	0.322849
$300$	0	0
$301$	1.58258	0.0912181
$302$	0	0
$303$	11.5826	0.665402
$304$	0	0
$305$	30.0000	1.71780
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	1.16515	0.0662831
$310$	0	0
$311$	−14.3303	−0.812597	−0.406298	−	0.913740i	$-0.633181\pi$
$311$	−14.3303	−0.812597	−0.406298	+	0.913740i	$0.633181\pi$
$312$	0	0
$313$	19.5826	1.10687	0.553436	−	0.832891i	$-0.313316\pi$
$313$	19.5826	1.10687	0.553436	+	0.832891i	$0.313316\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	−22.4174	−1.25909	−0.629544	−	0.776965i	$-0.716758\pi$
$317$	−22.4174	−1.25909	−0.629544	+	0.776965i	$0.716758\pi$
$318$	0	0
$319$	−8.16515	−0.457161
$320$	0	0
$321$	12.5826	0.702291
$322$	0	0
$323$	49.9129	2.77723
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	3.58258	0.198117
$328$	0	0
$329$	1.41742	0.0781451
$330$	0	0
$331$	3.16515	0.173972	0.0869862	−	0.996210i	$-0.472276\pi$
$331$	3.16515	0.173972	0.0869862	+	0.996210i	$0.472276\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	−25.7477	−1.40675
$336$	0	0
$337$	17.5826	0.957784	0.478892	−	0.877874i	$-0.341039\pi$
$337$	17.5826	0.957784	0.478892	+	0.877874i	$0.341039\pi$
$338$	0	0
$339$	9.16515	0.497783
$340$	0	0
$341$	−3.58258	−0.194007
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	16.7477	0.901667
$346$	0	0
$347$	−26.3303	−1.41348	−0.706742	−	0.707471i	$-0.749836\pi$
$347$	−26.3303	−1.41348	−0.706742	+	0.707471i	$0.749836\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	24.1652	1.28618	0.643091	−	0.765790i	$-0.277652\pi$
$353$	24.1652	1.28618	0.643091	+	0.765790i	$0.277652\pi$
$354$	0	0
$355$	−33.4955	−1.77775
$356$	0	0
$357$	−7.58258	−0.401312
$358$	0	0
$359$	8.83485	0.466285	0.233143	−	0.972443i	$-0.425099\pi$
$359$	8.83485	0.466285	0.233143	+	0.972443i	$0.425099\pi$
$360$	0	0
$361$	24.3303	1.28054
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	21.0000	1.09919
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	0	0
$369$	−11.1652	−0.581235
$370$	0	0
$371$	9.58258	0.497503
$372$	0	0
$373$	−34.7477	−1.79917	−0.899585	−	0.436747i	$-0.856131\pi$
$373$	−34.7477	−1.79917	−0.899585	+	0.436747i	$0.856131\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−8.16515	−0.420527
$378$	0	0
$379$	12.5826	0.646323	0.323162	−	0.946344i	$-0.395254\pi$
$379$	12.5826	0.646323	0.323162	+	0.946344i	$0.395254\pi$
$380$	0	0
$381$	11.5826	0.593393
$382$	0	0
$383$	−10.3303	−0.527854	−0.263927	−	0.964543i	$-0.585018\pi$
$383$	−10.3303	−0.527854	−0.263927	+	0.964543i	$0.585018\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	−1.58258	−0.0804468
$388$	0	0
$389$	−26.3303	−1.33500	−0.667500	−	0.744610i	$-0.732635\pi$
$389$	−26.3303	−1.33500	−0.667500	+	0.744610i	$0.732635\pi$
$390$	0	0
$391$	42.3303	2.14074
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	−21.4955	−1.08155
$396$	0	0
$397$	−31.5826	−1.58508	−0.792542	−	0.609817i	$-0.791243\pi$
$397$	−31.5826	−1.58508	−0.792542	+	0.609817i	$0.791243\pi$
$398$	0	0
$399$	−6.58258	−0.329541
$400$	0	0
$401$	31.9129	1.59365	0.796827	−	0.604208i	$-0.206510\pi$
$401$	31.9129	1.59365	0.796827	+	0.604208i	$0.206510\pi$
$402$	0	0
$403$	−3.58258	−0.178461
$404$	0	0
$405$	3.00000	0.149071
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	8.33030	0.411907	0.205953	−	0.978562i	$-0.433970\pi$
$409$	8.33030	0.411907	0.205953	+	0.978562i	$0.433970\pi$
$410$	0	0
$411$	−11.5826	−0.571326
$412$	0	0
$413$	4.58258	0.225494
$414$	0	0
$415$	34.7477	1.70570
$416$	0	0
$417$	−11.1652	−0.546760
$418$	0	0
$419$	−2.58258	−0.126167	−0.0630835	−	0.998008i	$-0.520093\pi$
$419$	−2.58258	−0.126167	−0.0630835	+	0.998008i	$0.520093\pi$
$420$	0	0
$421$	−33.6606	−1.64052	−0.820259	−	0.571993i	$-0.806171\pi$
$421$	−33.6606	−1.64052	−0.820259	+	0.571993i	$0.806171\pi$
$422$	0	0
$423$	−1.41742	−0.0689175
$424$	0	0
$425$	30.3303	1.47124
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−17.7477	−0.854878	−0.427439	−	0.904044i	$-0.640584\pi$
$431$	−17.7477	−0.854878	−0.427439	+	0.904044i	$0.640584\pi$
$432$	0	0
$433$	−11.1652	−0.536563	−0.268281	−	0.963341i	$-0.586456\pi$
$433$	−11.1652	−0.536563	−0.268281	+	0.963341i	$0.586456\pi$
$434$	0	0
$435$	−24.4955	−1.17447
$436$	0	0
$437$	36.7477	1.75788
$438$	0	0
$439$	−17.4174	−0.831288	−0.415644	−	0.909527i	$-0.636444\pi$
$439$	−17.4174	−0.831288	−0.415644	+	0.909527i	$0.636444\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	23.1652	1.10061	0.550305	−	0.834964i	$-0.314512\pi$
$443$	23.1652	1.10061	0.550305	+	0.834964i	$0.314512\pi$
$444$	0	0
$445$	27.4955	1.30341
$446$	0	0
$447$	6.16515	0.291602
$448$	0	0
$449$	−18.3303	−0.865060	−0.432530	−	0.901619i	$-0.642379\pi$
$449$	−18.3303	−0.865060	−0.432530	+	0.901619i	$0.642379\pi$
$450$	0	0
$451$	−11.1652	−0.525746
$452$	0	0
$453$	−3.58258	−0.168324
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−19.9129	−0.931485	−0.465743	−	0.884920i	$-0.654213\pi$
$457$	−19.9129	−0.931485	−0.465743	+	0.884920i	$0.654213\pi$
$458$	0	0
$459$	7.58258	0.353924
$460$	0	0
$461$	18.3303	0.853727	0.426864	−	0.904316i	$-0.359618\pi$
$461$	18.3303	0.853727	0.426864	+	0.904316i	$0.359618\pi$
$462$	0	0
$463$	−8.58258	−0.398866	−0.199433	−	0.979911i	$-0.563910\pi$
$463$	−8.58258	−0.398866	−0.199433	+	0.979911i	$0.563910\pi$
$464$	0	0
$465$	−10.7477	−0.498414
$466$	0	0
$467$	38.5826	1.78539	0.892694	−	0.450663i	$-0.148812\pi$
$467$	38.5826	1.78539	0.892694	+	0.450663i	$0.148812\pi$
$468$	0	0
$469$	8.58258	0.396307
$470$	0	0
$471$	19.1652	0.883084
$472$	0	0
$473$	−1.58258	−0.0727669
$474$	0	0
$475$	26.3303	1.20812
$476$	0	0
$477$	−9.58258	−0.438756
$478$	0	0
$479$	15.5826	0.711986	0.355993	−	0.934489i	$-0.384143\pi$
$479$	15.5826	0.711986	0.355993	+	0.934489i	$0.384143\pi$
$480$	0	0
$481$	1.00000	0.0455961
$482$	0	0
$483$	−5.58258	−0.254016
$484$	0	0
$485$	−7.25227	−0.329309
$486$	0	0
$487$	−10.3303	−0.468111	−0.234055	−	0.972223i	$-0.575200\pi$
$487$	−10.3303	−0.468111	−0.234055	+	0.972223i	$0.575200\pi$
$488$	0	0
$489$	−8.58258	−0.388117
$490$	0	0
$491$	22.9129	1.03404	0.517022	−	0.855972i	$-0.327041\pi$
$491$	22.9129	1.03404	0.517022	+	0.855972i	$0.327041\pi$
$492$	0	0
$493$	−61.9129	−2.78842
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	11.1652	0.500825
$498$	0	0
$499$	−41.7477	−1.86888	−0.934442	−	0.356114i	$-0.884101\pi$
$499$	−41.7477	−1.86888	−0.934442	+	0.356114i	$0.884101\pi$
$500$	0	0
$501$	4.74773	0.212113
$502$	0	0
$503$	0.747727	0.0333395	0.0166698	−	0.999861i	$-0.494694\pi$
$503$	0.747727	0.0333395	0.0166698	+	0.999861i	$0.494694\pi$
$504$	0	0
$505$	34.7477	1.54625
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	0	0
$513$	6.58258	0.290628
$514$	0	0
$515$	3.49545	0.154028
$516$	0	0
$517$	−1.41742	−0.0623382
$518$	0	0
$519$	−7.16515	−0.314515
$520$	0	0
$521$	15.8348	0.693737	0.346869	−	0.937914i	$-0.387245\pi$
$521$	15.8348	0.693737	0.346869	+	0.937914i	$0.387245\pi$
$522$	0	0
$523$	−15.4174	−0.674157	−0.337078	−	0.941477i	$-0.609439\pi$
$523$	−15.4174	−0.674157	−0.337078	+	0.941477i	$0.609439\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	−27.1652	−1.18333
$528$	0	0
$529$	8.16515	0.355007
$530$	0	0
$531$	−4.58258	−0.198867
$532$	0	0
$533$	−11.1652	−0.483616
$534$	0	0
$535$	37.7477	1.63198
$536$	0	0
$537$	−14.3303	−0.618398
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	18.3303	0.788081	0.394041	−	0.919093i	$-0.371077\pi$
$541$	18.3303	0.788081	0.394041	+	0.919093i	$0.371077\pi$
$542$	0	0
$543$	−5.58258	−0.239571
$544$	0	0
$545$	10.7477	0.460382
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−53.7477	−2.28973
$552$	0	0
$553$	7.16515	0.304693
$554$	0	0
$555$	3.00000	0.127343
$556$	0	0
$557$	9.33030	0.395338	0.197669	−	0.980269i	$-0.436663\pi$
$557$	9.33030	0.395338	0.197669	+	0.980269i	$0.436663\pi$
$558$	0	0
$559$	−1.58258	−0.0669358
$560$	0	0
$561$	7.58258	0.320137
$562$	0	0
$563$	37.5826	1.58392	0.791958	−	0.610575i	$-0.209062\pi$
$563$	37.5826	1.58392	0.791958	+	0.610575i	$0.209062\pi$
$564$	0	0
$565$	27.4955	1.15674
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−26.6606	−1.11767	−0.558835	−	0.829279i	$-0.688752\pi$
$569$	−26.6606	−1.11767	−0.558835	+	0.829279i	$0.688752\pi$
$570$	0	0
$571$	−28.8348	−1.20670	−0.603350	−	0.797476i	$-0.706168\pi$
$571$	−28.8348	−1.20670	−0.603350	+	0.797476i	$0.706168\pi$
$572$	0	0
$573$	−11.5826	−0.483869
$574$	0	0
$575$	22.3303	0.931238
$576$	0	0
$577$	−21.9129	−0.912245	−0.456123	−	0.889917i	$-0.650762\pi$
$577$	−21.9129	−0.912245	−0.456123	+	0.889917i	$0.650762\pi$
$578$	0	0
$579$	−2.41742	−0.100465
$580$	0	0
$581$	−11.5826	−0.480526
$582$	0	0
$583$	−9.58258	−0.396870
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−37.7477	−1.55802	−0.779008	−	0.627014i	$-0.784277\pi$
$587$	−37.7477	−1.55802	−0.779008	+	0.627014i	$0.784277\pi$
$588$	0	0
$589$	−23.5826	−0.971703
$590$	0	0
$591$	5.16515	0.212466
$592$	0	0
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	0	0
$595$	−22.7477	−0.932566
$596$	0	0
$597$	9.58258	0.392189
$598$	0	0
$599$	−7.16515	−0.292760	−0.146380	−	0.989228i	$-0.546762\pi$
$599$	−7.16515	−0.292760	−0.146380	+	0.989228i	$0.546762\pi$
$600$	0	0
$601$	24.4955	0.999190	0.499595	−	0.866259i	$-0.333482\pi$
$601$	24.4955	0.999190	0.499595	+	0.866259i	$0.333482\pi$
$602$	0	0
$603$	−8.58258	−0.349510
$604$	0	0
$605$	3.00000	0.121967
$606$	0	0
$607$	21.7477	0.882713	0.441357	−	0.897332i	$-0.354497\pi$
$607$	21.7477	0.882713	0.441357	+	0.897332i	$0.354497\pi$
$608$	0	0
$609$	8.16515	0.330869
$610$	0	0
$611$	−1.41742	−0.0573428
$612$	0	0
$613$	−26.7477	−1.08033	−0.540165	−	0.841559i	$-0.681638\pi$
$613$	−26.7477	−1.08033	−0.540165	+	0.841559i	$0.681638\pi$
$614$	0	0
$615$	−33.4955	−1.35067
$616$	0	0
$617$	2.83485	0.114127	0.0570634	−	0.998371i	$-0.481826\pi$
$617$	2.83485	0.114127	0.0570634	+	0.998371i	$0.481826\pi$
$618$	0	0
$619$	29.0780	1.16874	0.584372	−	0.811486i	$-0.301341\pi$
$619$	29.0780	1.16874	0.584372	+	0.811486i	$0.301341\pi$
$620$	0	0
$621$	5.58258	0.224021
$622$	0	0
$623$	−9.16515	−0.367194
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	6.58258	0.262883
$628$	0	0
$629$	7.58258	0.302337
$630$	0	0
$631$	−23.1652	−0.922190	−0.461095	−	0.887351i	$-0.652543\pi$
$631$	−23.1652	−0.922190	−0.461095	+	0.887351i	$0.652543\pi$
$632$	0	0
$633$	13.1652	0.523268
$634$	0	0
$635$	34.7477	1.37892
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−11.1652	−0.441687
$640$	0	0
$641$	43.5826	1.72141	0.860704	−	0.509106i	$-0.170024\pi$
$641$	43.5826	1.72141	0.860704	+	0.509106i	$0.170024\pi$
$642$	0	0
$643$	−38.2432	−1.50816	−0.754082	−	0.656780i	$-0.771918\pi$
$643$	−38.2432	−1.50816	−0.754082	+	0.656780i	$0.771918\pi$
$644$	0	0
$645$	−4.74773	−0.186942
$646$	0	0
$647$	10.9129	0.429030	0.214515	−	0.976721i	$-0.431183\pi$
$647$	10.9129	0.429030	0.214515	+	0.976721i	$0.431183\pi$
$648$	0	0
$649$	−4.58258	−0.179882
$650$	0	0
$651$	3.58258	0.140412
$652$	0	0
$653$	−30.3303	−1.18692	−0.593458	−	0.804865i	$-0.702238\pi$
$653$	−30.3303	−1.18692	−0.593458	+	0.804865i	$0.702238\pi$
$654$	0	0
$655$	48.0000	1.87552
$656$	0	0
$657$	7.00000	0.273096
$658$	0	0
$659$	28.5826	1.11342	0.556710	−	0.830707i	$-0.312064\pi$
$659$	28.5826	1.11342	0.556710	+	0.830707i	$0.312064\pi$
$660$	0	0
$661$	−39.0780	−1.51996	−0.759980	−	0.649947i	$-0.774791\pi$
$661$	−39.0780	−1.51996	−0.759980	+	0.649947i	$0.774791\pi$
$662$	0	0
$663$	7.58258	0.294483
$664$	0	0
$665$	−19.7477	−0.765784
$666$	0	0
$667$	−45.5826	−1.76496
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	11.2523	0.433743	0.216872	−	0.976200i	$-0.430415\pi$
$673$	11.2523	0.433743	0.216872	+	0.976200i	$0.430415\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−45.1652	−1.73584	−0.867919	−	0.496706i	$-0.834543\pi$
$677$	−45.1652	−1.73584	−0.867919	+	0.496706i	$0.834543\pi$
$678$	0	0
$679$	2.41742	0.0927722
$680$	0	0
$681$	22.0000	0.843042
$682$	0	0
$683$	33.0780	1.26570	0.632848	−	0.774276i	$-0.281886\pi$
$683$	33.0780	1.26570	0.632848	+	0.774276i	$0.281886\pi$
$684$	0	0
$685$	−34.7477	−1.32764
$686$	0	0
$687$	0.747727	0.0285276
$688$	0	0
$689$	−9.58258	−0.365067
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−33.4955	−1.27055
$696$	0	0
$697$	−84.6606	−3.20675
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	6.58258	0.248267
$704$	0	0
$705$	−4.25227	−0.160150
$706$	0	0
$707$	−11.5826	−0.435608
$708$	0	0
$709$	27.6606	1.03882	0.519408	−	0.854526i	$-0.326153\pi$
$709$	27.6606	1.03882	0.519408	+	0.854526i	$0.326153\pi$
$710$	0	0
$711$	−7.16515	−0.268714
$712$	0	0
$713$	−20.0000	−0.749006
$714$	0	0
$715$	3.00000	0.112194
$716$	0	0
$717$	−16.5826	−0.619288
$718$	0	0
$719$	14.0780	0.525022	0.262511	−	0.964929i	$-0.415449\pi$
$719$	14.0780	0.525022	0.262511	+	0.964929i	$0.415449\pi$
$720$	0	0
$721$	−1.16515	−0.0433925
$722$	0	0
$723$	−10.1652	−0.378046
$724$	0	0
$725$	−32.6606	−1.21298
$726$	0	0
$727$	15.9129	0.590176	0.295088	−	0.955470i	$-0.404651\pi$
$727$	15.9129	0.590176	0.295088	+	0.955470i	$0.404651\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−8.58258	−0.316143
$738$	0	0
$739$	31.9129	1.17393	0.586967	−	0.809611i	$-0.300322\pi$
$739$	31.9129	1.17393	0.586967	+	0.809611i	$0.300322\pi$
$740$	0	0
$741$	6.58258	0.241817
$742$	0	0
$743$	53.2432	1.95330	0.976651	−	0.214830i	$-0.0689198\pi$
$743$	53.2432	1.95330	0.976651	+	0.214830i	$0.0689198\pi$
$744$	0	0
$745$	18.4955	0.677621
$746$	0	0
$747$	11.5826	0.423784
$748$	0	0
$749$	−12.5826	−0.459757
$750$	0	0
$751$	8.91288	0.325236	0.162618	−	0.986689i	$-0.448006\pi$
$751$	8.91288	0.325236	0.162618	+	0.986689i	$0.448006\pi$
$752$	0	0
$753$	−7.41742	−0.270306
$754$	0	0
$755$	−10.7477	−0.391150
$756$	0	0
$757$	27.3303	0.993337	0.496668	−	0.867940i	$-0.334557\pi$
$757$	27.3303	0.993337	0.496668	+	0.867940i	$0.334557\pi$
$758$	0	0
$759$	5.58258	0.202635
$760$	0	0
$761$	42.3303	1.53447	0.767236	−	0.641365i	$-0.221631\pi$
$761$	42.3303	1.53447	0.767236	+	0.641365i	$0.221631\pi$
$762$	0	0
$763$	−3.58258	−0.129698
$764$	0	0
$765$	22.7477	0.822446
$766$	0	0
$767$	−4.58258	−0.165467
$768$	0	0
$769$	6.49545	0.234232	0.117116	−	0.993118i	$-0.462635\pi$
$769$	6.49545	0.234232	0.117116	+	0.993118i	$0.462635\pi$
$770$	0	0
$771$	19.0000	0.684268
$772$	0	0
$773$	6.16515	0.221745	0.110873	−	0.993835i	$-0.464635\pi$
$773$	6.16515	0.221745	0.110873	+	0.993835i	$0.464635\pi$
$774$	0	0
$775$	−14.3303	−0.514760
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−73.4955	−2.63325
$780$	0	0
$781$	−11.1652	−0.399521
$782$	0	0
$783$	−8.16515	−0.291799
$784$	0	0
$785$	57.4955	2.05210
$786$	0	0
$787$	−38.5826	−1.37532	−0.687660	−	0.726033i	$-0.741362\pi$
$787$	−38.5826	−1.37532	−0.687660	+	0.726033i	$0.741362\pi$
$788$	0	0
$789$	22.9129	0.815720
$790$	0	0
$791$	−9.16515	−0.325875
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−28.7477	−1.01958
$796$	0	0
$797$	−52.4955	−1.85948	−0.929742	−	0.368211i	$-0.879970\pi$
$797$	−52.4955	−1.85948	−0.929742	+	0.368211i	$0.879970\pi$
$798$	0	0
$799$	−10.7477	−0.380227
$800$	0	0
$801$	9.16515	0.323835
$802$	0	0
$803$	7.00000	0.247025
$804$	0	0
$805$	−16.7477	−0.590280
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	9.33030	0.328036	0.164018	−	0.986457i	$-0.447554\pi$
$809$	9.33030	0.328036	0.164018	+	0.986457i	$0.447554\pi$
$810$	0	0
$811$	2.25227	0.0790880	0.0395440	−	0.999218i	$-0.487409\pi$
$811$	2.25227	0.0790880	0.0395440	+	0.999218i	$0.487409\pi$
$812$	0	0
$813$	−5.41742	−0.189997
$814$	0	0
$815$	−25.7477	−0.901904
$816$	0	0
$817$	−10.4174	−0.364460
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−47.0000	−1.64031	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−47.0000	−1.64031	−0.820156	+	0.572140i	$0.806113\pi$
$822$	0	0
$823$	30.5826	1.06604	0.533021	−	0.846102i	$-0.321057\pi$
$823$	30.5826	1.06604	0.533021	+	0.846102i	$0.321057\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−8.91288	−0.309931	−0.154966	−	0.987920i	$-0.549527\pi$
$827$	−8.91288	−0.309931	−0.154966	+	0.987920i	$0.549527\pi$
$828$	0	0
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	−19.1652	−0.664832
$832$	0	0
$833$	7.58258	0.262721
$834$	0	0
$835$	14.2432	0.492906
$836$	0	0
$837$	−3.58258	−0.123832
$838$	0	0
$839$	−7.08712	−0.244675	−0.122337	−	0.992489i	$-0.539039\pi$
$839$	−7.08712	−0.244675	−0.122337	+	0.992489i	$0.539039\pi$
$840$	0	0
$841$	37.6697	1.29896
$842$	0	0
$843$	−27.3303	−0.941306
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−27.7477	−0.952300
$850$	0	0
$851$	5.58258	0.191368
$852$	0	0
$853$	17.1652	0.587724	0.293862	−	0.955848i	$-0.405059\pi$
$853$	17.1652	0.587724	0.293862	+	0.955848i	$0.405059\pi$
$854$	0	0
$855$	19.7477	0.675358
$856$	0	0
$857$	−7.66970	−0.261992	−0.130996	−	0.991383i	$-0.541817\pi$
$857$	−7.66970	−0.261992	−0.130996	+	0.991383i	$0.541817\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	11.1652	0.380507
$862$	0	0
$863$	−14.4174	−0.490775	−0.245387	−	0.969425i	$-0.578915\pi$
$863$	−14.4174	−0.490775	−0.245387	+	0.969425i	$0.578915\pi$
$864$	0	0
$865$	−21.4955	−0.730867
$866$	0	0
$867$	40.4955	1.37530
$868$	0	0
$869$	−7.16515	−0.243061
$870$	0	0
$871$	−8.58258	−0.290809
$872$	0	0
$873$	−2.41742	−0.0818174
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	9.49545	0.320639	0.160319	−	0.987065i	$-0.448748\pi$
$877$	9.49545	0.320639	0.160319	+	0.987065i	$0.448748\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.6606	−1.26882	−0.634409	−	0.772998i	$-0.718756\pi$
$881$	−37.6606	−1.26882	−0.634409	+	0.772998i	$0.718756\pi$
$882$	0	0
$883$	55.7477	1.87606	0.938030	−	0.346554i	$-0.112648\pi$
$883$	55.7477	1.87606	0.938030	+	0.346554i	$0.112648\pi$
$884$	0	0
$885$	−13.7477	−0.462125
$886$	0	0
$887$	−38.7477	−1.30102	−0.650511	−	0.759497i	$-0.725445\pi$
$887$	−38.7477	−1.30102	−0.650511	+	0.759497i	$0.725445\pi$
$888$	0	0
$889$	−11.5826	−0.388467
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−9.33030	−0.312227
$894$	0	0
$895$	−42.9909	−1.43703
$896$	0	0
$897$	5.58258	0.186397
$898$	0	0
$899$	29.2523	0.975618
$900$	0	0
$901$	−72.6606	−2.42068
$902$	0	0
$903$	1.58258	0.0526648
$904$	0	0
$905$	−16.7477	−0.556713
$906$	0	0
$907$	−42.3303	−1.40555	−0.702777	−	0.711410i	$-0.748057\pi$
$907$	−42.3303	−1.40555	−0.702777	+	0.711410i	$0.748057\pi$
$908$	0	0
$909$	11.5826	0.384170
$910$	0	0
$911$	3.49545	0.115810	0.0579048	−	0.998322i	$-0.481558\pi$
$911$	3.49545	0.115810	0.0579048	+	0.998322i	$0.481558\pi$
$912$	0	0
$913$	11.5826	0.383327
$914$	0	0
$915$	30.0000	0.991769
$916$	0	0
$917$	−16.0000	−0.528367
$918$	0	0
$919$	8.08712	0.266770	0.133385	−	0.991064i	$-0.457415\pi$
$919$	8.08712	0.266770	0.133385	+	0.991064i	$0.457415\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11.1652	−0.367505
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	1.16515	0.0382686
$928$	0	0
$929$	−21.3303	−0.699825	−0.349912	−	0.936782i	$-0.613789\pi$
$929$	−21.3303	−0.699825	−0.349912	+	0.936782i	$0.613789\pi$
$930$	0	0
$931$	6.58258	0.215735
$932$	0	0
$933$	−14.3303	−0.469153
$934$	0	0
$935$	22.7477	0.743930
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	19.5826	0.639053
$940$	0	0
$941$	35.1652	1.14635	0.573176	−	0.819433i	$-0.305711\pi$
$941$	35.1652	1.14635	0.573176	+	0.819433i	$0.305711\pi$
$942$	0	0
$943$	−62.3303	−2.02975
$944$	0	0
$945$	−3.00000	−0.0975900
$946$	0	0
$947$	−45.1652	−1.46767	−0.733835	−	0.679328i	$-0.762272\pi$
$947$	−45.1652	−1.46767	−0.733835	+	0.679328i	$0.762272\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	0	0
$951$	−22.4174	−0.726935
$952$	0	0
$953$	−28.1652	−0.912359	−0.456179	−	0.889888i	$-0.650782\pi$
$953$	−28.1652	−0.912359	−0.456179	+	0.889888i	$0.650782\pi$
$954$	0	0
$955$	−34.7477	−1.12441
$956$	0	0
$957$	−8.16515	−0.263942
$958$	0	0
$959$	11.5826	0.374021
$960$	0	0
$961$	−18.1652	−0.585973
$962$	0	0
$963$	12.5826	0.405468
$964$	0	0
$965$	−7.25227	−0.233459
$966$	0	0
$967$	13.1652	0.423363	0.211681	−	0.977339i	$-0.432106\pi$
$967$	13.1652	0.423363	0.211681	+	0.977339i	$0.432106\pi$
$968$	0	0
$969$	49.9129	1.60343
$970$	0	0
$971$	12.5826	0.403794	0.201897	−	0.979407i	$-0.435289\pi$
$971$	12.5826	0.403794	0.201897	+	0.979407i	$0.435289\pi$
$972$	0	0
$973$	11.1652	0.357938
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	23.2523	0.743906	0.371953	−	0.928252i	$-0.378688\pi$
$977$	23.2523	0.743906	0.371953	+	0.928252i	$0.378688\pi$
$978$	0	0
$979$	9.16515	0.292920
$980$	0	0
$981$	3.58258	0.114383
$982$	0	0
$983$	4.83485	0.154208	0.0771039	−	0.997023i	$-0.475433\pi$
$983$	4.83485	0.154208	0.0771039	+	0.997023i	$0.475433\pi$
$984$	0	0
$985$	15.4955	0.493726
$986$	0	0
$987$	1.41742	0.0451171
$988$	0	0
$989$	−8.83485	−0.280932
$990$	0	0
$991$	20.2523	0.643335	0.321667	−	0.946853i	$-0.395757\pi$
$991$	20.2523	0.643335	0.321667	+	0.946853i	$0.395757\pi$
$992$	0	0
$993$	3.16515	0.100443
$994$	0	0
$995$	28.7477	0.911364
$996$	0	0
$997$	10.6606	0.337625	0.168812	−	0.985648i	$-0.446007\pi$
$997$	10.6606	0.337625	0.168812	+	0.985648i	$0.446007\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bl.1.2		2
4.3	odd	2		231.2.a.b.1.2	✓	2
12.11	even	2		693.2.a.j.1.1		2
20.19	odd	2		5775.2.a.bn.1.1		2
28.27	even	2		1617.2.a.o.1.2		2
44.43	even	2		2541.2.a.z.1.1		2
84.83	odd	2		4851.2.a.ba.1.1		2
132.131	odd	2		7623.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.b.1.2	✓	2	4.3	odd	2
693.2.a.j.1.1		2	12.11	even	2
1617.2.a.o.1.2		2	28.27	even	2
2541.2.a.z.1.1		2	44.43	even	2
3696.2.a.bl.1.2		2	1.1	even	1	trivial
4851.2.a.ba.1.1		2	84.83	odd	2
5775.2.a.bn.1.1		2	20.19	odd	2
7623.2.a.bf.1.2		2	132.131	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:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} +7.58258 q^{17} +6.58258 q^{19} -1.00000 q^{21} +5.58258 q^{23} +4.00000 q^{25} +1.00000 q^{27} -8.16515 q^{29} -3.58258 q^{31} +1.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} +1.00000 q^{39} -11.1652 q^{41} -1.58258 q^{43} +3.00000 q^{45} -1.41742 q^{47} +1.00000 q^{49} +7.58258 q^{51} -9.58258 q^{53} +3.00000 q^{55} +6.58258 q^{57} -4.58258 q^{59} +10.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -8.58258 q^{67} +5.58258 q^{69} -11.1652 q^{71} +7.00000 q^{73} +4.00000 q^{75} -1.00000 q^{77} -7.16515 q^{79} +1.00000 q^{81} +11.5826 q^{83} +22.7477 q^{85} -8.16515 q^{87} +9.16515 q^{89} -1.00000 q^{91} -3.58258 q^{93} +19.7477 q^{95} -2.41742 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 + 6 * q^43 + 6 * q^45 - 12 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 6 * q^55 + 4 * q^57 + 20 * q^61 - 2 * q^63 + 6 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 + 14 * q^73 + 8 * q^75 - 2 * q^77 + 4 * q^79 + 2 * q^81 + 14 * q^83 + 18 * q^85 + 2 * q^87 - 2 * q^91 + 2 * q^93 + 12 * q^95 - 14 * q^97 + 2 * q^99

Introduction

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