LMFDB - Embedded newform 3640.2.a.l.1.2

Newspace parameters

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$29.0655463357$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

$f(q)$	$=$	$q+1.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.30278 q^{9} +O(q^{10})$	$q+1.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.30278 q^{9} -4.60555 q^{11} -1.00000 q^{13} +1.30278 q^{15} +6.30278 q^{17} -6.30278 q^{19} +1.30278 q^{21} -8.00000 q^{23} +1.00000 q^{25} -5.60555 q^{27} +1.90833 q^{29} -0.697224 q^{31} -6.00000 q^{33} +1.00000 q^{35} -3.90833 q^{37} -1.30278 q^{39} -4.30278 q^{41} -2.00000 q^{43} -1.30278 q^{45} -4.60555 q^{47} +1.00000 q^{49} +8.21110 q^{51} -2.60555 q^{53} -4.60555 q^{55} -8.21110 q^{57} +11.1194 q^{59} -1.21110 q^{61} -1.30278 q^{63} -1.00000 q^{65} +8.90833 q^{67} -10.4222 q^{69} -2.60555 q^{71} -8.60555 q^{73} +1.30278 q^{75} -4.60555 q^{77} +3.51388 q^{79} -3.39445 q^{81} -11.8167 q^{83} +6.30278 q^{85} +2.48612 q^{87} -7.30278 q^{89} -1.00000 q^{91} -0.908327 q^{93} -6.30278 q^{95} -13.2111 q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9	$ 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 9 q^{17} - 9 q^{19} - q^{21} - 16 q^{23} + 2 q^{25} - 4 q^{27} - 7 q^{29} - 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 5 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{55} - 2 q^{57} - 3 q^{59} + 12 q^{61} + q^{63} - 2 q^{65} + 7 q^{67} + 8 q^{69} + 2 q^{71} - 10 q^{73} - q^{75} - 2 q^{77} - 11 q^{79} - 14 q^{81} - 2 q^{83} + 9 q^{85} + 23 q^{87} - 11 q^{89} - 2 q^{91} + 9 q^{93} - 9 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9 - 2 * q^11 - 2 * q^13 - q^15 + 9 * q^17 - 9 * q^19 - q^21 - 16 * q^23 + 2 * q^25 - 4 * q^27 - 7 * q^29 - 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 5 * q^41 - 4 * q^43 + q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 + 2 * q^53 - 2 * q^55 - 2 * q^57 - 3 * q^59 + 12 * q^61 + q^63 - 2 * q^65 + 7 * q^67 + 8 * q^69 + 2 * q^71 - 10 * q^73 - q^75 - 2 * q^77 - 11 * q^79 - 14 * q^81 - 2 * q^83 + 9 * q^85 + 23 * q^87 - 11 * q^89 - 2 * q^91 + 9 * q^93 - 9 * q^95 - 12 * q^97 + 12 * 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
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	−4.60555	−1.38863	−0.694313	−	0.719673i	$-0.744292\pi$
$11$	−4.60555	−1.38863	−0.694313	+	0.719673i	$0.744292\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.30278	0.336375
$16$	0	0
$17$	6.30278	1.52865	0.764324	−	0.644833i	$-0.223073\pi$
$17$	6.30278	1.52865	0.764324	+	0.644833i	$0.223073\pi$
$18$	0	0
$19$	−6.30278	−1.44596	−0.722978	−	0.690871i	$-0.757227\pi$
$19$	−6.30278	−1.44596	−0.722978	+	0.690871i	$0.757227\pi$
$20$	0	0
$21$	1.30278	0.284289
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.60555	−1.07879
$28$	0	0
$29$	1.90833	0.354367	0.177184	−	0.984178i	$-0.443301\pi$
$29$	1.90833	0.354367	0.177184	+	0.984178i	$0.443301\pi$
$30$	0	0
$31$	−0.697224	−0.125225	−0.0626126	−	0.998038i	$-0.519943\pi$
$31$	−0.697224	−0.125225	−0.0626126	+	0.998038i	$0.519943\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.90833	−0.642525	−0.321262	−	0.946990i	$-0.604107\pi$
$37$	−3.90833	−0.642525	−0.321262	+	0.946990i	$0.604107\pi$
$38$	0	0
$39$	−1.30278	−0.208611
$40$	0	0
$41$	−4.30278	−0.671981	−0.335990	−	0.941865i	$-0.609071\pi$
$41$	−4.30278	−0.671981	−0.335990	+	0.941865i	$0.609071\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−1.30278	−0.194206
$46$	0	0
$47$	−4.60555	−0.671789	−0.335894	−	0.941900i	$-0.609039\pi$
$47$	−4.60555	−0.671789	−0.335894	+	0.941900i	$0.609039\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.21110	1.14978
$52$	0	0
$53$	−2.60555	−0.357900	−0.178950	−	0.983858i	$-0.557270\pi$
$53$	−2.60555	−0.357900	−0.178950	+	0.983858i	$0.557270\pi$
$54$	0	0
$55$	−4.60555	−0.621012
$56$	0	0
$57$	−8.21110	−1.08759
$58$	0	0
$59$	11.1194	1.44763	0.723813	−	0.689996i	$-0.242388\pi$
$59$	11.1194	1.44763	0.723813	+	0.689996i	$0.242388\pi$
$60$	0	0
$61$	−1.21110	−0.155066	−0.0775329	−	0.996990i	$-0.524704\pi$
$61$	−1.21110	−0.155066	−0.0775329	+	0.996990i	$0.524704\pi$
$62$	0	0
$63$	−1.30278	−0.164134
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	8.90833	1.08833	0.544163	−	0.838980i	$-0.316847\pi$
$67$	8.90833	1.08833	0.544163	+	0.838980i	$0.316847\pi$
$68$	0	0
$69$	−10.4222	−1.25469
$70$	0	0
$71$	−2.60555	−0.309222	−0.154611	−	0.987975i	$-0.549412\pi$
$71$	−2.60555	−0.309222	−0.154611	+	0.987975i	$0.549412\pi$
$72$	0	0
$73$	−8.60555	−1.00720	−0.503602	−	0.863936i	$-0.667992\pi$
$73$	−8.60555	−1.00720	−0.503602	+	0.863936i	$0.667992\pi$
$74$	0	0
$75$	1.30278	0.150432
$76$	0	0
$77$	−4.60555	−0.524851
$78$	0	0
$79$	3.51388	0.395342	0.197671	−	0.980268i	$-0.436662\pi$
$79$	3.51388	0.395342	0.197671	+	0.980268i	$0.436662\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−11.8167	−1.29705	−0.648523	−	0.761195i	$-0.724613\pi$
$83$	−11.8167	−1.29705	−0.648523	+	0.761195i	$0.724613\pi$
$84$	0	0
$85$	6.30278	0.683632
$86$	0	0
$87$	2.48612	0.266540
$88$	0	0
$89$	−7.30278	−0.774093	−0.387046	−	0.922060i	$-0.626505\pi$
$89$	−7.30278	−0.774093	−0.387046	+	0.922060i	$0.626505\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.908327	−0.0941891
$94$	0	0
$95$	−6.30278	−0.646651
$96$	0	0
$97$	−13.2111	−1.34138	−0.670692	−	0.741736i	$-0.734003\pi$
$97$	−13.2111	−1.34138	−0.670692	+	0.741736i	$0.734003\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−8.90833	−0.877764	−0.438882	−	0.898545i	$-0.644625\pi$
$103$	−8.90833	−0.877764	−0.438882	+	0.898545i	$0.644625\pi$
$104$	0	0
$105$	1.30278	0.127138
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−5.09167	−0.483280
$112$	0	0
$113$	7.81665	0.735329	0.367664	−	0.929959i	$-0.380157\pi$
$113$	7.81665	0.735329	0.367664	+	0.929959i	$0.380157\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	1.30278	0.120442
$118$	0	0
$119$	6.30278	0.577774
$120$	0	0
$121$	10.2111	0.928282
$122$	0	0
$123$	−5.60555	−0.505436
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.60555	−0.763619	−0.381810	−	0.924241i	$-0.624699\pi$
$127$	−8.60555	−0.763619	−0.381810	+	0.924241i	$0.624699\pi$
$128$	0	0
$129$	−2.60555	−0.229406
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−6.30278	−0.546520
$134$	0	0
$135$	−5.60555	−0.482449
$136$	0	0
$137$	2.69722	0.230439	0.115220	−	0.993340i	$-0.463243\pi$
$137$	2.69722	0.230439	0.115220	+	0.993340i	$0.463243\pi$
$138$	0	0
$139$	16.4222	1.39291	0.696457	−	0.717599i	$-0.254759\pi$
$139$	16.4222	1.39291	0.696457	+	0.717599i	$0.254759\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	4.60555	0.385136
$144$	0	0
$145$	1.90833	0.158478
$146$	0	0
$147$	1.30278	0.107451
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	−8.21110	−0.663828
$154$	0	0
$155$	−0.697224	−0.0560024
$156$	0	0
$157$	10.5139	0.839099	0.419549	−	0.907732i	$-0.362188\pi$
$157$	10.5139	0.839099	0.419549	+	0.907732i	$0.362188\pi$
$158$	0	0
$159$	−3.39445	−0.269197
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	25.1194	1.96751	0.983753	−	0.179528i	$-0.0574572\pi$
$163$	25.1194	1.96751	0.983753	+	0.179528i	$0.0574572\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−23.2111	−1.79613	−0.898065	−	0.439864i	$-0.855027\pi$
$167$	−23.2111	−1.79613	−0.898065	+	0.439864i	$0.855027\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.21110	0.627919
$172$	0	0
$173$	18.6972	1.42152	0.710762	−	0.703433i	$-0.248350\pi$
$173$	18.6972	1.42152	0.710762	+	0.703433i	$0.248350\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	14.4861	1.08884
$178$	0	0
$179$	−9.09167	−0.679544	−0.339772	−	0.940508i	$-0.610350\pi$
$179$	−9.09167	−0.679544	−0.339772	+	0.940508i	$0.610350\pi$
$180$	0	0
$181$	9.81665	0.729666	0.364833	−	0.931073i	$-0.381126\pi$
$181$	9.81665	0.729666	0.364833	+	0.931073i	$0.381126\pi$
$182$	0	0
$183$	−1.57779	−0.116634
$184$	0	0
$185$	−3.90833	−0.287346
$186$	0	0
$187$	−29.0278	−2.12272
$188$	0	0
$189$	−5.60555	−0.407744
$190$	0	0
$191$	−16.5139	−1.19490	−0.597451	−	0.801905i	$-0.703820\pi$
$191$	−16.5139	−1.19490	−0.597451	+	0.801905i	$0.703820\pi$
$192$	0	0
$193$	10.9083	0.785199	0.392599	−	0.919710i	$-0.371576\pi$
$193$	10.9083	0.785199	0.392599	+	0.919710i	$0.371576\pi$
$194$	0	0
$195$	−1.30278	−0.0932937
$196$	0	0
$197$	13.1194	0.934721	0.467360	−	0.884067i	$-0.345205\pi$
$197$	13.1194	0.934721	0.467360	+	0.884067i	$0.345205\pi$
$198$	0	0
$199$	1.21110	0.0858528	0.0429264	−	0.999078i	$-0.486332\pi$
$199$	1.21110	0.0858528	0.0429264	+	0.999078i	$0.486332\pi$
$200$	0	0
$201$	11.6056	0.818592
$202$	0	0
$203$	1.90833	0.133938
$204$	0	0
$205$	−4.30278	−0.300519
$206$	0	0
$207$	10.4222	0.724393
$208$	0	0
$209$	29.0278	2.00789
$210$	0	0
$211$	11.5139	0.792648	0.396324	−	0.918111i	$-0.370286\pi$
$211$	11.5139	0.792648	0.396324	+	0.918111i	$0.370286\pi$
$212$	0	0
$213$	−3.39445	−0.232584
$214$	0	0
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−0.697224	−0.0473307
$218$	0	0
$219$	−11.2111	−0.757576
$220$	0	0
$221$	−6.30278	−0.423971
$222$	0	0
$223$	13.2111	0.884681	0.442340	−	0.896847i	$-0.354148\pi$
$223$	13.2111	0.884681	0.442340	+	0.896847i	$0.354148\pi$
$224$	0	0
$225$	−1.30278	−0.0868517
$226$	0	0
$227$	−19.8167	−1.31528	−0.657639	−	0.753333i	$-0.728445\pi$
$227$	−19.8167	−1.31528	−0.657639	+	0.753333i	$0.728445\pi$
$228$	0	0
$229$	−0.302776	−0.0200080	−0.0100040	−	0.999950i	$-0.503184\pi$
$229$	−0.302776	−0.0200080	−0.0100040	+	0.999950i	$0.503184\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−1.81665	−0.119013	−0.0595065	−	0.998228i	$-0.518953\pi$
$233$	−1.81665	−0.119013	−0.0595065	+	0.998228i	$0.518953\pi$
$234$	0	0
$235$	−4.60555	−0.300433
$236$	0	0
$237$	4.57779	0.297360
$238$	0	0
$239$	21.6333	1.39934	0.699671	−	0.714465i	$-0.253330\pi$
$239$	21.6333	1.39934	0.699671	+	0.714465i	$0.253330\pi$
$240$	0	0
$241$	−16.9083	−1.08916	−0.544581	−	0.838709i	$-0.683311\pi$
$241$	−16.9083	−1.08916	−0.544581	+	0.838709i	$0.683311\pi$
$242$	0	0
$243$	12.3944	0.795104
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	6.30278	0.401036
$248$	0	0
$249$	−15.3944	−0.975584
$250$	0	0
$251$	−27.6333	−1.74420	−0.872099	−	0.489329i	$-0.837242\pi$
$251$	−27.6333	−1.74420	−0.872099	+	0.489329i	$0.837242\pi$
$252$	0	0
$253$	36.8444	2.31639
$254$	0	0
$255$	8.21110	0.514199
$256$	0	0
$257$	21.6333	1.34945	0.674724	−	0.738070i	$-0.264263\pi$
$257$	21.6333	1.34945	0.674724	+	0.738070i	$0.264263\pi$
$258$	0	0
$259$	−3.90833	−0.242852
$260$	0	0
$261$	−2.48612	−0.153887
$262$	0	0
$263$	15.6333	0.963991	0.481996	−	0.876174i	$-0.339912\pi$
$263$	15.6333	0.963991	0.481996	+	0.876174i	$0.339912\pi$
$264$	0	0
$265$	−2.60555	−0.160058
$266$	0	0
$267$	−9.51388	−0.582240
$268$	0	0
$269$	−17.8167	−1.08630	−0.543150	−	0.839636i	$-0.682769\pi$
$269$	−17.8167	−1.08630	−0.543150	+	0.839636i	$0.682769\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−1.30278	−0.0788476
$274$	0	0
$275$	−4.60555	−0.277725
$276$	0	0
$277$	1.21110	0.0727681	0.0363840	−	0.999338i	$-0.488416\pi$
$277$	1.21110	0.0727681	0.0363840	+	0.999338i	$0.488416\pi$
$278$	0	0
$279$	0.908327	0.0543801
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−15.0917	−0.897107	−0.448553	−	0.893756i	$-0.648061\pi$
$283$	−15.0917	−0.897107	−0.448553	+	0.893756i	$0.648061\pi$
$284$	0	0
$285$	−8.21110	−0.486384
$286$	0	0
$287$	−4.30278	−0.253985
$288$	0	0
$289$	22.7250	1.33676
$290$	0	0
$291$	−17.2111	−1.00893
$292$	0	0
$293$	2.42221	0.141507	0.0707534	−	0.997494i	$-0.477460\pi$
$293$	2.42221	0.141507	0.0707534	+	0.997494i	$0.477460\pi$
$294$	0	0
$295$	11.1194	0.647398
$296$	0	0
$297$	25.8167	1.49803
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	7.81665	0.449055
$304$	0	0
$305$	−1.21110	−0.0693475
$306$	0	0
$307$	7.39445	0.422023	0.211012	−	0.977484i	$-0.432324\pi$
$307$	7.39445	0.422023	0.211012	+	0.977484i	$0.432324\pi$
$308$	0	0
$309$	−11.6056	−0.660217
$310$	0	0
$311$	−23.0278	−1.30578	−0.652892	−	0.757451i	$-0.726445\pi$
$311$	−23.0278	−1.30578	−0.652892	+	0.757451i	$0.726445\pi$
$312$	0	0
$313$	−30.3305	−1.71438	−0.857192	−	0.514998i	$-0.827793\pi$
$313$	−30.3305	−1.71438	−0.857192	+	0.514998i	$0.827793\pi$
$314$	0	0
$315$	−1.30278	−0.0734031
$316$	0	0
$317$	−16.4222	−0.922363	−0.461181	−	0.887306i	$-0.652574\pi$
$317$	−16.4222	−0.922363	−0.461181	+	0.887306i	$0.652574\pi$
$318$	0	0
$319$	−8.78890	−0.492084
$320$	0	0
$321$	−7.81665	−0.436283
$322$	0	0
$323$	−39.7250	−2.21036
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−5.21110	−0.288175
$328$	0	0
$329$	−4.60555	−0.253912
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	0	0
$333$	5.09167	0.279022
$334$	0	0
$335$	8.90833	0.486714
$336$	0	0
$337$	−24.4222	−1.33036	−0.665181	−	0.746682i	$-0.731646\pi$
$337$	−24.4222	−1.33036	−0.665181	+	0.746682i	$0.731646\pi$
$338$	0	0
$339$	10.1833	0.553083
$340$	0	0
$341$	3.21110	0.173891
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−10.4222	−0.561113
$346$	0	0
$347$	−29.8167	−1.60064	−0.800321	−	0.599572i	$-0.795338\pi$
$347$	−29.8167	−1.60064	−0.800321	+	0.599572i	$0.795338\pi$
$348$	0	0
$349$	−22.9361	−1.22774	−0.613870	−	0.789407i	$-0.710388\pi$
$349$	−22.9361	−1.22774	−0.613870	+	0.789407i	$0.710388\pi$
$350$	0	0
$351$	5.60555	0.299202
$352$	0	0
$353$	17.6333	0.938526	0.469263	−	0.883058i	$-0.344520\pi$
$353$	17.6333	0.938526	0.469263	+	0.883058i	$0.344520\pi$
$354$	0	0
$355$	−2.60555	−0.138288
$356$	0	0
$357$	8.21110	0.434578
$358$	0	0
$359$	25.0278	1.32091	0.660457	−	0.750864i	$-0.270362\pi$
$359$	25.0278	1.32091	0.660457	+	0.750864i	$0.270362\pi$
$360$	0	0
$361$	20.7250	1.09079
$362$	0	0
$363$	13.3028	0.698215
$364$	0	0
$365$	−8.60555	−0.450435
$366$	0	0
$367$	30.4222	1.58803	0.794013	−	0.607901i	$-0.207988\pi$
$367$	30.4222	1.58803	0.794013	+	0.607901i	$0.207988\pi$
$368$	0	0
$369$	5.60555	0.291813
$370$	0	0
$371$	−2.60555	−0.135273
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	1.30278	0.0672750
$376$	0	0
$377$	−1.90833	−0.0982838
$378$	0	0
$379$	−29.6333	−1.52216	−0.761080	−	0.648658i	$-0.775331\pi$
$379$	−29.6333	−1.52216	−0.761080	+	0.648658i	$0.775331\pi$
$380$	0	0
$381$	−11.2111	−0.574362
$382$	0	0
$383$	−24.2389	−1.23855	−0.619274	−	0.785175i	$-0.712573\pi$
$383$	−24.2389	−1.23855	−0.619274	+	0.785175i	$0.712573\pi$
$384$	0	0
$385$	−4.60555	−0.234721
$386$	0	0
$387$	2.60555	0.132448
$388$	0	0
$389$	9.51388	0.482373	0.241186	−	0.970479i	$-0.422463\pi$
$389$	9.51388	0.482373	0.241186	+	0.970479i	$0.422463\pi$
$390$	0	0
$391$	−50.4222	−2.54996
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.51388	0.176802
$396$	0	0
$397$	29.8167	1.49645	0.748227	−	0.663442i	$-0.230905\pi$
$397$	29.8167	1.49645	0.748227	+	0.663442i	$0.230905\pi$
$398$	0	0
$399$	−8.21110	−0.411069
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	0.697224	0.0347312
$404$	0	0
$405$	−3.39445	−0.168672
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	3.51388	0.173327
$412$	0	0
$413$	11.1194	0.547151
$414$	0	0
$415$	−11.8167	−0.580057
$416$	0	0
$417$	21.3944	1.04769
$418$	0	0
$419$	27.6333	1.34998	0.674988	−	0.737829i	$-0.264149\pi$
$419$	27.6333	1.34998	0.674988	+	0.737829i	$0.264149\pi$
$420$	0	0
$421$	16.4222	0.800369	0.400185	−	0.916435i	$-0.368946\pi$
$421$	16.4222	0.800369	0.400185	+	0.916435i	$0.368946\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	6.30278	0.305730
$426$	0	0
$427$	−1.21110	−0.0586094
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	21.9361	1.05418	0.527090	−	0.849809i	$-0.323283\pi$
$433$	21.9361	1.05418	0.527090	+	0.849809i	$0.323283\pi$
$434$	0	0
$435$	2.48612	0.119200
$436$	0	0
$437$	50.4222	2.41202
$438$	0	0
$439$	4.78890	0.228562	0.114281	−	0.993448i	$-0.463544\pi$
$439$	4.78890	0.228562	0.114281	+	0.993448i	$0.463544\pi$
$440$	0	0
$441$	−1.30278	−0.0620369
$442$	0	0
$443$	9.39445	0.446344	0.223172	−	0.974779i	$-0.428359\pi$
$443$	9.39445	0.446344	0.223172	+	0.974779i	$0.428359\pi$
$444$	0	0
$445$	−7.30278	−0.346185
$446$	0	0
$447$	10.4222	0.492953
$448$	0	0
$449$	1.02776	0.0485028	0.0242514	−	0.999706i	$-0.492280\pi$
$449$	1.02776	0.0485028	0.0242514	+	0.999706i	$0.492280\pi$
$450$	0	0
$451$	19.8167	0.933130
$452$	0	0
$453$	−13.0278	−0.612097
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	9.48612	0.443742	0.221871	−	0.975076i	$-0.428784\pi$
$457$	9.48612	0.443742	0.221871	+	0.975076i	$0.428784\pi$
$458$	0	0
$459$	−35.3305	−1.64909
$460$	0	0
$461$	14.0917	0.656315	0.328157	−	0.944623i	$-0.393572\pi$
$461$	14.0917	0.656315	0.328157	+	0.944623i	$0.393572\pi$
$462$	0	0
$463$	−26.7250	−1.24202	−0.621008	−	0.783805i	$-0.713276\pi$
$463$	−26.7250	−1.24202	−0.621008	+	0.783805i	$0.713276\pi$
$464$	0	0
$465$	−0.908327	−0.0421227
$466$	0	0
$467$	−37.5139	−1.73594	−0.867968	−	0.496621i	$-0.834574\pi$
$467$	−37.5139	−1.73594	−0.867968	+	0.496621i	$0.834574\pi$
$468$	0	0
$469$	8.90833	0.411348
$470$	0	0
$471$	13.6972	0.631135
$472$	0	0
$473$	9.21110	0.423527
$474$	0	0
$475$	−6.30278	−0.289191
$476$	0	0
$477$	3.39445	0.155421
$478$	0	0
$479$	30.0917	1.37492	0.687462	−	0.726221i	$-0.258725\pi$
$479$	30.0917	1.37492	0.687462	+	0.726221i	$0.258725\pi$
$480$	0	0
$481$	3.90833	0.178204
$482$	0	0
$483$	−10.4222	−0.474227
$484$	0	0
$485$	−13.2111	−0.599885
$486$	0	0
$487$	26.6972	1.20977	0.604883	−	0.796314i	$-0.293220\pi$
$487$	26.6972	1.20977	0.604883	+	0.796314i	$0.293220\pi$
$488$	0	0
$489$	32.7250	1.47987
$490$	0	0
$491$	−1.57779	−0.0712049	−0.0356024	−	0.999366i	$-0.511335\pi$
$491$	−1.57779	−0.0712049	−0.0356024	+	0.999366i	$0.511335\pi$
$492$	0	0
$493$	12.0278	0.541703
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	−2.60555	−0.116875
$498$	0	0
$499$	26.0000	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$499$	26.0000	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$500$	0	0
$501$	−30.2389	−1.35097
$502$	0	0
$503$	15.6333	0.697055	0.348527	−	0.937299i	$-0.386682\pi$
$503$	15.6333	0.697055	0.348527	+	0.937299i	$0.386682\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	1.30278	0.0578583
$508$	0	0
$509$	15.3305	0.679514	0.339757	−	0.940513i	$-0.389655\pi$
$509$	15.3305	0.679514	0.339757	+	0.940513i	$0.389655\pi$
$510$	0	0
$511$	−8.60555	−0.380687
$512$	0	0
$513$	35.3305	1.55988
$514$	0	0
$515$	−8.90833	−0.392548
$516$	0	0
$517$	21.2111	0.932863
$518$	0	0
$519$	24.3583	1.06921
$520$	0	0
$521$	28.2389	1.23717	0.618583	−	0.785719i	$-0.287707\pi$
$521$	28.2389	1.23717	0.618583	+	0.785719i	$0.287707\pi$
$522$	0	0
$523$	5.57779	0.243900	0.121950	−	0.992536i	$-0.461085\pi$
$523$	5.57779	0.243900	0.121950	+	0.992536i	$0.461085\pi$
$524$	0	0
$525$	1.30278	0.0568578
$526$	0	0
$527$	−4.39445	−0.191425
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−14.4861	−0.628644
$532$	0	0
$533$	4.30278	0.186374
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−11.8444	−0.511124
$538$	0	0
$539$	−4.60555	−0.198375
$540$	0	0
$541$	−34.0555	−1.46416	−0.732080	−	0.681218i	$-0.761450\pi$
$541$	−34.0555	−1.46416	−0.732080	+	0.681218i	$0.761450\pi$
$542$	0	0
$543$	12.7889	0.548824
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	5.63331	0.240863	0.120431	−	0.992722i	$-0.461572\pi$
$547$	5.63331	0.240863	0.120431	+	0.992722i	$0.461572\pi$
$548$	0	0
$549$	1.57779	0.0673386
$550$	0	0
$551$	−12.0278	−0.512400
$552$	0	0
$553$	3.51388	0.149425
$554$	0	0
$555$	−5.09167	−0.216129
$556$	0	0
$557$	14.9083	0.631686	0.315843	−	0.948811i	$-0.397713\pi$
$557$	14.9083	0.631686	0.315843	+	0.948811i	$0.397713\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	−37.8167	−1.59662
$562$	0	0
$563$	−12.5416	−0.528567	−0.264283	−	0.964445i	$-0.585135\pi$
$563$	−12.5416	−0.528567	−0.264283	+	0.964445i	$0.585135\pi$
$564$	0	0
$565$	7.81665	0.328849
$566$	0	0
$567$	−3.39445	−0.142553
$568$	0	0
$569$	−22.7527	−0.953844	−0.476922	−	0.878946i	$-0.658248\pi$
$569$	−22.7527	−0.953844	−0.476922	+	0.878946i	$0.658248\pi$
$570$	0	0
$571$	21.7250	0.909162	0.454581	−	0.890705i	$-0.349789\pi$
$571$	21.7250	0.909162	0.454581	+	0.890705i	$0.349789\pi$
$572$	0	0
$573$	−21.5139	−0.898755
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	−7.39445	−0.307835	−0.153917	−	0.988084i	$-0.549189\pi$
$577$	−7.39445	−0.307835	−0.153917	+	0.988084i	$0.549189\pi$
$578$	0	0
$579$	14.2111	0.590593
$580$	0	0
$581$	−11.8167	−0.490237
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	1.30278	0.0538631
$586$	0	0
$587$	−19.8167	−0.817921	−0.408960	−	0.912552i	$-0.634109\pi$
$587$	−19.8167	−0.817921	−0.408960	+	0.912552i	$0.634109\pi$
$588$	0	0
$589$	4.39445	0.181070
$590$	0	0
$591$	17.0917	0.703057
$592$	0	0
$593$	−43.8167	−1.79933	−0.899667	−	0.436576i	$-0.856191\pi$
$593$	−43.8167	−1.79933	−0.899667	+	0.436576i	$0.856191\pi$
$594$	0	0
$595$	6.30278	0.258389
$596$	0	0
$597$	1.57779	0.0645748
$598$	0	0
$599$	−10.4222	−0.425840	−0.212920	−	0.977070i	$-0.568297\pi$
$599$	−10.4222	−0.425840	−0.212920	+	0.977070i	$0.568297\pi$
$600$	0	0
$601$	22.7889	0.929579	0.464789	−	0.885421i	$-0.346130\pi$
$601$	22.7889	0.929579	0.464789	+	0.885421i	$0.346130\pi$
$602$	0	0
$603$	−11.6056	−0.472615
$604$	0	0
$605$	10.2111	0.415140
$606$	0	0
$607$	6.51388	0.264390	0.132195	−	0.991224i	$-0.457797\pi$
$607$	6.51388	0.264390	0.132195	+	0.991224i	$0.457797\pi$
$608$	0	0
$609$	2.48612	0.100743
$610$	0	0
$611$	4.60555	0.186321
$612$	0	0
$613$	38.8444	1.56891	0.784455	−	0.620185i	$-0.212943\pi$
$613$	38.8444	1.56891	0.784455	+	0.620185i	$0.212943\pi$
$614$	0	0
$615$	−5.60555	−0.226038
$616$	0	0
$617$	−6.90833	−0.278119	−0.139059	−	0.990284i	$-0.544408\pi$
$617$	−6.90833	−0.278119	−0.139059	+	0.990284i	$0.544408\pi$
$618$	0	0
$619$	−21.7250	−0.873201	−0.436600	−	0.899656i	$-0.643818\pi$
$619$	−21.7250	−0.873201	−0.436600	+	0.899656i	$0.643818\pi$
$620$	0	0
$621$	44.8444	1.79954
$622$	0	0
$623$	−7.30278	−0.292580
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	37.8167	1.51025
$628$	0	0
$629$	−24.6333	−0.982194
$630$	0	0
$631$	−15.8167	−0.629651	−0.314826	−	0.949150i	$-0.601946\pi$
$631$	−15.8167	−0.629651	−0.314826	+	0.949150i	$0.601946\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	−8.60555	−0.341501
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	3.39445	0.134282
$640$	0	0
$641$	−6.27502	−0.247848	−0.123924	−	0.992292i	$-0.539548\pi$
$641$	−6.27502	−0.247848	−0.123924	+	0.992292i	$0.539548\pi$
$642$	0	0
$643$	−14.8444	−0.585406	−0.292703	−	0.956203i	$-0.594555\pi$
$643$	−14.8444	−0.585406	−0.292703	+	0.956203i	$0.594555\pi$
$644$	0	0
$645$	−2.60555	−0.102593
$646$	0	0
$647$	18.5139	0.727856	0.363928	−	0.931427i	$-0.381435\pi$
$647$	18.5139	0.727856	0.363928	+	0.931427i	$0.381435\pi$
$648$	0	0
$649$	−51.2111	−2.01021
$650$	0	0
$651$	−0.908327	−0.0356001
$652$	0	0
$653$	−26.4222	−1.03398	−0.516990	−	0.855991i	$-0.672948\pi$
$653$	−26.4222	−1.03398	−0.516990	+	0.855991i	$0.672948\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.2111	0.437387
$658$	0	0
$659$	31.1472	1.21332	0.606661	−	0.794961i	$-0.292509\pi$
$659$	31.1472	1.21332	0.606661	+	0.794961i	$0.292509\pi$
$660$	0	0
$661$	33.5139	1.30354	0.651769	−	0.758417i	$-0.274027\pi$
$661$	33.5139	1.30354	0.651769	+	0.758417i	$0.274027\pi$
$662$	0	0
$663$	−8.21110	−0.318893
$664$	0	0
$665$	−6.30278	−0.244411
$666$	0	0
$667$	−15.2666	−0.591126
$668$	0	0
$669$	17.2111	0.665420
$670$	0	0
$671$	5.57779	0.215328
$672$	0	0
$673$	−13.8167	−0.532593	−0.266296	−	0.963891i	$-0.585800\pi$
$673$	−13.8167	−0.532593	−0.266296	+	0.963891i	$0.585800\pi$
$674$	0	0
$675$	−5.60555	−0.215758
$676$	0	0
$677$	20.4222	0.784889	0.392445	−	0.919776i	$-0.371629\pi$
$677$	20.4222	0.784889	0.392445	+	0.919776i	$0.371629\pi$
$678$	0	0
$679$	−13.2111	−0.506996
$680$	0	0
$681$	−25.8167	−0.989296
$682$	0	0
$683$	−43.9361	−1.68117	−0.840584	−	0.541682i	$-0.817788\pi$
$683$	−43.9361	−1.68117	−0.840584	+	0.541682i	$0.817788\pi$
$684$	0	0
$685$	2.69722	0.103056
$686$	0	0
$687$	−0.394449	−0.0150492
$688$	0	0
$689$	2.60555	0.0992636
$690$	0	0
$691$	−13.9083	−0.529098	−0.264549	−	0.964372i	$-0.585223\pi$
$691$	−13.9083	−0.529098	−0.264549	+	0.964372i	$0.585223\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	16.4222	0.622930
$696$	0	0
$697$	−27.1194	−1.02722
$698$	0	0
$699$	−2.36669	−0.0895165
$700$	0	0
$701$	−9.88057	−0.373184	−0.186592	−	0.982437i	$-0.559744\pi$
$701$	−9.88057	−0.373184	−0.186592	+	0.982437i	$0.559744\pi$
$702$	0	0
$703$	24.6333	0.929063
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	14.7889	0.555409	0.277704	−	0.960667i	$-0.410426\pi$
$709$	14.7889	0.555409	0.277704	+	0.960667i	$0.410426\pi$
$710$	0	0
$711$	−4.57779	−0.171681
$712$	0	0
$713$	5.57779	0.208890
$714$	0	0
$715$	4.60555	0.172238
$716$	0	0
$717$	28.1833	1.05253
$718$	0	0
$719$	26.4222	0.985382	0.492691	−	0.870204i	$-0.336013\pi$
$719$	26.4222	0.985382	0.492691	+	0.870204i	$0.336013\pi$
$720$	0	0
$721$	−8.90833	−0.331763
$722$	0	0
$723$	−22.0278	−0.819221
$724$	0	0
$725$	1.90833	0.0708735
$726$	0	0
$727$	−29.3028	−1.08678	−0.543390	−	0.839480i	$-0.682859\pi$
$727$	−29.3028	−1.08678	−0.543390	+	0.839480i	$0.682859\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−12.6056	−0.466233
$732$	0	0
$733$	12.0000	0.443230	0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000	0.443230	0.221615	+	0.975134i	$0.428867\pi$
$734$	0	0
$735$	1.30278	0.0480536
$736$	0	0
$737$	−41.0278	−1.51128
$738$	0	0
$739$	−26.0555	−0.958468	−0.479234	−	0.877687i	$-0.659085\pi$
$739$	−26.0555	−0.958468	−0.479234	+	0.877687i	$0.659085\pi$
$740$	0	0
$741$	8.21110	0.301642
$742$	0	0
$743$	36.9083	1.35404	0.677018	−	0.735967i	$-0.263272\pi$
$743$	36.9083	1.35404	0.677018	+	0.735967i	$0.263272\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	15.3944	0.563253
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	12.5139	0.456638	0.228319	−	0.973586i	$-0.426677\pi$
$751$	12.5139	0.456638	0.228319	+	0.973586i	$0.426677\pi$
$752$	0	0
$753$	−36.0000	−1.31191
$754$	0	0
$755$	−10.0000	−0.363937
$756$	0	0
$757$	45.8167	1.66523	0.832617	−	0.553849i	$-0.186841\pi$
$757$	45.8167	1.66523	0.832617	+	0.553849i	$0.186841\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	−24.5139	−0.888627	−0.444314	−	0.895871i	$-0.646552\pi$
$761$	−24.5139	−0.888627	−0.444314	+	0.895871i	$0.646552\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	0	0
$765$	−8.21110	−0.296873
$766$	0	0
$767$	−11.1194	−0.401499
$768$	0	0
$769$	−15.5778	−0.561750	−0.280875	−	0.959744i	$-0.590625\pi$
$769$	−15.5778	−0.561750	−0.280875	+	0.959744i	$0.590625\pi$
$770$	0	0
$771$	28.1833	1.01500
$772$	0	0
$773$	16.1833	0.582075	0.291037	−	0.956712i	$-0.406000\pi$
$773$	16.1833	0.582075	0.291037	+	0.956712i	$0.406000\pi$
$774$	0	0
$775$	−0.697224	−0.0250450
$776$	0	0
$777$	−5.09167	−0.182663
$778$	0	0
$779$	27.1194	0.971654
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	−10.6972	−0.382288
$784$	0	0
$785$	10.5139	0.375256
$786$	0	0
$787$	−14.4222	−0.514096	−0.257048	−	0.966399i	$-0.582750\pi$
$787$	−14.4222	−0.514096	−0.257048	+	0.966399i	$0.582750\pi$
$788$	0	0
$789$	20.3667	0.725073
$790$	0	0
$791$	7.81665	0.277928
$792$	0	0
$793$	1.21110	0.0430075
$794$	0	0
$795$	−3.39445	−0.120389
$796$	0	0
$797$	−43.5416	−1.54232	−0.771162	−	0.636639i	$-0.780324\pi$
$797$	−43.5416	−1.54232	−0.771162	+	0.636639i	$0.780324\pi$
$798$	0	0
$799$	−29.0278	−1.02693
$800$	0	0
$801$	9.51388	0.336156
$802$	0	0
$803$	39.6333	1.39863
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	−23.2111	−0.817070
$808$	0	0
$809$	−21.5139	−0.756388	−0.378194	−	0.925726i	$-0.623455\pi$
$809$	−21.5139	−0.756388	−0.378194	+	0.925726i	$0.623455\pi$
$810$	0	0
$811$	−31.6333	−1.11080	−0.555398	−	0.831585i	$-0.687434\pi$
$811$	−31.6333	−1.11080	−0.555398	+	0.831585i	$0.687434\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	25.1194	0.879895
$816$	0	0
$817$	12.6056	0.441012
$818$	0	0
$819$	1.30278	0.0455227
$820$	0	0
$821$	46.4777	1.62208	0.811042	−	0.584988i	$-0.198901\pi$
$821$	46.4777	1.62208	0.811042	+	0.584988i	$0.198901\pi$
$822$	0	0
$823$	−3.02776	−0.105541	−0.0527705	−	0.998607i	$-0.516805\pi$
$823$	−3.02776	−0.105541	−0.0527705	+	0.998607i	$0.516805\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−3.63331	−0.126342	−0.0631712	−	0.998003i	$-0.520121\pi$
$827$	−3.63331	−0.126342	−0.0631712	+	0.998003i	$0.520121\pi$
$828$	0	0
$829$	9.21110	0.319915	0.159957	−	0.987124i	$-0.448864\pi$
$829$	9.21110	0.319915	0.159957	+	0.987124i	$0.448864\pi$
$830$	0	0
$831$	1.57779	0.0547331
$832$	0	0
$833$	6.30278	0.218378
$834$	0	0
$835$	−23.2111	−0.803253
$836$	0	0
$837$	3.90833	0.135092
$838$	0	0
$839$	−36.8444	−1.27201	−0.636005	−	0.771685i	$-0.719414\pi$
$839$	−36.8444	−1.27201	−0.636005	+	0.771685i	$0.719414\pi$
$840$	0	0
$841$	−25.3583	−0.874424
$842$	0	0
$843$	−41.6888	−1.43584
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	10.2111	0.350858
$848$	0	0
$849$	−19.6611	−0.674766
$850$	0	0
$851$	31.2666	1.07181
$852$	0	0
$853$	24.1833	0.828022	0.414011	−	0.910272i	$-0.364128\pi$
$853$	24.1833	0.828022	0.414011	+	0.910272i	$0.364128\pi$
$854$	0	0
$855$	8.21110	0.280814
$856$	0	0
$857$	49.3028	1.68415	0.842075	−	0.539360i	$-0.181334\pi$
$857$	49.3028	1.68415	0.842075	+	0.539360i	$0.181334\pi$
$858$	0	0
$859$	18.4222	0.628558	0.314279	−	0.949331i	$-0.398237\pi$
$859$	18.4222	0.628558	0.314279	+	0.949331i	$0.398237\pi$
$860$	0	0
$861$	−5.60555	−0.191037
$862$	0	0
$863$	50.7250	1.72670	0.863349	−	0.504607i	$-0.168363\pi$
$863$	50.7250	1.72670	0.863349	+	0.504607i	$0.168363\pi$
$864$	0	0
$865$	18.6972	0.635725
$866$	0	0
$867$	29.6056	1.00546
$868$	0	0
$869$	−16.1833	−0.548982
$870$	0	0
$871$	−8.90833	−0.301847
$872$	0	0
$873$	17.2111	0.582508
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	21.0917	0.712215	0.356108	−	0.934445i	$-0.384104\pi$
$877$	21.0917	0.712215	0.356108	+	0.934445i	$0.384104\pi$
$878$	0	0
$879$	3.15559	0.106435
$880$	0	0
$881$	29.4500	0.992194	0.496097	−	0.868267i	$-0.334766\pi$
$881$	29.4500	0.992194	0.496097	+	0.868267i	$0.334766\pi$
$882$	0	0
$883$	28.6056	0.962653	0.481327	−	0.876541i	$-0.340155\pi$
$883$	28.6056	0.962653	0.481327	+	0.876541i	$0.340155\pi$
$884$	0	0
$885$	14.4861	0.486946
$886$	0	0
$887$	26.1472	0.877937	0.438968	−	0.898503i	$-0.355344\pi$
$887$	26.1472	0.877937	0.438968	+	0.898503i	$0.355344\pi$
$888$	0	0
$889$	−8.60555	−0.288621
$890$	0	0
$891$	15.6333	0.523736
$892$	0	0
$893$	29.0278	0.971377
$894$	0	0
$895$	−9.09167	−0.303901
$896$	0	0
$897$	10.4222	0.347987
$898$	0	0
$899$	−1.33053	−0.0443757
$900$	0	0
$901$	−16.4222	−0.547103
$902$	0	0
$903$	−2.60555	−0.0867073
$904$	0	0
$905$	9.81665	0.326317
$906$	0	0
$907$	2.60555	0.0865159	0.0432580	−	0.999064i	$-0.486226\pi$
$907$	2.60555	0.0865159	0.0432580	+	0.999064i	$0.486226\pi$
$908$	0	0
$909$	−7.81665	−0.259262
$910$	0	0
$911$	−13.0917	−0.433746	−0.216873	−	0.976200i	$-0.569586\pi$
$911$	−13.0917	−0.433746	−0.216873	+	0.976200i	$0.569586\pi$
$912$	0	0
$913$	54.4222	1.80111
$914$	0	0
$915$	−1.57779	−0.0521603
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21.5416	0.710593	0.355296	−	0.934754i	$-0.384380\pi$
$919$	21.5416	0.710593	0.355296	+	0.934754i	$0.384380\pi$
$920$	0	0
$921$	9.63331	0.317428
$922$	0	0
$923$	2.60555	0.0857628
$924$	0	0
$925$	−3.90833	−0.128505
$926$	0	0
$927$	11.6056	0.381176
$928$	0	0
$929$	34.7527	1.14020	0.570100	−	0.821575i	$-0.306904\pi$
$929$	34.7527	1.14020	0.570100	+	0.821575i	$0.306904\pi$
$930$	0	0
$931$	−6.30278	−0.206565
$932$	0	0
$933$	−30.0000	−0.982156
$934$	0	0
$935$	−29.0278	−0.949309
$936$	0	0
$937$	−33.6972	−1.10084	−0.550420	−	0.834888i	$-0.685532\pi$
$937$	−33.6972	−1.10084	−0.550420	+	0.834888i	$0.685532\pi$
$938$	0	0
$939$	−39.5139	−1.28949
$940$	0	0
$941$	−1.88057	−0.0613048	−0.0306524	−	0.999530i	$-0.509758\pi$
$941$	−1.88057	−0.0613048	−0.0306524	+	0.999530i	$0.509758\pi$
$942$	0	0
$943$	34.4222	1.12094
$944$	0	0
$945$	−5.60555	−0.182349
$946$	0	0
$947$	5.14719	0.167261	0.0836305	−	0.996497i	$-0.473348\pi$
$947$	5.14719	0.167261	0.0836305	+	0.996497i	$0.473348\pi$
$948$	0	0
$949$	8.60555	0.279348
$950$	0	0
$951$	−21.3944	−0.693763
$952$	0	0
$953$	−25.0278	−0.810729	−0.405364	−	0.914155i	$-0.632855\pi$
$953$	−25.0278	−0.810729	−0.405364	+	0.914155i	$0.632855\pi$
$954$	0	0
$955$	−16.5139	−0.534377
$956$	0	0
$957$	−11.4500	−0.370125
$958$	0	0
$959$	2.69722	0.0870979
$960$	0	0
$961$	−30.5139	−0.984319
$962$	0	0
$963$	7.81665	0.251888
$964$	0	0
$965$	10.9083	0.351151
$966$	0	0
$967$	30.5139	0.981260	0.490630	−	0.871368i	$-0.336767\pi$
$967$	30.5139	0.981260	0.490630	+	0.871368i	$0.336767\pi$
$968$	0	0
$969$	−51.7527	−1.66254
$970$	0	0
$971$	12.9722	0.416299	0.208150	−	0.978097i	$-0.433256\pi$
$971$	12.9722	0.416299	0.208150	+	0.978097i	$0.433256\pi$
$972$	0	0
$973$	16.4222	0.526472
$974$	0	0
$975$	−1.30278	−0.0417222
$976$	0	0
$977$	2.90833	0.0930456	0.0465228	−	0.998917i	$-0.485186\pi$
$977$	2.90833	0.0930456	0.0465228	+	0.998917i	$0.485186\pi$
$978$	0	0
$979$	33.6333	1.07493
$980$	0	0
$981$	5.21110	0.166378
$982$	0	0
$983$	11.4500	0.365197	0.182599	−	0.983188i	$-0.441549\pi$
$983$	11.4500	0.365197	0.182599	+	0.983188i	$0.441549\pi$
$984$	0	0
$985$	13.1194	0.418020
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	27.1472	0.862359	0.431179	−	0.902266i	$-0.358098\pi$
$991$	27.1472	0.862359	0.431179	+	0.902266i	$0.358098\pi$
$992$	0	0
$993$	23.4500	0.744162
$994$	0	0
$995$	1.21110	0.0383945
$996$	0	0
$997$	55.1472	1.74653	0.873264	−	0.487247i	$-0.161999\pi$
$997$	55.1472	1.74653	0.873264	+	0.487247i	$0.161999\pi$
$998$	0	0
$999$	21.9083	0.693149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.l.1.2	✓	2
4.3	odd	2		7280.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.l.1.2	✓	2	1.1	even	1	trivial
7280.2.a.bh.1.1		2	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.30278 q^{9} +O(q^{10})\)	\(q+1.30278 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.30278 q^{9} -4.60555 q^{11} -1.00000 q^{13} +1.30278 q^{15} +6.30278 q^{17} -6.30278 q^{19} +1.30278 q^{21} -8.00000 q^{23} +1.00000 q^{25} -5.60555 q^{27} +1.90833 q^{29} -0.697224 q^{31} -6.00000 q^{33} +1.00000 q^{35} -3.90833 q^{37} -1.30278 q^{39} -4.30278 q^{41} -2.00000 q^{43} -1.30278 q^{45} -4.60555 q^{47} +1.00000 q^{49} +8.21110 q^{51} -2.60555 q^{53} -4.60555 q^{55} -8.21110 q^{57} +11.1194 q^{59} -1.21110 q^{61} -1.30278 q^{63} -1.00000 q^{65} +8.90833 q^{67} -10.4222 q^{69} -2.60555 q^{71} -8.60555 q^{73} +1.30278 q^{75} -4.60555 q^{77} +3.51388 q^{79} -3.39445 q^{81} -11.8167 q^{83} +6.30278 q^{85} +2.48612 q^{87} -7.30278 q^{89} -1.00000 q^{91} -0.908327 q^{93} -6.30278 q^{95} -13.2111 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9	\( 2 q - q^{3} + 2 q^{5} + 2 q^{7} + q^{9} - 2 q^{11} - 2 q^{13} - q^{15} + 9 q^{17} - 9 q^{19} - q^{21} - 16 q^{23} + 2 q^{25} - 4 q^{27} - 7 q^{29} - 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 5 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 2 q^{53} - 2 q^{55} - 2 q^{57} - 3 q^{59} + 12 q^{61} + q^{63} - 2 q^{65} + 7 q^{67} + 8 q^{69} + 2 q^{71} - 10 q^{73} - q^{75} - 2 q^{77} - 11 q^{79} - 14 q^{81} - 2 q^{83} + 9 q^{85} + 23 q^{87} - 11 q^{89} - 2 q^{91} + 9 q^{93} - 9 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 + 2 * q^7 + q^9 - 2 * q^11 - 2 * q^13 - q^15 + 9 * q^17 - 9 * q^19 - q^21 - 16 * q^23 + 2 * q^25 - 4 * q^27 - 7 * q^29 - 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 5 * q^41 - 4 * q^43 + q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 + 2 * q^53 - 2 * q^55 - 2 * q^57 - 3 * q^59 + 12 * q^61 + q^63 - 2 * q^65 + 7 * q^67 + 8 * q^69 + 2 * q^71 - 10 * q^73 - q^75 - 2 * q^77 - 11 * q^79 - 14 * q^81 - 2 * q^83 + 9 * q^85 + 23 * q^87 - 11 * q^89 - 2 * q^91 + 9 * q^93 - 9 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

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