LMFDB - Embedded newform 1860.2.a.i.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,2,Mod(1,1860)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1860.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.8521747760$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.224148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 9x + 14 $ x^4 - x^3 - 11x^2 + 9x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.13833$ of defining polynomial
Character	$\chi$	$=$	1860.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +3.47073 q^{7} +1.00000 q^{9} +6.27666 q^{11} -0.805928 q^{13} -1.00000 q^{15} -0.710785 q^{17} -3.47073 q^{21} +0.710785 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.51671 q^{29} +1.00000 q^{31} -6.27666 q^{33} +3.47073 q^{35} +5.47073 q^{37} +0.805928 q^{39} -4.27666 q^{41} -6.27666 q^{43} +1.00000 q^{45} -9.65225 q^{47} +5.04598 q^{49} +0.710785 q^{51} -11.6522 q^{53} +6.27666 q^{55} -13.8797 q^{59} -11.9749 q^{61} +3.47073 q^{63} -0.805928 q^{65} +15.1690 q^{67} -0.710785 q^{69} +6.18152 q^{71} +11.0826 q^{73} -1.00000 q^{75} +21.7846 q^{77} +9.37559 q^{79} +1.00000 q^{81} -4.71079 q^{83} -0.710785 q^{85} -7.51671 q^{87} +12.1815 q^{89} -2.79716 q^{91} -1.00000 q^{93} +6.85509 q^{97} +6.27666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * q^99

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.47073	1.31181	0.655907	−	0.754842i	$-0.272286\pi$
$7$	3.47073	1.31181	0.655907	+	0.754842i	$0.272286\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.27666	1.89248	0.946242	−	0.323459i	$-0.104846\pi$
$11$	6.27666	1.89248	0.946242	+	0.323459i	$0.104846\pi$
$12$	0	0
$13$	−0.805928	−0.223524	−0.111762	−	0.993735i	$-0.535649\pi$
$13$	−0.805928	−0.223524	−0.111762	+	0.993735i	$0.535649\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.710785	−0.172391	−0.0861954	−	0.996278i	$-0.527471\pi$
$17$	−0.710785	−0.172391	−0.0861954	+	0.996278i	$0.527471\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−3.47073	−0.757376
$22$	0	0
$23$	0.710785	0.148209	0.0741045	−	0.997250i	$-0.476390\pi$
$23$	0.710785	0.148209	0.0741045	+	0.997250i	$0.476390\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.51671	1.39582	0.697909	−	0.716186i	$-0.254114\pi$
$29$	7.51671	1.39582	0.697909	+	0.716186i	$0.254114\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−6.27666	−1.09263
$34$	0	0
$35$	3.47073	0.586661
$36$	0	0
$37$	5.47073	0.899383	0.449691	−	0.893184i	$-0.351534\pi$
$37$	5.47073	0.899383	0.449691	+	0.893184i	$0.351534\pi$
$38$	0	0
$39$	0.805928	0.129052
$40$	0	0
$41$	−4.27666	−0.667902	−0.333951	−	0.942590i	$-0.608382\pi$
$41$	−4.27666	−0.667902	−0.333951	+	0.942590i	$0.608382\pi$
$42$	0	0
$43$	−6.27666	−0.957182	−0.478591	−	0.878038i	$-0.658852\pi$
$43$	−6.27666	−0.957182	−0.478591	+	0.878038i	$0.658852\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.65225	−1.40793	−0.703963	−	0.710237i	$-0.748588\pi$
$47$	−9.65225	−1.40793	−0.703963	+	0.710237i	$0.748588\pi$
$48$	0	0
$49$	5.04598	0.720854
$50$	0	0
$51$	0.710785	0.0995299
$52$	0	0
$53$	−11.6522	−1.60056	−0.800280	−	0.599627i	$-0.795316\pi$
$53$	−11.6522	−1.60056	−0.800280	+	0.599627i	$0.795316\pi$
$54$	0	0
$55$	6.27666	0.846345
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.8797	−1.80699	−0.903495	−	0.428599i	$-0.859007\pi$
$59$	−13.8797	−1.80699	−0.903495	+	0.428599i	$0.859007\pi$
$60$	0	0
$61$	−11.9749	−1.53323	−0.766614	−	0.642108i	$-0.778060\pi$
$61$	−11.9749	−1.53323	−0.766614	+	0.642108i	$0.778060\pi$
$62$	0	0
$63$	3.47073	0.437271
$64$	0	0
$65$	−0.805928	−0.0999631
$66$	0	0
$67$	15.1690	1.85318	0.926592	−	0.376068i	$-0.122724\pi$
$67$	15.1690	1.85318	0.926592	+	0.376068i	$0.122724\pi$
$68$	0	0
$69$	−0.710785	−0.0855685
$70$	0	0
$71$	6.18152	0.733611	0.366806	−	0.930298i	$-0.380451\pi$
$71$	6.18152	0.733611	0.366806	+	0.930298i	$0.380451\pi$
$72$	0	0
$73$	11.0826	1.29712	0.648559	−	0.761164i	$-0.275372\pi$
$73$	11.0826	1.29712	0.648559	+	0.761164i	$0.275372\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	21.7846	2.48259
$78$	0	0
$79$	9.37559	1.05484	0.527418	−	0.849606i	$-0.323160\pi$
$79$	9.37559	1.05484	0.527418	+	0.849606i	$0.323160\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.71079	−0.517076	−0.258538	−	0.966001i	$-0.583241\pi$
$83$	−4.71079	−0.517076	−0.258538	+	0.966001i	$0.583241\pi$
$84$	0	0
$85$	−0.710785	−0.0770955
$86$	0	0
$87$	−7.51671	−0.805876
$88$	0	0
$89$	12.1815	1.29124	0.645619	−	0.763660i	$-0.276599\pi$
$89$	12.1815	1.29124	0.645619	+	0.763660i	$0.276599\pi$
$90$	0	0
$91$	−2.79716	−0.293222
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.85509	0.696029	0.348014	−	0.937489i	$-0.386856\pi$
$97$	6.85509	0.696029	0.348014	+	0.937489i	$0.386856\pi$
$98$	0	0
$99$	6.27666	0.630828
$100$	0	0
$101$	13.6982	1.36302	0.681512	−	0.731807i	$-0.261323\pi$
$101$	13.6982	1.36302	0.681512	+	0.731807i	$0.261323\pi$
$102$	0	0
$103$	−9.74739	−0.960439	−0.480220	−	0.877148i	$-0.659443\pi$
$103$	−9.74739	−0.960439	−0.480220	+	0.877148i	$0.659443\pi$
$104$	0	0
$105$	−3.47073	−0.338709
$106$	0	0
$107$	2.71079	0.262062	0.131031	−	0.991378i	$-0.458171\pi$
$107$	2.71079	0.262062	0.131031	+	0.991378i	$0.458171\pi$
$108$	0	0
$109$	16.9874	1.62710	0.813551	−	0.581493i	$-0.197531\pi$
$109$	16.9874	1.62710	0.813551	+	0.581493i	$0.197531\pi$
$110$	0	0
$111$	−5.47073	−0.519259
$112$	0	0
$113$	−0.388144	−0.0365135	−0.0182568	−	0.999833i	$-0.505812\pi$
$113$	−0.388144	−0.0365135	−0.0182568	+	0.999833i	$0.505812\pi$
$114$	0	0
$115$	0.710785	0.0662811
$116$	0	0
$117$	−0.805928	−0.0745081
$118$	0	0
$119$	−2.46695	−0.226145
$120$	0	0
$121$	28.3965	2.58150
$122$	0	0
$123$	4.27666	0.385613
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.61186	0.497972	0.248986	−	0.968507i	$-0.419903\pi$
$127$	5.61186	0.497972	0.248986	+	0.968507i	$0.419903\pi$
$128$	0	0
$129$	6.27666	0.552629
$130$	0	0
$131$	1.42475	0.124481	0.0622405	−	0.998061i	$-0.480175\pi$
$131$	1.42475	0.124481	0.0622405	+	0.998061i	$0.480175\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.6857	−0.912939	−0.456469	−	0.889739i	$-0.650886\pi$
$137$	−10.6857	−0.912939	−0.456469	+	0.889739i	$0.650886\pi$
$138$	0	0
$139$	−12.9164	−1.09555	−0.547775	−	0.836625i	$-0.684525\pi$
$139$	−12.9164	−1.09555	−0.547775	+	0.836625i	$0.684525\pi$
$140$	0	0
$141$	9.65225	0.812866
$142$	0	0
$143$	−5.05854	−0.423016
$144$	0	0
$145$	7.51671	0.624229
$146$	0	0
$147$	−5.04598	−0.416186
$148$	0	0
$149$	−11.1198	−0.910970	−0.455485	−	0.890244i	$-0.650534\pi$
$149$	−11.1198	−0.910970	−0.455485	+	0.890244i	$0.650534\pi$
$150$	0	0
$151$	−2.98745	−0.243115	−0.121557	−	0.992584i	$-0.538789\pi$
$151$	−2.98745	−0.243115	−0.121557	+	0.992584i	$0.538789\pi$
$152$	0	0
$153$	−0.710785	−0.0574636
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	13.6063	1.08590	0.542949	−	0.839766i	$-0.317308\pi$
$157$	13.6063	1.08590	0.542949	+	0.839766i	$0.317308\pi$
$158$	0	0
$159$	11.6522	0.924083
$160$	0	0
$161$	2.46695	0.194423
$162$	0	0
$163$	−2.80593	−0.219777	−0.109889	−	0.993944i	$-0.535049\pi$
$163$	−2.80593	−0.219777	−0.109889	+	0.993944i	$0.535049\pi$
$164$	0	0
$165$	−6.27666	−0.488637
$166$	0	0
$167$	−23.4948	−1.81808	−0.909040	−	0.416708i	$-0.863184\pi$
$167$	−23.4948	−1.81808	−0.909040	+	0.416708i	$0.863184\pi$
$168$	0	0
$169$	−12.3505	−0.950037
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.94146	−0.679807	−0.339903	−	0.940460i	$-0.610394\pi$
$173$	−8.94146	−0.679807	−0.339903	+	0.940460i	$0.610394\pi$
$174$	0	0
$175$	3.47073	0.262363
$176$	0	0
$177$	13.8797	1.04327
$178$	0	0
$179$	17.9749	1.34351	0.671753	−	0.740775i	$-0.265542\pi$
$179$	17.9749	1.34351	0.671753	+	0.740775i	$0.265542\pi$
$180$	0	0
$181$	16.7512	1.24511	0.622553	−	0.782578i	$-0.286096\pi$
$181$	16.7512	1.24511	0.622553	+	0.782578i	$0.286096\pi$
$182$	0	0
$183$	11.9749	0.885209
$184$	0	0
$185$	5.47073	0.402216
$186$	0	0
$187$	−4.46136	−0.326247
$188$	0	0
$189$	−3.47073	−0.252459
$190$	0	0
$191$	−22.0700	−1.59693	−0.798466	−	0.602040i	$-0.794355\pi$
$191$	−22.0700	−1.59693	−0.798466	+	0.602040i	$0.794355\pi$
$192$	0	0
$193$	−1.05295	−0.0757929	−0.0378964	−	0.999282i	$-0.512066\pi$
$193$	−1.05295	−0.0757929	−0.0378964	+	0.999282i	$0.512066\pi$
$194$	0	0
$195$	0.805928	0.0577137
$196$	0	0
$197$	15.8425	1.12873	0.564367	−	0.825524i	$-0.309120\pi$
$197$	15.8425	1.12873	0.564367	+	0.825524i	$0.309120\pi$
$198$	0	0
$199$	−7.54077	−0.534551	−0.267275	−	0.963620i	$-0.586123\pi$
$199$	−7.54077	−0.534551	−0.267275	+	0.963620i	$0.586123\pi$
$200$	0	0
$201$	−15.1690	−1.06994
$202$	0	0
$203$	26.0885	1.83105
$204$	0	0
$205$	−4.27666	−0.298695
$206$	0	0
$207$	0.710785	0.0494030
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−6.84950	−0.471539	−0.235770	−	0.971809i	$-0.575761\pi$
$211$	−6.84950	−0.471539	−0.235770	+	0.971809i	$0.575761\pi$
$212$	0	0
$213$	−6.18152	−0.423551
$214$	0	0
$215$	−6.27666	−0.428065
$216$	0	0
$217$	3.47073	0.235609
$218$	0	0
$219$	−11.0826	−0.748892
$220$	0	0
$221$	0.572842	0.0385335
$222$	0	0
$223$	−3.33520	−0.223341	−0.111671	−	0.993745i	$-0.535620\pi$
$223$	−3.33520	−0.223341	−0.111671	+	0.993745i	$0.535620\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−16.5937	−1.10136	−0.550682	−	0.834715i	$-0.685632\pi$
$227$	−16.5937	−1.10136	−0.550682	+	0.834715i	$0.685632\pi$
$228$	0	0
$229$	23.6926	1.56565	0.782827	−	0.622239i	$-0.213777\pi$
$229$	23.6926	1.56565	0.782827	+	0.622239i	$0.213777\pi$
$230$	0	0
$231$	−21.7846	−1.43332
$232$	0	0
$233$	1.74421	0.114267	0.0571336	−	0.998367i	$-0.481804\pi$
$233$	1.74421	0.114267	0.0571336	+	0.998367i	$0.481804\pi$
$234$	0	0
$235$	−9.65225	−0.629643
$236$	0	0
$237$	−9.37559	−0.609010
$238$	0	0
$239$	−7.24323	−0.468526	−0.234263	−	0.972173i	$-0.575268\pi$
$239$	−7.24323	−0.468526	−0.234263	+	0.972173i	$0.575268\pi$
$240$	0	0
$241$	−16.1652	−1.04129	−0.520645	−	0.853773i	$-0.674309\pi$
$241$	−16.1652	−1.04129	−0.520645	+	0.853773i	$0.674309\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.04598	0.322376
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.71079	0.298534
$250$	0	0
$251$	14.5282	0.917013	0.458506	−	0.888691i	$-0.348385\pi$
$251$	14.5282	0.917013	0.458506	+	0.888691i	$0.348385\pi$
$252$	0	0
$253$	4.46136	0.280483
$254$	0	0
$255$	0.710785	0.0445111
$256$	0	0
$257$	−4.23068	−0.263902	−0.131951	−	0.991256i	$-0.542124\pi$
$257$	−4.23068	−0.263902	−0.131951	+	0.991256i	$0.542124\pi$
$258$	0	0
$259$	18.9874	1.17982
$260$	0	0
$261$	7.51671	0.465273
$262$	0	0
$263$	−25.6926	−1.58428	−0.792138	−	0.610342i	$-0.791032\pi$
$263$	−25.6926	−1.58428	−0.792138	+	0.610342i	$0.791032\pi$
$264$	0	0
$265$	−11.6522	−0.715792
$266$	0	0
$267$	−12.1815	−0.745497
$268$	0	0
$269$	29.9530	1.82626	0.913132	−	0.407664i	$-0.133656\pi$
$269$	29.9530	1.82626	0.913132	+	0.407664i	$0.133656\pi$
$270$	0	0
$271$	15.5867	0.946827	0.473414	−	0.880840i	$-0.343022\pi$
$271$	15.5867	0.946827	0.473414	+	0.880840i	$0.343022\pi$
$272$	0	0
$273$	2.79716	0.169292
$274$	0	0
$275$	6.27666	0.378497
$276$	0	0
$277$	−6.80034	−0.408593	−0.204296	−	0.978909i	$-0.565491\pi$
$277$	−6.80034	−0.408593	−0.204296	+	0.978909i	$0.565491\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	17.4948	1.04365	0.521826	−	0.853052i	$-0.325251\pi$
$281$	17.4948	1.04365	0.521826	+	0.853052i	$0.325251\pi$
$282$	0	0
$283$	−3.66102	−0.217625	−0.108812	−	0.994062i	$-0.534705\pi$
$283$	−3.66102	−0.217625	−0.108812	+	0.994062i	$0.534705\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.8431	−0.876163
$288$	0	0
$289$	−16.4948	−0.970281
$290$	0	0
$291$	−6.85509	−0.401852
$292$	0	0
$293$	−14.9011	−0.870530	−0.435265	−	0.900302i	$-0.643345\pi$
$293$	−14.9011	−0.870530	−0.435265	+	0.900302i	$0.643345\pi$
$294$	0	0
$295$	−13.8797	−0.808110
$296$	0	0
$297$	−6.27666	−0.364209
$298$	0	0
$299$	−0.572842	−0.0331283
$300$	0	0
$301$	−21.7846	−1.25564
$302$	0	0
$303$	−13.6982	−0.786943
$304$	0	0
$305$	−11.9749	−0.685680
$306$	0	0
$307$	−14.3139	−0.816936	−0.408468	−	0.912773i	$-0.633937\pi$
$307$	−14.3139	−0.816936	−0.408468	+	0.912773i	$0.633937\pi$
$308$	0	0
$309$	9.74739	0.554510
$310$	0	0
$311$	−24.7599	−1.40401	−0.702004	−	0.712173i	$-0.747711\pi$
$311$	−24.7599	−1.40401	−0.702004	+	0.712173i	$0.747711\pi$
$312$	0	0
$313$	−6.41778	−0.362755	−0.181377	−	0.983414i	$-0.558056\pi$
$313$	−6.41778	−0.362755	−0.181377	+	0.983414i	$0.558056\pi$
$314$	0	0
$315$	3.47073	0.195554
$316$	0	0
$317$	−19.5352	−1.09720	−0.548602	−	0.836083i	$-0.684840\pi$
$317$	−19.5352	−1.09720	−0.548602	+	0.836083i	$0.684840\pi$
$318$	0	0
$319$	47.1799	2.64156
$320$	0	0
$321$	−2.71079	−0.151301
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.805928	−0.0447048
$326$	0	0
$327$	−16.9874	−0.939408
$328$	0	0
$329$	−33.5004	−1.84694
$330$	0	0
$331$	5.46755	0.300524	0.150262	−	0.988646i	$-0.451988\pi$
$331$	5.46755	0.300524	0.150262	+	0.988646i	$0.451988\pi$
$332$	0	0
$333$	5.47073	0.299794
$334$	0	0
$335$	15.1690	0.828769
$336$	0	0
$337$	6.80034	0.370438	0.185219	−	0.982697i	$-0.440701\pi$
$337$	6.80034	0.370438	0.185219	+	0.982697i	$0.440701\pi$
$338$	0	0
$339$	0.388144	0.0210811
$340$	0	0
$341$	6.27666	0.339900
$342$	0	0
$343$	−6.78188	−0.366187
$344$	0	0
$345$	−0.710785	−0.0382674
$346$	0	0
$347$	21.0083	1.12779	0.563893	−	0.825848i	$-0.309303\pi$
$347$	21.0083	1.12779	0.563893	+	0.825848i	$0.309303\pi$
$348$	0	0
$349$	22.9623	1.22915	0.614573	−	0.788860i	$-0.289328\pi$
$349$	22.9623	1.22915	0.614573	+	0.788860i	$0.289328\pi$
$350$	0	0
$351$	0.805928	0.0430173
$352$	0	0
$353$	−29.2641	−1.55757	−0.778786	−	0.627290i	$-0.784164\pi$
$353$	−29.2641	−1.55757	−0.778786	+	0.627290i	$0.784164\pi$
$354$	0	0
$355$	6.18152	0.328081
$356$	0	0
$357$	2.46695	0.130565
$358$	0	0
$359$	−10.9383	−0.577301	−0.288650	−	0.957435i	$-0.593206\pi$
$359$	−10.9383	−0.577301	−0.288650	+	0.957435i	$0.593206\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−28.3965	−1.49043
$364$	0	0
$365$	11.0826	0.580089
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−4.27666	−0.222634
$370$	0	0
$371$	−40.4418	−2.09964
$372$	0	0
$373$	−10.3630	−0.536578	−0.268289	−	0.963339i	$-0.586458\pi$
$373$	−10.3630	−0.536578	−0.268289	+	0.963339i	$0.586458\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−6.05793	−0.311999
$378$	0	0
$379$	30.4362	1.56341	0.781703	−	0.623651i	$-0.214352\pi$
$379$	30.4362	1.56341	0.781703	+	0.623651i	$0.214352\pi$
$380$	0	0
$381$	−5.61186	−0.287504
$382$	0	0
$383$	5.17214	0.264284	0.132142	−	0.991231i	$-0.457814\pi$
$383$	5.17214	0.264284	0.132142	+	0.991231i	$0.457814\pi$
$384$	0	0
$385$	21.7846	1.11025
$386$	0	0
$387$	−6.27666	−0.319061
$388$	0	0
$389$	−5.24005	−0.265681	−0.132841	−	0.991137i	$-0.542410\pi$
$389$	−5.24005	−0.265681	−0.132841	+	0.991137i	$0.542410\pi$
$390$	0	0
$391$	−0.505216	−0.0255499
$392$	0	0
$393$	−1.42475	−0.0718692
$394$	0	0
$395$	9.37559	0.471737
$396$	0	0
$397$	−33.6851	−1.69061	−0.845303	−	0.534288i	$-0.820580\pi$
$397$	−33.6851	−1.69061	−0.845303	+	0.534288i	$0.820580\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.1815	−0.608316	−0.304158	−	0.952622i	$-0.598375\pi$
$401$	−12.1815	−0.608316	−0.304158	+	0.952622i	$0.598375\pi$
$402$	0	0
$403$	−0.805928	−0.0401461
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	34.3379	1.70207
$408$	0	0
$409$	22.4614	1.11064	0.555321	−	0.831636i	$-0.312595\pi$
$409$	22.4614	1.11064	0.555321	+	0.831636i	$0.312595\pi$
$410$	0	0
$411$	10.6857	0.527086
$412$	0	0
$413$	−48.1729	−2.37043
$414$	0	0
$415$	−4.71079	−0.231243
$416$	0	0
$417$	12.9164	0.632517
$418$	0	0
$419$	−21.3133	−1.04122	−0.520611	−	0.853794i	$-0.674296\pi$
$419$	−21.3133	−1.04122	−0.520611	+	0.853794i	$0.674296\pi$
$420$	0	0
$421$	2.04598	0.0997150	0.0498575	−	0.998756i	$-0.484123\pi$
$421$	2.04598	0.0997150	0.0498575	+	0.998756i	$0.484123\pi$
$422$	0	0
$423$	−9.65225	−0.469308
$424$	0	0
$425$	−0.710785	−0.0344782
$426$	0	0
$427$	−41.5616	−2.01131
$428$	0	0
$429$	5.05854	0.244228
$430$	0	0
$431$	−18.8268	−0.906855	−0.453428	−	0.891293i	$-0.649799\pi$
$431$	−18.8268	−0.906855	−0.453428	+	0.891293i	$0.649799\pi$
$432$	0	0
$433$	−2.62759	−0.126274	−0.0631370	−	0.998005i	$-0.520111\pi$
$433$	−2.62759	−0.126274	−0.0631370	+	0.998005i	$0.520111\pi$
$434$	0	0
$435$	−7.51671	−0.360399
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.84314	−0.135696	−0.0678479	−	0.997696i	$-0.521613\pi$
$439$	−2.84314	−0.135696	−0.0678479	+	0.997696i	$0.521613\pi$
$440$	0	0
$441$	5.04598	0.240285
$442$	0	0
$443$	−7.35607	−0.349497	−0.174749	−	0.984613i	$-0.555911\pi$
$443$	−7.35607	−0.349497	−0.174749	+	0.984613i	$0.555911\pi$
$444$	0	0
$445$	12.1815	0.577459
$446$	0	0
$447$	11.1198	0.525949
$448$	0	0
$449$	−18.0644	−0.852514	−0.426257	−	0.904602i	$-0.640168\pi$
$449$	−18.0644	−0.852514	−0.426257	+	0.904602i	$0.640168\pi$
$450$	0	0
$451$	−26.8431	−1.26399
$452$	0	0
$453$	2.98745	0.140362
$454$	0	0
$455$	−2.79716	−0.131133
$456$	0	0
$457$	21.6108	1.01091	0.505455	−	0.862853i	$-0.331325\pi$
$457$	21.6108	1.01091	0.505455	+	0.862853i	$0.331325\pi$
$458$	0	0
$459$	0.710785	0.0331766
$460$	0	0
$461$	22.5501	1.05026	0.525132	−	0.851021i	$-0.324016\pi$
$461$	22.5501	1.05026	0.525132	+	0.851021i	$0.324016\pi$
$462$	0	0
$463$	35.0947	1.63099	0.815494	−	0.578765i	$-0.196465\pi$
$463$	35.0947	1.63099	0.815494	+	0.578765i	$0.196465\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	4.13872	0.191517	0.0957585	−	0.995405i	$-0.469472\pi$
$467$	4.13872	0.191517	0.0957585	+	0.995405i	$0.469472\pi$
$468$	0	0
$469$	52.6474	2.43103
$470$	0	0
$471$	−13.6063	−0.626944
$472$	0	0
$473$	−39.3965	−1.81145
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.6522	−0.533520
$478$	0	0
$479$	18.8268	0.860218	0.430109	−	0.902777i	$-0.358475\pi$
$479$	18.8268	0.860218	0.430109	+	0.902777i	$0.358475\pi$
$480$	0	0
$481$	−4.40902	−0.201034
$482$	0	0
$483$	−2.46695	−0.112250
$484$	0	0
$485$	6.85509	0.311274
$486$	0	0
$487$	0.363035	0.0164507	0.00822534	−	0.999966i	$-0.497382\pi$
$487$	0.363035	0.0164507	0.00822534	+	0.999966i	$0.497382\pi$
$488$	0	0
$489$	2.80593	0.126888
$490$	0	0
$491$	−22.2460	−1.00395	−0.501973	−	0.864883i	$-0.667392\pi$
$491$	−22.2460	−1.00395	−0.501973	+	0.864883i	$0.667392\pi$
$492$	0	0
$493$	−5.34277	−0.240626
$494$	0	0
$495$	6.27666	0.282115
$496$	0	0
$497$	21.4544	0.962361
$498$	0	0
$499$	8.77629	0.392881	0.196440	−	0.980516i	$-0.437062\pi$
$499$	8.77629	0.392881	0.196440	+	0.980516i	$0.437062\pi$
$500$	0	0
$501$	23.4948	1.04967
$502$	0	0
$503$	−18.8091	−0.838657	−0.419328	−	0.907835i	$-0.637734\pi$
$503$	−18.8091	−0.838657	−0.419328	+	0.907835i	$0.637734\pi$
$504$	0	0
$505$	13.6982	0.609563
$506$	0	0
$507$	12.3505	0.548504
$508$	0	0
$509$	31.6763	1.40403	0.702014	−	0.712163i	$-0.252285\pi$
$509$	31.6763	1.40403	0.702014	+	0.712163i	$0.252285\pi$
$510$	0	0
$511$	38.4647	1.70158
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.74739	−0.429521
$516$	0	0
$517$	−60.5839	−2.66448
$518$	0	0
$519$	8.94146	0.392487
$520$	0	0
$521$	6.94705	0.304356	0.152178	−	0.988353i	$-0.451371\pi$
$521$	6.94705	0.304356	0.152178	+	0.988353i	$0.451371\pi$
$522$	0	0
$523$	39.5867	1.73101	0.865504	−	0.500902i	$-0.166998\pi$
$523$	39.5867	1.73101	0.865504	+	0.500902i	$0.166998\pi$
$524$	0	0
$525$	−3.47073	−0.151475
$526$	0	0
$527$	−0.710785	−0.0309623
$528$	0	0
$529$	−22.4948	−0.978034
$530$	0	0
$531$	−13.8797	−0.602330
$532$	0	0
$533$	3.44668	0.149292
$534$	0	0
$535$	2.71079	0.117197
$536$	0	0
$537$	−17.9749	−0.775674
$538$	0	0
$539$	31.6719	1.36421
$540$	0	0
$541$	14.2187	0.611311	0.305655	−	0.952142i	$-0.401124\pi$
$541$	14.2187	0.611311	0.305655	+	0.952142i	$0.401124\pi$
$542$	0	0
$543$	−16.7512	−0.718862
$544$	0	0
$545$	16.9874	0.727662
$546$	0	0
$547$	7.07700	0.302591	0.151295	−	0.988489i	$-0.451656\pi$
$547$	7.07700	0.302591	0.151295	+	0.988489i	$0.451656\pi$
$548$	0	0
$549$	−11.9749	−0.511076
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.5402	1.38375
$554$	0	0
$555$	−5.47073	−0.232220
$556$	0	0
$557$	−10.6857	−0.452767	−0.226383	−	0.974038i	$-0.572690\pi$
$557$	−10.6857	−0.452767	−0.226383	+	0.974038i	$0.572690\pi$
$558$	0	0
$559$	5.05854	0.213953
$560$	0	0
$561$	4.46136	0.188359
$562$	0	0
$563$	−19.2641	−0.811885	−0.405943	−	0.913899i	$-0.633057\pi$
$563$	−19.2641	−0.811885	−0.405943	+	0.913899i	$0.633057\pi$
$564$	0	0
$565$	−0.388144	−0.0163293
$566$	0	0
$567$	3.47073	0.145757
$568$	0	0
$569$	−18.8212	−0.789026	−0.394513	−	0.918890i	$-0.629087\pi$
$569$	−18.8212	−0.789026	−0.394513	+	0.918890i	$0.629087\pi$
$570$	0	0
$571$	−14.2460	−0.596175	−0.298088	−	0.954539i	$-0.596349\pi$
$571$	−14.2460	−0.596175	−0.298088	+	0.954539i	$0.596349\pi$
$572$	0	0
$573$	22.0700	0.921989
$574$	0	0
$575$	0.710785	0.0296418
$576$	0	0
$577$	−44.6397	−1.85837	−0.929187	−	0.369609i	$-0.879492\pi$
$577$	−44.6397	−1.85837	−0.929187	+	0.369609i	$0.879492\pi$
$578$	0	0
$579$	1.05295	0.0437590
$580$	0	0
$581$	−16.3499	−0.678307
$582$	0	0
$583$	−73.1372	−3.02903
$584$	0	0
$585$	−0.805928	−0.0333210
$586$	0	0
$587$	19.0334	0.785594	0.392797	−	0.919625i	$-0.371508\pi$
$587$	19.0334	0.785594	0.392797	+	0.919625i	$0.371508\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−15.8425	−0.651675
$592$	0	0
$593$	−20.0732	−0.824308	−0.412154	−	0.911114i	$-0.635223\pi$
$593$	−20.0732	−0.824308	−0.412154	+	0.911114i	$0.635223\pi$
$594$	0	0
$595$	−2.46695	−0.101135
$596$	0	0
$597$	7.54077	0.308623
$598$	0	0
$599$	2.55573	0.104424	0.0522121	−	0.998636i	$-0.483373\pi$
$599$	2.55573	0.104424	0.0522121	+	0.998636i	$0.483373\pi$
$600$	0	0
$601$	−13.1254	−0.535396	−0.267698	−	0.963503i	$-0.586263\pi$
$601$	−13.1254	−0.535396	−0.267698	+	0.963503i	$0.586263\pi$
$602$	0	0
$603$	15.1690	0.617728
$604$	0	0
$605$	28.3965	1.15448
$606$	0	0
$607$	10.8059	0.438599	0.219300	−	0.975658i	$-0.429623\pi$
$607$	10.8059	0.438599	0.219300	+	0.975658i	$0.429623\pi$
$608$	0	0
$609$	−26.0885	−1.05716
$610$	0	0
$611$	7.77902	0.314705
$612$	0	0
$613$	−8.90425	−0.359639	−0.179820	−	0.983700i	$-0.557551\pi$
$613$	−8.90425	−0.359639	−0.179820	+	0.983700i	$0.557551\pi$
$614$	0	0
$615$	4.27666	0.172452
$616$	0	0
$617$	−5.77703	−0.232575	−0.116287	−	0.993216i	$-0.537099\pi$
$617$	−5.77703	−0.232575	−0.116287	+	0.993216i	$0.537099\pi$
$618$	0	0
$619$	26.1128	1.04956	0.524782	−	0.851237i	$-0.324147\pi$
$619$	26.1128	1.04956	0.524782	+	0.851237i	$0.324147\pi$
$620$	0	0
$621$	−0.710785	−0.0285228
$622$	0	0
$623$	42.2788	1.69386
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.88852	−0.155045
$630$	0	0
$631$	10.6592	0.424337	0.212168	−	0.977233i	$-0.431947\pi$
$631$	10.6592	0.424337	0.212168	+	0.977233i	$0.431947\pi$
$632$	0	0
$633$	6.84950	0.272243
$634$	0	0
$635$	5.61186	0.222700
$636$	0	0
$637$	−4.06670	−0.161128
$638$	0	0
$639$	6.18152	0.244537
$640$	0	0
$641$	−5.30768	−0.209641	−0.104820	−	0.994491i	$-0.533427\pi$
$641$	−5.30768	−0.209641	−0.104820	+	0.994491i	$0.533427\pi$
$642$	0	0
$643$	−32.3128	−1.27429	−0.637147	−	0.770743i	$-0.719885\pi$
$643$	−32.3128	−1.27429	−0.637147	+	0.770743i	$0.719885\pi$
$644$	0	0
$645$	6.27666	0.247143
$646$	0	0
$647$	27.3045	1.07345	0.536725	−	0.843757i	$-0.319661\pi$
$647$	27.3045	1.07345	0.536725	+	0.843757i	$0.319661\pi$
$648$	0	0
$649$	−87.1185	−3.41970
$650$	0	0
$651$	−3.47073	−0.136029
$652$	0	0
$653$	38.0746	1.48997	0.744986	−	0.667080i	$-0.232456\pi$
$653$	38.0746	1.48997	0.744986	+	0.667080i	$0.232456\pi$
$654$	0	0
$655$	1.42475	0.0556696
$656$	0	0
$657$	11.0826	0.432373
$658$	0	0
$659$	−43.8415	−1.70782	−0.853911	−	0.520419i	$-0.825776\pi$
$659$	−43.8415	−1.70782	−0.853911	+	0.520419i	$0.825776\pi$
$660$	0	0
$661$	−12.5609	−0.488562	−0.244281	−	0.969704i	$-0.578552\pi$
$661$	−12.5609	−0.488562	−0.244281	+	0.969704i	$0.578552\pi$
$662$	0	0
$663$	−0.572842	−0.0222473
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.34277	0.206873
$668$	0	0
$669$	3.33520	0.128946
$670$	0	0
$671$	−75.1623	−2.90161
$672$	0	0
$673$	46.4595	1.79088	0.895442	−	0.445179i	$-0.146860\pi$
$673$	46.4595	1.79088	0.895442	+	0.445179i	$0.146860\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	46.2307	1.77679	0.888395	−	0.459081i	$-0.151821\pi$
$677$	46.2307	1.77679	0.888395	+	0.459081i	$0.151821\pi$
$678$	0	0
$679$	23.7922	0.913060
$680$	0	0
$681$	16.5937	0.635872
$682$	0	0
$683$	17.3700	0.664645	0.332322	−	0.943166i	$-0.392168\pi$
$683$	17.3700	0.664645	0.332322	+	0.943166i	$0.392168\pi$
$684$	0	0
$685$	−10.6857	−0.408279
$686$	0	0
$687$	−23.6926	−0.903931
$688$	0	0
$689$	9.39087	0.357764
$690$	0	0
$691$	30.0732	1.14404	0.572019	−	0.820240i	$-0.306160\pi$
$691$	30.0732	1.14404	0.572019	+	0.820240i	$0.306160\pi$
$692$	0	0
$693$	21.7846	0.827529
$694$	0	0
$695$	−12.9164	−0.489945
$696$	0	0
$697$	3.03979	0.115140
$698$	0	0
$699$	−1.74421	−0.0659722
$700$	0	0
$701$	5.88852	0.222406	0.111203	−	0.993798i	$-0.464530\pi$
$701$	5.88852	0.222406	0.111203	+	0.993798i	$0.464530\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	9.65225	0.363525
$706$	0	0
$707$	47.5429	1.78803
$708$	0	0
$709$	−29.1735	−1.09563	−0.547817	−	0.836598i	$-0.684541\pi$
$709$	−29.1735	−1.09563	−0.547817	+	0.836598i	$0.684541\pi$
$710$	0	0
$711$	9.37559	0.351612
$712$	0	0
$713$	0.710785	0.0266191
$714$	0	0
$715$	−5.05854	−0.189179
$716$	0	0
$717$	7.24323	0.270504
$718$	0	0
$719$	−22.2264	−0.828906	−0.414453	−	0.910071i	$-0.636027\pi$
$719$	−22.2264	−0.828906	−0.414453	+	0.910071i	$0.636027\pi$
$720$	0	0
$721$	−33.8306	−1.25992
$722$	0	0
$723$	16.1652	0.601189
$724$	0	0
$725$	7.51671	0.279164
$726$	0	0
$727$	−45.5069	−1.68776	−0.843879	−	0.536534i	$-0.819733\pi$
$727$	−45.5069	−1.68776	−0.843879	+	0.536534i	$0.819733\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.46136	0.165009
$732$	0	0
$733$	−33.7770	−1.24758	−0.623792	−	0.781591i	$-0.714409\pi$
$733$	−33.7770	−1.24758	−0.623792	+	0.781591i	$0.714409\pi$
$734$	0	0
$735$	−5.04598	−0.186124
$736$	0	0
$737$	95.2104	3.50712
$738$	0	0
$739$	7.68295	0.282622	0.141311	−	0.989965i	$-0.454868\pi$
$739$	7.68295	0.282622	0.141311	+	0.989965i	$0.454868\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.8495	−1.42525	−0.712625	−	0.701545i	$-0.752494\pi$
$743$	−38.8495	−1.42525	−0.712625	+	0.701545i	$0.752494\pi$
$744$	0	0
$745$	−11.1198	−0.407398
$746$	0	0
$747$	−4.71079	−0.172359
$748$	0	0
$749$	9.40841	0.343776
$750$	0	0
$751$	−49.0083	−1.78834	−0.894169	−	0.447729i	$-0.852233\pi$
$751$	−49.0083	−1.78834	−0.894169	+	0.447729i	$0.852233\pi$
$752$	0	0
$753$	−14.5282	−0.529437
$754$	0	0
$755$	−2.98745	−0.108724
$756$	0	0
$757$	−2.04357	−0.0742750	−0.0371375	−	0.999310i	$-0.511824\pi$
$757$	−2.04357	−0.0742750	−0.0371375	+	0.999310i	$0.511824\pi$
$758$	0	0
$759$	−4.46136	−0.161937
$760$	0	0
$761$	25.7627	0.933896	0.466948	−	0.884285i	$-0.345353\pi$
$761$	25.7627	0.933896	0.466948	+	0.884285i	$0.345353\pi$
$762$	0	0
$763$	58.9589	2.13445
$764$	0	0
$765$	−0.710785	−0.0256985
$766$	0	0
$767$	11.1861	0.403906
$768$	0	0
$769$	19.0543	0.687116	0.343558	−	0.939132i	$-0.388368\pi$
$769$	19.0543	0.687116	0.343558	+	0.939132i	$0.388368\pi$
$770$	0	0
$771$	4.23068	0.152364
$772$	0	0
$773$	31.2390	1.12359	0.561794	−	0.827277i	$-0.310111\pi$
$773$	31.2390	1.12359	0.561794	+	0.827277i	$0.310111\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−18.9874	−0.681171
$778$	0	0
$779$	0	0
$780$	0	0
$781$	38.7993	1.38835
$782$	0	0
$783$	−7.51671	−0.268625
$784$	0	0
$785$	13.6063	0.485629
$786$	0	0
$787$	39.5497	1.40979	0.704897	−	0.709310i	$-0.250993\pi$
$787$	39.5497	1.40979	0.704897	+	0.709310i	$0.250993\pi$
$788$	0	0
$789$	25.6926	0.914682
$790$	0	0
$791$	−1.34714	−0.0478989
$792$	0	0
$793$	9.65090	0.342713
$794$	0	0
$795$	11.6522	0.413263
$796$	0	0
$797$	−16.8027	−0.595184	−0.297592	−	0.954693i	$-0.596183\pi$
$797$	−16.8027	−0.595184	−0.297592	+	0.954693i	$0.596183\pi$
$798$	0	0
$799$	6.86068	0.242713
$800$	0	0
$801$	12.1815	0.430413
$802$	0	0
$803$	69.5616	2.45478
$804$	0	0
$805$	2.46695	0.0869484
$806$	0	0
$807$	−29.9530	−1.05439
$808$	0	0
$809$	−7.14052	−0.251047	−0.125524	−	0.992091i	$-0.540061\pi$
$809$	−7.14052	−0.251047	−0.125524	+	0.992091i	$0.540061\pi$
$810$	0	0
$811$	4.45500	0.156436	0.0782181	−	0.996936i	$-0.475077\pi$
$811$	4.45500	0.156436	0.0782181	+	0.996936i	$0.475077\pi$
$812$	0	0
$813$	−15.5867	−0.546651
$814$	0	0
$815$	−2.80593	−0.0982874
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.79716	−0.0977407
$820$	0	0
$821$	−19.0728	−0.665644	−0.332822	−	0.942990i	$-0.608001\pi$
$821$	−19.0728	−0.665644	−0.332822	+	0.942990i	$0.608001\pi$
$822$	0	0
$823$	−25.8041	−0.899475	−0.449738	−	0.893161i	$-0.648483\pi$
$823$	−25.8041	−0.899475	−0.449738	+	0.893161i	$0.648483\pi$
$824$	0	0
$825$	−6.27666	−0.218525
$826$	0	0
$827$	−9.99017	−0.347392	−0.173696	−	0.984799i	$-0.555571\pi$
$827$	−9.99017	−0.347392	−0.173696	+	0.984799i	$0.555571\pi$
$828$	0	0
$829$	−53.5756	−1.86076	−0.930378	−	0.366601i	$-0.880521\pi$
$829$	−53.5756	−1.86076	−0.930378	+	0.366601i	$0.880521\pi$
$830$	0	0
$831$	6.80034	0.235901
$832$	0	0
$833$	−3.58661	−0.124269
$834$	0	0
$835$	−23.4948	−0.813071
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−50.5314	−1.74454	−0.872269	−	0.489027i	$-0.837352\pi$
$839$	−50.5314	−1.74454	−0.872269	+	0.489027i	$0.837352\pi$
$840$	0	0
$841$	27.5010	0.948310
$842$	0	0
$843$	−17.4948	−0.602552
$844$	0	0
$845$	−12.3505	−0.424869
$846$	0	0
$847$	98.5565	3.38644
$848$	0	0
$849$	3.66102	0.125646
$850$	0	0
$851$	3.88852	0.133297
$852$	0	0
$853$	−4.78188	−0.163728	−0.0818642	−	0.996643i	$-0.526087\pi$
$853$	−4.78188	−0.163728	−0.0818642	+	0.996643i	$0.526087\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.3777	1.20848	0.604240	−	0.796803i	$-0.293477\pi$
$857$	35.3777	1.20848	0.604240	+	0.796803i	$0.293477\pi$
$858$	0	0
$859$	−4.72607	−0.161251	−0.0806257	−	0.996744i	$-0.525692\pi$
$859$	−4.72607	−0.161251	−0.0806257	+	0.996744i	$0.525692\pi$
$860$	0	0
$861$	14.8431	0.505853
$862$	0	0
$863$	31.7770	1.08170	0.540851	−	0.841118i	$-0.318102\pi$
$863$	31.7770	1.08170	0.540851	+	0.841118i	$0.318102\pi$
$864$	0	0
$865$	−8.94146	−0.304019
$866$	0	0
$867$	16.4948	0.560192
$868$	0	0
$869$	58.8474	1.99626
$870$	0	0
$871$	−12.2251	−0.414231
$872$	0	0
$873$	6.85509	0.232010
$874$	0	0
$875$	3.47073	0.117332
$876$	0	0
$877$	−35.6795	−1.20481	−0.602405	−	0.798190i	$-0.705791\pi$
$877$	−35.6795	−1.20481	−0.602405	+	0.798190i	$0.705791\pi$
$878$	0	0
$879$	14.9011	0.502601
$880$	0	0
$881$	57.7495	1.94563	0.972815	−	0.231582i	$-0.0743903\pi$
$881$	57.7495	1.94563	0.972815	+	0.231582i	$0.0743903\pi$
$882$	0	0
$883$	−14.2591	−0.479858	−0.239929	−	0.970790i	$-0.577124\pi$
$883$	−14.2591	−0.479858	−0.239929	+	0.970790i	$0.577124\pi$
$884$	0	0
$885$	13.8797	0.466563
$886$	0	0
$887$	−29.7330	−0.998338	−0.499169	−	0.866505i	$-0.666361\pi$
$887$	−29.7330	−0.998338	−0.499169	+	0.866505i	$0.666361\pi$
$888$	0	0
$889$	19.4772	0.653246
$890$	0	0
$891$	6.27666	0.210276
$892$	0	0
$893$	0	0
$894$	0	0
$895$	17.9749	0.600834
$896$	0	0
$897$	0.572842	0.0191266
$898$	0	0
$899$	7.51671	0.250696
$900$	0	0
$901$	8.28225	0.275922
$902$	0	0
$903$	21.7846	0.724946
$904$	0	0
$905$	16.7512	0.556828
$906$	0	0
$907$	−21.9989	−0.730463	−0.365231	−	0.930917i	$-0.619010\pi$
$907$	−21.9989	−0.730463	−0.365231	+	0.930917i	$0.619010\pi$
$908$	0	0
$909$	13.6982	0.454342
$910$	0	0
$911$	−15.0529	−0.498726	−0.249363	−	0.968410i	$-0.580221\pi$
$911$	−15.0529	−0.498726	−0.249363	+	0.968410i	$0.580221\pi$
$912$	0	0
$913$	−29.5680	−0.978558
$914$	0	0
$915$	11.9749	0.395878
$916$	0	0
$917$	4.94493	0.163296
$918$	0	0
$919$	19.3045	0.636797	0.318398	−	0.947957i	$-0.396855\pi$
$919$	19.3045	0.636797	0.318398	+	0.947957i	$0.396855\pi$
$920$	0	0
$921$	14.3139	0.471658
$922$	0	0
$923$	−4.98186	−0.163980
$924$	0	0
$925$	5.47073	0.179877
$926$	0	0
$927$	−9.74739	−0.320146
$928$	0	0
$929$	19.5055	0.639956	0.319978	−	0.947425i	$-0.396324\pi$
$929$	19.5055	0.639956	0.319978	+	0.947425i	$0.396324\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.7599	0.810604
$934$	0	0
$935$	−4.46136	−0.145902
$936$	0	0
$937$	−34.8551	−1.13867	−0.569333	−	0.822107i	$-0.692799\pi$
$937$	−34.8551	−1.13867	−0.569333	+	0.822107i	$0.692799\pi$
$938$	0	0
$939$	6.41778	0.209436
$940$	0	0
$941$	−23.7683	−0.774823	−0.387412	−	0.921907i	$-0.626631\pi$
$941$	−23.7683	−0.774823	−0.387412	+	0.921907i	$0.626631\pi$
$942$	0	0
$943$	−3.03979	−0.0989891
$944$	0	0
$945$	−3.47073	−0.112903
$946$	0	0
$947$	40.4878	1.31568	0.657839	−	0.753159i	$-0.271471\pi$
$947$	40.4878	1.31568	0.657839	+	0.753159i	$0.271471\pi$
$948$	0	0
$949$	−8.93177	−0.289937
$950$	0	0
$951$	19.5352	0.633472
$952$	0	0
$953$	−33.7672	−1.09383	−0.546914	−	0.837189i	$-0.684197\pi$
$953$	−33.7672	−1.09383	−0.546914	+	0.837189i	$0.684197\pi$
$954$	0	0
$955$	−22.0700	−0.714170
$956$	0	0
$957$	−47.1799	−1.52511
$958$	0	0
$959$	−37.0871	−1.19761
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	2.71079	0.0873538
$964$	0	0
$965$	−1.05295	−0.0338956
$966$	0	0
$967$	−45.5485	−1.46474	−0.732370	−	0.680907i	$-0.761586\pi$
$967$	−45.5485	−1.46474	−0.732370	+	0.680907i	$0.761586\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.4387	0.655908	0.327954	−	0.944694i	$-0.393641\pi$
$971$	20.4387	0.655908	0.327954	+	0.944694i	$0.393641\pi$
$972$	0	0
$973$	−44.8292	−1.43716
$974$	0	0
$975$	0.805928	0.0258104
$976$	0	0
$977$	−20.0669	−0.641996	−0.320998	−	0.947080i	$-0.604018\pi$
$977$	−20.0669	−0.641996	−0.320998	+	0.947080i	$0.604018\pi$
$978$	0	0
$979$	76.4592	2.44365
$980$	0	0
$981$	16.9874	0.542367
$982$	0	0
$983$	−4.09196	−0.130513	−0.0652567	−	0.997869i	$-0.520787\pi$
$983$	−4.09196	−0.130513	−0.0652567	+	0.997869i	$0.520787\pi$
$984$	0	0
$985$	15.8425	0.504785
$986$	0	0
$987$	33.5004	1.06633
$988$	0	0
$989$	−4.46136	−0.141863
$990$	0	0
$991$	−31.5867	−1.00339	−0.501693	−	0.865046i	$-0.667289\pi$
$991$	−31.5867	−1.00339	−0.501693	+	0.865046i	$0.667289\pi$
$992$	0	0
$993$	−5.46755	−0.173508
$994$	0	0
$995$	−7.54077	−0.239058
$996$	0	0
$997$	−53.0223	−1.67923	−0.839616	−	0.543181i	$-0.817220\pi$
$997$	−53.0223	−1.67923	−0.839616	+	0.543181i	$0.817220\pi$
$998$	0	0
$999$	−5.47073	−0.173086

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1860.2.a.i.1.4	✓	4
3.2	odd	2		5580.2.a.m.1.4		4
4.3	odd	2		7440.2.a.cb.1.1		4
5.2	odd	4		9300.2.g.s.3349.8		8
5.3	odd	4		9300.2.g.s.3349.1		8
5.4	even	2		9300.2.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.i.1.4	✓	4	1.1	even	1	trivial
5580.2.a.m.1.4		4	3.2	odd	2
7440.2.a.cb.1.1		4	4.3	odd	2
9300.2.a.x.1.1		4	5.4	even	2
9300.2.g.s.3349.1		8	5.3	odd	4
9300.2.g.s.3349.8		8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.47073 q^{7} +1.00000 q^{9} +6.27666 q^{11} -0.805928 q^{13} -1.00000 q^{15} -0.710785 q^{17} -3.47073 q^{21} +0.710785 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.51671 q^{29} +1.00000 q^{31} -6.27666 q^{33} +3.47073 q^{35} +5.47073 q^{37} +0.805928 q^{39} -4.27666 q^{41} -6.27666 q^{43} +1.00000 q^{45} -9.65225 q^{47} +5.04598 q^{49} +0.710785 q^{51} -11.6522 q^{53} +6.27666 q^{55} -13.8797 q^{59} -11.9749 q^{61} +3.47073 q^{63} -0.805928 q^{65} +15.1690 q^{67} -0.710785 q^{69} +6.18152 q^{71} +11.0826 q^{73} -1.00000 q^{75} +21.7846 q^{77} +9.37559 q^{79} +1.00000 q^{81} -4.71079 q^{83} -0.710785 q^{85} -7.51671 q^{87} +12.1815 q^{89} -2.79716 q^{91} -1.00000 q^{93} +6.85509 q^{97} +6.27666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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