LMFDB - Embedded newform 1840.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1,1840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.69353$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +2.74252 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{5} +2.74252 q^{7} +3.56155 q^{9} +3.38705 q^{11} -2.46356 q^{13} +2.56155 q^{15} -2.64453 q^{17} -3.38705 q^{19} +7.02511 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} +9.20608 q^{29} +5.10809 q^{31} +8.67611 q^{33} +2.74252 q^{35} +5.50369 q^{37} -6.31054 q^{39} -1.20608 q^{41} -3.02511 q^{43} +3.56155 q^{45} -8.21255 q^{47} +0.521423 q^{49} -6.77410 q^{51} -10.5417 q^{53} +3.38705 q^{55} -8.67611 q^{57} -8.28259 q^{59} +0.263945 q^{61} +9.76763 q^{63} -2.46356 q^{65} +7.66964 q^{67} +2.56155 q^{69} +0.0150161 q^{71} -5.53644 q^{73} +2.56155 q^{75} +9.28906 q^{77} +8.67611 q^{79} -7.00000 q^{81} +3.52142 q^{83} -2.64453 q^{85} +23.5819 q^{87} -4.66318 q^{89} -6.75637 q^{91} +13.0846 q^{93} -3.38705 q^{95} -4.09799 q^{97} +12.0632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9	$ 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - q^{17} + 4 q^{19} + 10 q^{21} + 4 q^{23} + 4 q^{25} + 14 q^{27} + 19 q^{29} + q^{31} - 2 q^{33} + 3 q^{35} - 3 q^{37} + 13 q^{41} + 6 q^{43} + 6 q^{45} - 6 q^{47} + 9 q^{49} + 8 q^{51} + 19 q^{53} - 4 q^{55} + 2 q^{57} - 23 q^{59} + 13 q^{63} + 3 q^{67} + 2 q^{69} + 3 q^{71} - 32 q^{73} + 2 q^{75} + 18 q^{77} - 2 q^{79} - 28 q^{81} + 21 q^{83} - q^{85} + 18 q^{87} + 40 q^{91} - 8 q^{93} + 4 q^{95} - 18 q^{97} - 6 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - q^17 + 4 * q^19 + 10 * q^21 + 4 * q^23 + 4 * q^25 + 14 * q^27 + 19 * q^29 + q^31 - 2 * q^33 + 3 * q^35 - 3 * q^37 + 13 * q^41 + 6 * q^43 + 6 * q^45 - 6 * q^47 + 9 * q^49 + 8 * q^51 + 19 * q^53 - 4 * q^55 + 2 * q^57 - 23 * q^59 + 13 * q^63 + 3 * q^67 + 2 * q^69 + 3 * q^71 - 32 * q^73 + 2 * q^75 + 18 * q^77 - 2 * q^79 - 28 * q^81 + 21 * q^83 - q^85 + 18 * q^87 + 40 * q^91 - 8 * q^93 + 4 * q^95 - 18 * q^97 - 6 * 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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.74252	1.03658	0.518288	−	0.855206i	$-0.326570\pi$
$7$	2.74252	1.03658	0.518288	+	0.855206i	$0.326570\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	3.38705	1.02123	0.510617	−	0.859808i	$-0.329417\pi$
$11$	3.38705	1.02123	0.510617	+	0.859808i	$0.329417\pi$
$12$	0	0
$13$	−2.46356	−0.683269	−0.341634	−	0.939833i	$-0.610980\pi$
$13$	−2.46356	−0.683269	−0.341634	+	0.939833i	$0.610980\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	−2.64453	−0.641393	−0.320696	−	0.947182i	$-0.603917\pi$
$17$	−2.64453	−0.641393	−0.320696	+	0.947182i	$0.603917\pi$
$18$	0	0
$19$	−3.38705	−0.777043	−0.388521	−	0.921440i	$-0.627014\pi$
$19$	−3.38705	−0.777043	−0.388521	+	0.921440i	$0.627014\pi$
$20$	0	0
$21$	7.02511	1.53301
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	9.20608	1.70953	0.854763	−	0.519018i	$-0.173702\pi$
$29$	9.20608	1.70953	0.854763	+	0.519018i	$0.173702\pi$
$30$	0	0
$31$	5.10809	0.917440	0.458720	−	0.888581i	$-0.348308\pi$
$31$	5.10809	0.917440	0.458720	+	0.888581i	$0.348308\pi$
$32$	0	0
$33$	8.67611	1.51032
$34$	0	0
$35$	2.74252	0.463571
$36$	0	0
$37$	5.50369	0.904801	0.452401	−	0.891815i	$-0.350568\pi$
$37$	5.50369	0.904801	0.452401	+	0.891815i	$0.350568\pi$
$38$	0	0
$39$	−6.31054	−1.01050
$40$	0	0
$41$	−1.20608	−0.188358	−0.0941792	−	0.995555i	$-0.530023\pi$
$41$	−1.20608	−0.188358	−0.0941792	+	0.995555i	$0.530023\pi$
$42$	0	0
$43$	−3.02511	−0.461325	−0.230663	−	0.973034i	$-0.574089\pi$
$43$	−3.02511	−0.461325	−0.230663	+	0.973034i	$0.574089\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	−8.21255	−1.19792	−0.598962	−	0.800778i	$-0.704420\pi$
$47$	−8.21255	−1.19792	−0.598962	+	0.800778i	$0.704420\pi$
$48$	0	0
$49$	0.521423	0.0744891
$50$	0	0
$51$	−6.77410	−0.948564
$52$	0	0
$53$	−10.5417	−1.44802	−0.724009	−	0.689790i	$-0.757703\pi$
$53$	−10.5417	−1.44802	−0.724009	+	0.689790i	$0.757703\pi$
$54$	0	0
$55$	3.38705	0.456710
$56$	0	0
$57$	−8.67611	−1.14918
$58$	0	0
$59$	−8.28259	−1.07830	−0.539151	−	0.842209i	$-0.681255\pi$
$59$	−8.28259	−1.07830	−0.539151	+	0.842209i	$0.681255\pi$
$60$	0	0
$61$	0.263945	0.0337947	0.0168973	−	0.999857i	$-0.494621\pi$
$61$	0.263945	0.0337947	0.0168973	+	0.999857i	$0.494621\pi$
$62$	0	0
$63$	9.76763	1.23061
$64$	0	0
$65$	−2.46356	−0.305567
$66$	0	0
$67$	7.66964	0.936996	0.468498	−	0.883465i	$-0.344795\pi$
$67$	7.66964	0.936996	0.468498	+	0.883465i	$0.344795\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	0.0150161	0.00178208	0.000891041	−	1.00000i	$-0.499716\pi$
$71$	0.0150161	0.00178208	0.000891041		1.00000i	$0.499716\pi$
$72$	0	0
$73$	−5.53644	−0.647991	−0.323996	−	0.946059i	$-0.605026\pi$
$73$	−5.53644	−0.647991	−0.323996	+	0.946059i	$0.605026\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	9.28906	1.05859
$78$	0	0
$79$	8.67611	0.976138	0.488069	−	0.872805i	$-0.337701\pi$
$79$	8.67611	0.976138	0.488069	+	0.872805i	$0.337701\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	3.52142	0.386526	0.193263	−	0.981147i	$-0.438093\pi$
$83$	3.52142	0.386526	0.193263	+	0.981147i	$0.438093\pi$
$84$	0	0
$85$	−2.64453	−0.286839
$86$	0	0
$87$	23.5819	2.52824
$88$	0	0
$89$	−4.66318	−0.494296	−0.247148	−	0.968978i	$-0.579493\pi$
$89$	−4.66318	−0.494296	−0.247148	+	0.968978i	$0.579493\pi$
$90$	0	0
$91$	−6.75637	−0.708260
$92$	0	0
$93$	13.0846	1.35681
$94$	0	0
$95$	−3.38705	−0.347504
$96$	0	0
$97$	−4.09799	−0.416088	−0.208044	−	0.978119i	$-0.566710\pi$
$97$	−4.09799	−0.416088	−0.208044	+	0.978119i	$0.566710\pi$
$98$	0	0
$99$	12.0632	1.21239
$100$	0	0
$101$	12.2955	1.22345	0.611725	−	0.791070i	$-0.290476\pi$
$101$	12.2955	1.22345	0.611725	+	0.791070i	$0.290476\pi$
$102$	0	0
$103$	−6.05023	−0.596147	−0.298073	−	0.954543i	$-0.596344\pi$
$103$	−6.05023	−0.596147	−0.298073	+	0.954543i	$0.596344\pi$
$104$	0	0
$105$	7.02511	0.685581
$106$	0	0
$107$	−13.1547	−1.27171	−0.635856	−	0.771808i	$-0.719353\pi$
$107$	−13.1547	−1.27171	−0.635856	+	0.771808i	$0.719353\pi$
$108$	0	0
$109$	−17.1183	−1.63964	−0.819818	−	0.572624i	$-0.805925\pi$
$109$	−17.1183	−1.63964	−0.819818	+	0.572624i	$0.805925\pi$
$110$	0	0
$111$	14.0980	1.33812
$112$	0	0
$113$	4.38058	0.412091	0.206045	−	0.978542i	$-0.433941\pi$
$113$	4.38058	0.412091	0.206045	+	0.978542i	$0.433941\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−8.77410	−0.811166
$118$	0	0
$119$	−7.25268	−0.664852
$120$	0	0
$121$	0.472110	0.0429191
$122$	0	0
$123$	−3.08944	−0.278566
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.4588	1.63795	0.818975	−	0.573829i	$-0.194543\pi$
$127$	18.4588	1.63795	0.818975	+	0.573829i	$0.194543\pi$
$128$	0	0
$129$	−7.74899	−0.682260
$130$	0	0
$131$	3.86834	0.337979	0.168989	−	0.985618i	$-0.445950\pi$
$131$	3.86834	0.337979	0.168989	+	0.985618i	$0.445950\pi$
$132$	0	0
$133$	−9.28906	−0.805463
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	20.6713	1.76607	0.883034	−	0.469308i	$-0.155497\pi$
$137$	20.6713	1.76607	0.883034	+	0.469308i	$0.155497\pi$
$138$	0	0
$139$	−12.5802	−1.06704	−0.533519	−	0.845788i	$-0.679131\pi$
$139$	−12.5802	−1.06704	−0.533519	+	0.845788i	$0.679131\pi$
$140$	0	0
$141$	−21.0369	−1.77162
$142$	0	0
$143$	−8.34420	−0.697777
$144$	0	0
$145$	9.20608	0.764523
$146$	0	0
$147$	1.33565	0.110163
$148$	0	0
$149$	11.4850	0.940891	0.470446	−	0.882429i	$-0.344093\pi$
$149$	11.4850	0.940891	0.470446	+	0.882429i	$0.344093\pi$
$150$	0	0
$151$	5.58667	0.454636	0.227318	−	0.973821i	$-0.427004\pi$
$151$	5.58667	0.454636	0.227318	+	0.973821i	$0.427004\pi$
$152$	0	0
$153$	−9.41863	−0.761451
$154$	0	0
$155$	5.10809	0.410292
$156$	0	0
$157$	5.86563	0.468128	0.234064	−	0.972221i	$-0.424797\pi$
$157$	5.86563	0.468128	0.234064	+	0.972221i	$0.424797\pi$
$158$	0	0
$159$	−27.0032	−2.14149
$160$	0	0
$161$	2.74252	0.216141
$162$	0	0
$163$	−0.0465955	−0.00364964	−0.00182482	−	0.999998i	$-0.500581\pi$
$163$	−0.0465955	−0.00364964	−0.00182482	+	0.999998i	$0.500581\pi$
$164$	0	0
$165$	8.67611	0.675434
$166$	0	0
$167$	2.24621	0.173817	0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621	0.173817	0.0869085	+	0.996216i	$0.472301\pi$
$168$	0	0
$169$	−6.93087	−0.533144
$170$	0	0
$171$	−12.0632	−0.922493
$172$	0	0
$173$	8.01293	0.609212	0.304606	−	0.952478i	$-0.401475\pi$
$173$	8.01293	0.609212	0.304606	+	0.952478i	$0.401475\pi$
$174$	0	0
$175$	2.74252	0.207315
$176$	0	0
$177$	−21.2163	−1.59471
$178$	0	0
$179$	11.9615	0.894047	0.447024	−	0.894522i	$-0.352484\pi$
$179$	11.9615	0.894047	0.447024	+	0.894522i	$0.352484\pi$
$180$	0	0
$181$	−17.4802	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$181$	−17.4802	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$182$	0	0
$183$	0.676108	0.0499794
$184$	0	0
$185$	5.50369	0.404639
$186$	0	0
$187$	−8.95715	−0.655012
$188$	0	0
$189$	3.94497	0.286954
$190$	0	0
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	0	0
$193$	−12.5438	−0.902924	−0.451462	−	0.892290i	$-0.649097\pi$
$193$	−12.5438	−0.902924	−0.451462	+	0.892290i	$0.649097\pi$
$194$	0	0
$195$	−6.31054	−0.451907
$196$	0	0
$197$	3.88544	0.276826	0.138413	−	0.990375i	$-0.455800\pi$
$197$	3.88544	0.276826	0.138413	+	0.990375i	$0.455800\pi$
$198$	0	0
$199$	−22.1612	−1.57096	−0.785481	−	0.618886i	$-0.787584\pi$
$199$	−22.1612	−1.57096	−0.785481	+	0.618886i	$0.787584\pi$
$200$	0	0
$201$	19.6462	1.38574
$202$	0	0
$203$	25.2479	1.77205
$204$	0	0
$205$	−1.20608	−0.0842364
$206$	0	0
$207$	3.56155	0.247545
$208$	0	0
$209$	−11.4721	−0.793542
$210$	0	0
$211$	−4.81048	−0.331167	−0.165584	−	0.986196i	$-0.552951\pi$
$211$	−4.81048	−0.331167	−0.165584	+	0.986196i	$0.552951\pi$
$212$	0	0
$213$	0.0384645	0.00263554
$214$	0	0
$215$	−3.02511	−0.206311
$216$	0	0
$217$	14.0090	0.950996
$218$	0	0
$219$	−14.1819	−0.958323
$220$	0	0
$221$	6.51496	0.438243
$222$	0	0
$223$	−13.7815	−0.922876	−0.461438	−	0.887172i	$-0.652666\pi$
$223$	−13.7815	−0.922876	−0.461438	+	0.887172i	$0.652666\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	−29.1012	−1.93151	−0.965757	−	0.259447i	$-0.916460\pi$
$227$	−29.1012	−1.93151	−0.965757	+	0.259447i	$0.916460\pi$
$228$	0	0
$229$	−9.38705	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$229$	−9.38705	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$230$	0	0
$231$	23.7944	1.56556
$232$	0	0
$233$	−12.6725	−0.830202	−0.415101	−	0.909775i	$-0.636254\pi$
$233$	−12.6725	−0.830202	−0.415101	+	0.909775i	$0.636254\pi$
$234$	0	0
$235$	−8.21255	−0.535728
$236$	0	0
$237$	22.2243	1.44362
$238$	0	0
$239$	−19.1883	−1.24119	−0.620596	−	0.784131i	$-0.713109\pi$
$239$	−19.1883	−1.24119	−0.620596	+	0.784131i	$0.713109\pi$
$240$	0	0
$241$	16.7741	1.08051	0.540257	−	0.841500i	$-0.318327\pi$
$241$	16.7741	1.08051	0.540257	+	0.841500i	$0.318327\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	0.521423	0.0333125
$246$	0	0
$247$	8.34420	0.530929
$248$	0	0
$249$	9.02031	0.571639
$250$	0	0
$251$	22.8114	1.43984	0.719921	−	0.694056i	$-0.244178\pi$
$251$	22.8114	1.43984	0.719921	+	0.694056i	$0.244178\pi$
$252$	0	0
$253$	3.38705	0.212942
$254$	0	0
$255$	−6.77410	−0.424211
$256$	0	0
$257$	−23.7526	−1.48165	−0.740824	−	0.671699i	$-0.765565\pi$
$257$	−23.7526	−1.48165	−0.740824	+	0.671699i	$0.765565\pi$
$258$	0	0
$259$	15.0940	0.937895
$260$	0	0
$261$	32.7879	2.02952
$262$	0	0
$263$	−16.5895	−1.02295	−0.511476	−	0.859297i	$-0.670901\pi$
$263$	−16.5895	−1.02295	−0.511476	+	0.859297i	$0.670901\pi$
$264$	0	0
$265$	−10.5417	−0.647574
$266$	0	0
$267$	−11.9450	−0.731020
$268$	0	0
$269$	−8.47495	−0.516727	−0.258363	−	0.966048i	$-0.583183\pi$
$269$	−8.47495	−0.516727	−0.258363	+	0.966048i	$0.583183\pi$
$270$	0	0
$271$	21.3029	1.29406	0.647030	−	0.762465i	$-0.276011\pi$
$271$	21.3029	1.29406	0.647030	+	0.762465i	$0.276011\pi$
$272$	0	0
$273$	−17.3068	−1.04745
$274$	0	0
$275$	3.38705	0.204247
$276$	0	0
$277$	21.9615	1.31954	0.659770	−	0.751467i	$-0.270654\pi$
$277$	21.9615	1.31954	0.659770	+	0.751467i	$0.270654\pi$
$278$	0	0
$279$	18.1927	1.08917
$280$	0	0
$281$	19.9020	1.18725	0.593627	−	0.804740i	$-0.297695\pi$
$281$	19.9020	1.18725	0.593627	+	0.804740i	$0.297695\pi$
$282$	0	0
$283$	8.87781	0.527731	0.263865	−	0.964559i	$-0.415003\pi$
$283$	8.87781	0.527731	0.263865	+	0.964559i	$0.415003\pi$
$284$	0	0
$285$	−8.67611	−0.513928
$286$	0	0
$287$	−3.30771	−0.195248
$288$	0	0
$289$	−10.0065	−0.588616
$290$	0	0
$291$	−10.4972	−0.615358
$292$	0	0
$293$	23.9709	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$293$	23.9709	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$294$	0	0
$295$	−8.28259	−0.482231
$296$	0	0
$297$	4.87209	0.282708
$298$	0	0
$299$	−2.46356	−0.142471
$300$	0	0
$301$	−8.29644	−0.478199
$302$	0	0
$303$	31.4956	1.80938
$304$	0	0
$305$	0.263945	0.0151134
$306$	0	0
$307$	−12.0632	−0.688481	−0.344240	−	0.938882i	$-0.611864\pi$
$307$	−12.0632	−0.688481	−0.344240	+	0.938882i	$0.611864\pi$
$308$	0	0
$309$	−15.4980	−0.881649
$310$	0	0
$311$	6.81257	0.386305	0.193153	−	0.981169i	$-0.438129\pi$
$311$	6.81257	0.386305	0.193153	+	0.981169i	$0.438129\pi$
$312$	0	0
$313$	−6.38539	−0.360923	−0.180462	−	0.983582i	$-0.557759\pi$
$313$	−6.38539	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$314$	0	0
$315$	9.76763	0.550344
$316$	0	0
$317$	16.8114	0.944222	0.472111	−	0.881539i	$-0.343492\pi$
$317$	16.8114	0.944222	0.472111	+	0.881539i	$0.343492\pi$
$318$	0	0
$319$	31.1815	1.74583
$320$	0	0
$321$	−33.6964	−1.88075
$322$	0	0
$323$	8.95715	0.498389
$324$	0	0
$325$	−2.46356	−0.136654
$326$	0	0
$327$	−43.8494	−2.42488
$328$	0	0
$329$	−22.5231	−1.24174
$330$	0	0
$331$	−3.29114	−0.180898	−0.0904488	−	0.995901i	$-0.528830\pi$
$331$	−3.29114	−0.180898	−0.0904488	+	0.995901i	$0.528830\pi$
$332$	0	0
$333$	19.6017	1.07417
$334$	0	0
$335$	7.66964	0.419037
$336$	0	0
$337$	−7.90201	−0.430450	−0.215225	−	0.976565i	$-0.569048\pi$
$337$	−7.90201	−0.430450	−0.215225	+	0.976565i	$0.569048\pi$
$338$	0	0
$339$	11.2211	0.609446
$340$	0	0
$341$	17.3014	0.936921
$342$	0	0
$343$	−17.7676	−0.959362
$344$	0	0
$345$	2.56155	0.137909
$346$	0	0
$347$	19.4024	1.04158	0.520789	−	0.853686i	$-0.325638\pi$
$347$	19.4024	1.04158	0.520789	+	0.853686i	$0.325638\pi$
$348$	0	0
$349$	−33.1664	−1.77536	−0.887680	−	0.460462i	$-0.847684\pi$
$349$	−33.1664	−1.77536	−0.887680	+	0.460462i	$0.847684\pi$
$350$	0	0
$351$	−3.54370	−0.189149
$352$	0	0
$353$	−29.8960	−1.59121	−0.795603	−	0.605819i	$-0.792846\pi$
$353$	−29.8960	−1.59121	−0.795603	+	0.605819i	$0.792846\pi$
$354$	0	0
$355$	0.0150161	0.000796971 0
$356$	0	0
$357$	−18.5781	−0.983258
$358$	0	0
$359$	24.9725	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$359$	24.9725	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$360$	0	0
$361$	−7.52789	−0.396205
$362$	0	0
$363$	1.20934	0.0634737
$364$	0	0
$365$	−5.53644	−0.289790
$366$	0	0
$367$	−15.8179	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$367$	−15.8179	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$368$	0	0
$369$	−4.29552	−0.223616
$370$	0	0
$371$	−28.9109	−1.50098
$372$	0	0
$373$	22.6283	1.17165	0.585826	−	0.810437i	$-0.300770\pi$
$373$	22.6283	1.17165	0.585826	+	0.810437i	$0.300770\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	−22.6797	−1.16807
$378$	0	0
$379$	−10.2721	−0.527641	−0.263821	−	0.964572i	$-0.584983\pi$
$379$	−10.2721	−0.527641	−0.263821	+	0.964572i	$0.584983\pi$
$380$	0	0
$381$	47.2831	2.42239
$382$	0	0
$383$	−6.21463	−0.317553	−0.158776	−	0.987315i	$-0.550755\pi$
$383$	−6.21463	−0.317553	−0.158776	+	0.987315i	$0.550755\pi$
$384$	0	0
$385$	9.28906	0.473414
$386$	0	0
$387$	−10.7741	−0.547678
$388$	0	0
$389$	27.6414	1.40147	0.700737	−	0.713420i	$-0.252855\pi$
$389$	27.6414	1.40147	0.700737	+	0.713420i	$0.252855\pi$
$390$	0	0
$391$	−2.64453	−0.133740
$392$	0	0
$393$	9.90897	0.499841
$394$	0	0
$395$	8.67611	0.436542
$396$	0	0
$397$	1.07171	0.0537875	0.0268938	−	0.999638i	$-0.491438\pi$
$397$	1.07171	0.0537875	0.0268938	+	0.999638i	$0.491438\pi$
$398$	0	0
$399$	−23.7944	−1.19121
$400$	0	0
$401$	1.49798	0.0748053	0.0374027	−	0.999300i	$-0.488092\pi$
$401$	1.49798	0.0748053	0.0374027	+	0.999300i	$0.488092\pi$
$402$	0	0
$403$	−12.5841	−0.626858
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	18.6413	0.924014
$408$	0	0
$409$	20.9373	1.03528	0.517642	−	0.855597i	$-0.326810\pi$
$409$	20.9373	1.03528	0.517642	+	0.855597i	$0.326810\pi$
$410$	0	0
$411$	52.9506	2.61186
$412$	0	0
$413$	−22.7152	−1.11774
$414$	0	0
$415$	3.52142	0.172860
$416$	0	0
$417$	−32.2248	−1.57806
$418$	0	0
$419$	18.7564	0.916309	0.458154	−	0.888873i	$-0.348511\pi$
$419$	18.7564	0.916309	0.458154	+	0.888873i	$0.348511\pi$
$420$	0	0
$421$	20.6163	1.00478	0.502388	−	0.864642i	$-0.332455\pi$
$421$	20.6163	1.00478	0.502388	+	0.864642i	$0.332455\pi$
$422$	0	0
$423$	−29.2494	−1.42216
$424$	0	0
$425$	−2.64453	−0.128279
$426$	0	0
$427$	0.723874	0.0350307
$428$	0	0
$429$	−21.3741	−1.03195
$430$	0	0
$431$	−27.0155	−1.30129	−0.650646	−	0.759381i	$-0.725502\pi$
$431$	−27.0155	−1.30129	−0.650646	+	0.759381i	$0.725502\pi$
$432$	0	0
$433$	−25.2900	−1.21536	−0.607679	−	0.794183i	$-0.707899\pi$
$433$	−25.2900	−1.21536	−0.607679	+	0.794183i	$0.707899\pi$
$434$	0	0
$435$	23.5819	1.13066
$436$	0	0
$437$	−3.38705	−0.162025
$438$	0	0
$439$	−34.4110	−1.64235	−0.821174	−	0.570679i	$-0.806680\pi$
$439$	−34.4110	−1.64235	−0.821174	+	0.570679i	$0.806680\pi$
$440$	0	0
$441$	1.85708	0.0884322
$442$	0	0
$443$	29.8281	1.41717	0.708587	−	0.705623i	$-0.249333\pi$
$443$	29.8281	1.41717	0.708587	+	0.705623i	$0.249333\pi$
$444$	0	0
$445$	−4.66318	−0.221056
$446$	0	0
$447$	29.4195	1.39150
$448$	0	0
$449$	−2.67456	−0.126220	−0.0631102	−	0.998007i	$-0.520102\pi$
$449$	−2.67456	−0.126220	−0.0631102	+	0.998007i	$0.520102\pi$
$450$	0	0
$451$	−4.08506	−0.192358
$452$	0	0
$453$	14.3105	0.672368
$454$	0	0
$455$	−6.75637	−0.316743
$456$	0	0
$457$	−30.9037	−1.44561	−0.722806	−	0.691051i	$-0.757148\pi$
$457$	−30.9037	−1.44561	−0.722806	+	0.691051i	$0.757148\pi$
$458$	0	0
$459$	−3.80402	−0.177556
$460$	0	0
$461$	−26.6022	−1.23899	−0.619493	−	0.785002i	$-0.712662\pi$
$461$	−26.6022	−1.23899	−0.619493	+	0.785002i	$0.712662\pi$
$462$	0	0
$463$	−26.7013	−1.24092	−0.620458	−	0.784239i	$-0.713053\pi$
$463$	−26.7013	−1.24092	−0.620458	+	0.784239i	$0.713053\pi$
$464$	0	0
$465$	13.0846	0.606786
$466$	0	0
$467$	−35.2503	−1.63119	−0.815596	−	0.578622i	$-0.803590\pi$
$467$	−35.2503	−1.63119	−0.815596	+	0.578622i	$0.803590\pi$
$468$	0	0
$469$	21.0342	0.971267
$470$	0	0
$471$	15.0251	0.692321
$472$	0	0
$473$	−10.2462	−0.471121
$474$	0	0
$475$	−3.38705	−0.155409
$476$	0	0
$477$	−37.5449	−1.71907
$478$	0	0
$479$	39.0705	1.78518	0.892589	−	0.450871i	$-0.148886\pi$
$479$	39.0705	1.78518	0.892589	+	0.450871i	$0.148886\pi$
$480$	0	0
$481$	−13.5587	−0.618222
$482$	0	0
$483$	7.02511	0.319654
$484$	0	0
$485$	−4.09799	−0.186080
$486$	0	0
$487$	−24.0839	−1.09135	−0.545673	−	0.837998i	$-0.683726\pi$
$487$	−24.0839	−1.09135	−0.545673	+	0.837998i	$0.683726\pi$
$488$	0	0
$489$	−0.119357	−0.00539751
$490$	0	0
$491$	−5.60051	−0.252748	−0.126374	−	0.991983i	$-0.540334\pi$
$491$	−5.60051	−0.252748	−0.126374	+	0.991983i	$0.540334\pi$
$492$	0	0
$493$	−24.3458	−1.09648
$494$	0	0
$495$	12.0632	0.542199
$496$	0	0
$497$	0.0411819	0.00184726
$498$	0	0
$499$	−9.53319	−0.426764	−0.213382	−	0.976969i	$-0.568448\pi$
$499$	−9.53319	−0.426764	−0.213382	+	0.976969i	$0.568448\pi$
$500$	0	0
$501$	5.75379	0.257060
$502$	0	0
$503$	−14.0568	−0.626762	−0.313381	−	0.949627i	$-0.601462\pi$
$503$	−14.0568	−0.626762	−0.313381	+	0.949627i	$0.601462\pi$
$504$	0	0
$505$	12.2955	0.547144
$506$	0	0
$507$	−17.7538	−0.788473
$508$	0	0
$509$	21.7105	0.962302	0.481151	−	0.876638i	$-0.340219\pi$
$509$	21.7105	0.962302	0.481151	+	0.876638i	$0.340219\pi$
$510$	0	0
$511$	−15.1838	−0.671692
$512$	0	0
$513$	−4.87209	−0.215108
$514$	0	0
$515$	−6.05023	−0.266605
$516$	0	0
$517$	−27.8163	−1.22336
$518$	0	0
$519$	20.5255	0.900972
$520$	0	0
$521$	−8.87689	−0.388904	−0.194452	−	0.980912i	$-0.562293\pi$
$521$	−8.87689	−0.388904	−0.194452	+	0.980912i	$0.562293\pi$
$522$	0	0
$523$	0.0550279	0.00240620	0.00120310	−	0.999999i	$-0.499617\pi$
$523$	0.0550279	0.00240620	0.00120310	+	0.999999i	$0.499617\pi$
$524$	0	0
$525$	7.02511	0.306601
$526$	0	0
$527$	−13.5085	−0.588439
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−29.4989	−1.28014
$532$	0	0
$533$	2.97126	0.128699
$534$	0	0
$535$	−13.1547	−0.568727
$536$	0	0
$537$	30.6401	1.32222
$538$	0	0
$539$	1.76609	0.0760708
$540$	0	0
$541$	5.99070	0.257560	0.128780	−	0.991673i	$-0.458894\pi$
$541$	5.99070	0.257560	0.128780	+	0.991673i	$0.458894\pi$
$542$	0	0
$543$	−44.7766	−1.92155
$544$	0	0
$545$	−17.1183	−0.733268
$546$	0	0
$547$	0.408533	0.0174676	0.00873380	−	0.999962i	$-0.497220\pi$
$547$	0.408533	0.0174676	0.00873380	+	0.999962i	$0.497220\pi$
$548$	0	0
$549$	0.940053	0.0401205
$550$	0	0
$551$	−31.1815	−1.32837
$552$	0	0
$553$	23.7944	1.01184
$554$	0	0
$555$	14.0980	0.598426
$556$	0	0
$557$	−5.46406	−0.231519	−0.115760	−	0.993277i	$-0.536930\pi$
$557$	−5.46406	−0.231519	−0.115760	+	0.993277i	$0.536930\pi$
$558$	0	0
$559$	7.45255	0.315209
$560$	0	0
$561$	−22.9442	−0.968706
$562$	0	0
$563$	41.8212	1.76255	0.881277	−	0.472601i	$-0.156685\pi$
$563$	41.8212	1.76255	0.881277	+	0.472601i	$0.156685\pi$
$564$	0	0
$565$	4.38058	0.184293
$566$	0	0
$567$	−19.1976	−0.806225
$568$	0	0
$569$	39.5127	1.65646	0.828230	−	0.560388i	$-0.189348\pi$
$569$	39.5127	1.65646	0.828230	+	0.560388i	$0.189348\pi$
$570$	0	0
$571$	24.4924	1.02498	0.512488	−	0.858694i	$-0.328724\pi$
$571$	24.4924	1.02498	0.512488	+	0.858694i	$0.328724\pi$
$572$	0	0
$573$	33.6155	1.40431
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−36.3382	−1.51278	−0.756390	−	0.654121i	$-0.773039\pi$
$577$	−36.3382	−1.51278	−0.756390	+	0.654121i	$0.773039\pi$
$578$	0	0
$579$	−32.1317	−1.33535
$580$	0	0
$581$	9.65758	0.400664
$582$	0	0
$583$	−35.7054	−1.47877
$584$	0	0
$585$	−8.77410	−0.362764
$586$	0	0
$587$	−5.89358	−0.243254	−0.121627	−	0.992576i	$-0.538811\pi$
$587$	−5.89358	−0.243254	−0.121627	+	0.992576i	$0.538811\pi$
$588$	0	0
$589$	−17.3014	−0.712890
$590$	0	0
$591$	9.95277	0.409402
$592$	0	0
$593$	−11.0931	−0.455538	−0.227769	−	0.973715i	$-0.573143\pi$
$593$	−11.0931	−0.455538	−0.227769	+	0.973715i	$0.573143\pi$
$594$	0	0
$595$	−7.25268	−0.297331
$596$	0	0
$597$	−56.7670	−2.32332
$598$	0	0
$599$	−40.1864	−1.64197	−0.820986	−	0.570949i	$-0.806575\pi$
$599$	−40.1864	−1.64197	−0.820986	+	0.570949i	$0.806575\pi$
$600$	0	0
$601$	41.1965	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$601$	41.1965	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$602$	0	0
$603$	27.3158	1.11239
$604$	0	0
$605$	0.472110	0.0191940
$606$	0	0
$607$	16.6283	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$607$	16.6283	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$608$	0	0
$609$	64.6738	2.62071
$610$	0	0
$611$	20.2321	0.818504
$612$	0	0
$613$	25.5052	1.03015	0.515073	−	0.857146i	$-0.327765\pi$
$613$	25.5052	1.03015	0.515073	+	0.857146i	$0.327765\pi$
$614$	0	0
$615$	−3.08944	−0.124578
$616$	0	0
$617$	−3.14742	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$617$	−3.14742	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$618$	0	0
$619$	39.2665	1.57825	0.789127	−	0.614230i	$-0.210533\pi$
$619$	39.2665	1.57825	0.789127	+	0.614230i	$0.210533\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	−12.7889	−0.512375
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.3864	−1.17358
$628$	0	0
$629$	−14.5547	−0.580333
$630$	0	0
$631$	−18.2916	−0.728179	−0.364089	−	0.931364i	$-0.618620\pi$
$631$	−18.2916	−0.728179	−0.364089	+	0.931364i	$0.618620\pi$
$632$	0	0
$633$	−12.3223	−0.489768
$634$	0	0
$635$	18.4588	0.732514
$636$	0	0
$637$	−1.28456	−0.0508960
$638$	0	0
$639$	0.0534806	0.00211566
$640$	0	0
$641$	28.5304	1.12688	0.563441	−	0.826157i	$-0.309477\pi$
$641$	28.5304	1.12688	0.563441	+	0.826157i	$0.309477\pi$
$642$	0	0
$643$	12.2430	0.482815	0.241408	−	0.970424i	$-0.422391\pi$
$643$	12.2430	0.482815	0.241408	+	0.970424i	$0.422391\pi$
$644$	0	0
$645$	−7.74899	−0.305116
$646$	0	0
$647$	−3.11947	−0.122639	−0.0613196	−	0.998118i	$-0.519531\pi$
$647$	−3.11947	−0.122639	−0.0613196	+	0.998118i	$0.519531\pi$
$648$	0	0
$649$	−28.0536	−1.10120
$650$	0	0
$651$	35.8849	1.40644
$652$	0	0
$653$	−21.7301	−0.850364	−0.425182	−	0.905108i	$-0.639790\pi$
$653$	−21.7301	−0.850364	−0.425182	+	0.905108i	$0.639790\pi$
$654$	0	0
$655$	3.86834	0.151149
$656$	0	0
$657$	−19.7183	−0.769285
$658$	0	0
$659$	12.2333	0.476541	0.238270	−	0.971199i	$-0.423420\pi$
$659$	12.2333	0.476541	0.238270	+	0.971199i	$0.423420\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	16.6884	0.648124
$664$	0	0
$665$	−9.28906	−0.360214
$666$	0	0
$667$	9.20608	0.356461
$668$	0	0
$669$	−35.3020	−1.36485
$670$	0	0
$671$	0.893994	0.0345123
$672$	0	0
$673$	10.7900	0.415925	0.207963	−	0.978137i	$-0.433317\pi$
$673$	10.7900	0.415925	0.207963	+	0.978137i	$0.433317\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	23.6721	0.909793	0.454896	−	0.890544i	$-0.349676\pi$
$677$	23.6721	0.909793	0.454896	+	0.890544i	$0.349676\pi$
$678$	0	0
$679$	−11.2388	−0.431307
$680$	0	0
$681$	−74.5443	−2.85654
$682$	0	0
$683$	10.3583	0.396350	0.198175	−	0.980167i	$-0.436499\pi$
$683$	10.3583	0.396350	0.198175	+	0.980167i	$0.436499\pi$
$684$	0	0
$685$	20.6713	0.789810
$686$	0	0
$687$	−24.0454	−0.917390
$688$	0	0
$689$	25.9702	0.989386
$690$	0	0
$691$	−13.5578	−0.515763	−0.257882	−	0.966177i	$-0.583024\pi$
$691$	−13.5578	−0.515763	−0.257882	+	0.966177i	$0.583024\pi$
$692$	0	0
$693$	33.0835	1.25674
$694$	0	0
$695$	−12.5802	−0.477194
$696$	0	0
$697$	3.18952	0.120812
$698$	0	0
$699$	−32.4612	−1.22780
$700$	0	0
$701$	−21.3774	−0.807415	−0.403708	−	0.914888i	$-0.632279\pi$
$701$	−21.3774	−0.807415	−0.403708	+	0.914888i	$0.632279\pi$
$702$	0	0
$703$	−18.6413	−0.703069
$704$	0	0
$705$	−21.0369	−0.792295
$706$	0	0
$707$	33.7207	1.26820
$708$	0	0
$709$	30.3719	1.14064	0.570320	−	0.821422i	$-0.306819\pi$
$709$	30.3719	1.14064	0.570320	+	0.821422i	$0.306819\pi$
$710$	0	0
$711$	30.9004	1.15886
$712$	0	0
$713$	5.10809	0.191299
$714$	0	0
$715$	−8.34420	−0.312056
$716$	0	0
$717$	−49.1520	−1.83561
$718$	0	0
$719$	−8.41851	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$719$	−8.41851	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$720$	0	0
$721$	−16.5929	−0.617951
$722$	0	0
$723$	42.9677	1.59799
$724$	0	0
$725$	9.20608	0.341905
$726$	0	0
$727$	−19.7505	−0.732507	−0.366253	−	0.930515i	$-0.619360\pi$
$727$	−19.7505	−0.732507	−0.366253	+	0.930515i	$0.619360\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	19.7280	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$733$	19.7280	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$734$	0	0
$735$	1.33565	0.0492663
$736$	0	0
$737$	25.9775	0.956892
$738$	0	0
$739$	0.180969	0.00665704	0.00332852	−	0.999994i	$-0.498940\pi$
$739$	0.180969	0.00665704	0.00332852	+	0.999994i	$0.498940\pi$
$740$	0	0
$741$	21.3741	0.785198
$742$	0	0
$743$	−6.05023	−0.221961	−0.110981	−	0.993823i	$-0.535399\pi$
$743$	−6.05023	−0.221961	−0.110981	+	0.993823i	$0.535399\pi$
$744$	0	0
$745$	11.4850	0.420779
$746$	0	0
$747$	12.5417	0.458878
$748$	0	0
$749$	−36.0770	−1.31823
$750$	0	0
$751$	21.5659	0.786952	0.393476	−	0.919335i	$-0.371272\pi$
$751$	21.5659	0.786952	0.393476	+	0.919335i	$0.371272\pi$
$752$	0	0
$753$	58.4326	2.12940
$754$	0	0
$755$	5.58667	0.203320
$756$	0	0
$757$	−24.7798	−0.900638	−0.450319	−	0.892868i	$-0.648690\pi$
$757$	−24.7798	−0.900638	−0.450319	+	0.892868i	$0.648690\pi$
$758$	0	0
$759$	8.67611	0.314923
$760$	0	0
$761$	2.29916	0.0833443	0.0416722	−	0.999131i	$-0.486731\pi$
$761$	2.29916	0.0833443	0.0416722	+	0.999131i	$0.486731\pi$
$762$	0	0
$763$	−46.9473	−1.69961
$764$	0	0
$765$	−9.41863	−0.340531
$766$	0	0
$767$	20.4047	0.736770
$768$	0	0
$769$	−10.3692	−0.373923	−0.186961	−	0.982367i	$-0.559864\pi$
$769$	−10.3692	−0.373923	−0.186961	+	0.982367i	$0.559864\pi$
$770$	0	0
$771$	−60.8436	−2.19123
$772$	0	0
$773$	12.7864	0.459895	0.229947	−	0.973203i	$-0.426145\pi$
$773$	12.7864	0.459895	0.229947	+	0.973203i	$0.426145\pi$
$774$	0	0
$775$	5.10809	0.183488
$776$	0	0
$777$	38.6640	1.38706
$778$	0	0
$779$	4.08506	0.146362
$780$	0	0
$781$	0.0508602	0.00181992
$782$	0	0
$783$	13.2425	0.473247
$784$	0	0
$785$	5.86563	0.209353
$786$	0	0
$787$	34.9239	1.24490	0.622451	−	0.782659i	$-0.286137\pi$
$787$	34.9239	1.24490	0.622451	+	0.782659i	$0.286137\pi$
$788$	0	0
$789$	−42.4949	−1.51286
$790$	0	0
$791$	12.0138	0.427163
$792$	0	0
$793$	−0.650244	−0.0230908
$794$	0	0
$795$	−27.0032	−0.957705
$796$	0	0
$797$	4.87844	0.172803	0.0864016	−	0.996260i	$-0.472463\pi$
$797$	4.87844	0.172803	0.0864016	+	0.996260i	$0.472463\pi$
$798$	0	0
$799$	21.7183	0.768339
$800$	0	0
$801$	−16.6081	−0.586820
$802$	0	0
$803$	−18.7522	−0.661751
$804$	0	0
$805$	2.74252	0.0966612
$806$	0	0
$807$	−21.7090	−0.764194
$808$	0	0
$809$	18.1498	0.638112	0.319056	−	0.947736i	$-0.396634\pi$
$809$	18.1498	0.638112	0.319056	+	0.947736i	$0.396634\pi$
$810$	0	0
$811$	−17.8019	−0.625110	−0.312555	−	0.949900i	$-0.601185\pi$
$811$	−17.8019	−0.625110	−0.312555	+	0.949900i	$0.601185\pi$
$812$	0	0
$813$	54.5685	1.91380
$814$	0	0
$815$	−0.0465955	−0.00163217
$816$	0	0
$817$	10.2462	0.358470
$818$	0	0
$819$	−24.0632	−0.840835
$820$	0	0
$821$	32.9701	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$821$	32.9701	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$822$	0	0
$823$	39.5819	1.37974	0.689869	−	0.723935i	$-0.257668\pi$
$823$	39.5819	1.37974	0.689869	+	0.723935i	$0.257668\pi$
$824$	0	0
$825$	8.67611	0.302063
$826$	0	0
$827$	1.50849	0.0524554	0.0262277	−	0.999656i	$-0.491651\pi$
$827$	1.50849	0.0524554	0.0262277	+	0.999656i	$0.491651\pi$
$828$	0	0
$829$	−7.66718	−0.266292	−0.133146	−	0.991096i	$-0.542508\pi$
$829$	−7.66718	−0.266292	−0.133146	+	0.991096i	$0.542508\pi$
$830$	0	0
$831$	56.2556	1.95149
$832$	0	0
$833$	−1.37892	−0.0477767
$834$	0	0
$835$	2.24621	0.0777333
$836$	0	0
$837$	7.34772	0.253974
$838$	0	0
$839$	8.53602	0.294696	0.147348	−	0.989085i	$-0.452926\pi$
$839$	8.53602	0.294696	0.147348	+	0.989085i	$0.452926\pi$
$840$	0	0
$841$	55.7519	1.92248
$842$	0	0
$843$	50.9800	1.75585
$844$	0	0
$845$	−6.93087	−0.238429
$846$	0	0
$847$	1.29477	0.0444889
$848$	0	0
$849$	22.7410	0.780468
$850$	0	0
$851$	5.50369	0.188664
$852$	0	0
$853$	−48.0665	−1.64577	−0.822883	−	0.568211i	$-0.807636\pi$
$853$	−48.0665	−1.64577	−0.822883	+	0.568211i	$0.807636\pi$
$854$	0	0
$855$	−12.0632	−0.412551
$856$	0	0
$857$	29.4313	1.00535	0.502677	−	0.864474i	$-0.332348\pi$
$857$	29.4313	1.00535	0.502677	+	0.864474i	$0.332348\pi$
$858$	0	0
$859$	33.5308	1.14406	0.572029	−	0.820234i	$-0.306157\pi$
$859$	33.5308	1.14406	0.572029	+	0.820234i	$0.306157\pi$
$860$	0	0
$861$	−8.47286	−0.288754
$862$	0	0
$863$	−6.79483	−0.231299	−0.115649	−	0.993290i	$-0.536895\pi$
$863$	−6.79483	−0.231299	−0.115649	+	0.993290i	$0.536895\pi$
$864$	0	0
$865$	8.01293	0.272448
$866$	0	0
$867$	−25.6321	−0.870511
$868$	0	0
$869$	29.3864	0.996866
$870$	0	0
$871$	−18.8946	−0.640220
$872$	0	0
$873$	−14.5952	−0.493973
$874$	0	0
$875$	2.74252	0.0927141
$876$	0	0
$877$	−21.9904	−0.742563	−0.371281	−	0.928520i	$-0.621082\pi$
$877$	−21.9904	−0.742563	−0.371281	+	0.928520i	$0.621082\pi$
$878$	0	0
$879$	61.4027	2.07106
$880$	0	0
$881$	3.66242	0.123390	0.0616951	−	0.998095i	$-0.480349\pi$
$881$	3.66242	0.123390	0.0616951	+	0.998095i	$0.480349\pi$
$882$	0	0
$883$	−10.6640	−0.358874	−0.179437	−	0.983769i	$-0.557428\pi$
$883$	−10.6640	−0.358874	−0.179437	+	0.983769i	$0.557428\pi$
$884$	0	0
$885$	−21.2163	−0.713178
$886$	0	0
$887$	−0.481294	−0.0161603	−0.00808014	−	0.999967i	$-0.502572\pi$
$887$	−0.481294	−0.0161603	−0.00808014	+	0.999967i	$0.502572\pi$
$888$	0	0
$889$	50.6235	1.69786
$890$	0	0
$891$	−23.7094	−0.794293
$892$	0	0
$893$	27.8163	0.930837
$894$	0	0
$895$	11.9615	0.399830
$896$	0	0
$897$	−6.31054	−0.210703
$898$	0	0
$899$	47.0255	1.56839
$900$	0	0
$901$	27.8779	0.928748
$902$	0	0
$903$	−21.2518	−0.707214
$904$	0	0
$905$	−17.4802	−0.581063
$906$	0	0
$907$	50.3078	1.67044	0.835222	−	0.549913i	$-0.185339\pi$
$907$	50.3078	1.67044	0.835222	+	0.549913i	$0.185339\pi$
$908$	0	0
$909$	43.7912	1.45246
$910$	0	0
$911$	39.5103	1.30903	0.654517	−	0.756047i	$-0.272872\pi$
$911$	39.5103	1.30903	0.654517	+	0.756047i	$0.272872\pi$
$912$	0	0
$913$	11.9272	0.394734
$914$	0	0
$915$	0.676108	0.0223515
$916$	0	0
$917$	10.6090	0.350341
$918$	0	0
$919$	10.0307	0.330881	0.165441	−	0.986220i	$-0.447095\pi$
$919$	10.0307	0.330881	0.165441	+	0.986220i	$0.447095\pi$
$920$	0	0
$921$	−30.9004	−1.01820
$922$	0	0
$923$	−0.0369930	−0.00121764
$924$	0	0
$925$	5.50369	0.180960
$926$	0	0
$927$	−21.5482	−0.707736
$928$	0	0
$929$	5.83676	0.191498	0.0957490	−	0.995406i	$-0.469475\pi$
$929$	5.83676	0.191498	0.0957490	+	0.995406i	$0.469475\pi$
$930$	0	0
$931$	−1.76609	−0.0578812
$932$	0	0
$933$	17.4507	0.571312
$934$	0	0
$935$	−8.95715	−0.292930
$936$	0	0
$937$	37.5175	1.22564	0.612822	−	0.790221i	$-0.290034\pi$
$937$	37.5175	1.22564	0.612822	+	0.790221i	$0.290034\pi$
$938$	0	0
$939$	−16.3565	−0.533774
$940$	0	0
$941$	45.6189	1.48713	0.743566	−	0.668662i	$-0.233133\pi$
$941$	45.6189	1.48713	0.743566	+	0.668662i	$0.233133\pi$
$942$	0	0
$943$	−1.20608	−0.0392754
$944$	0	0
$945$	3.94497	0.128330
$946$	0	0
$947$	17.3357	0.563333	0.281667	−	0.959512i	$-0.409113\pi$
$947$	17.3357	0.563333	0.281667	+	0.959512i	$0.409113\pi$
$948$	0	0
$949$	13.6394	0.442752
$950$	0	0
$951$	43.0633	1.39642
$952$	0	0
$953$	14.2706	0.462269	0.231135	−	0.972922i	$-0.425756\pi$
$953$	14.2706	0.462269	0.231135	+	0.972922i	$0.425756\pi$
$954$	0	0
$955$	13.1231	0.424654
$956$	0	0
$957$	79.8730	2.58193
$958$	0	0
$959$	56.6915	1.83066
$960$	0	0
$961$	−4.90742	−0.158304
$962$	0	0
$963$	−46.8511	−1.50976
$964$	0	0
$965$	−12.5438	−0.403800
$966$	0	0
$967$	4.07663	0.131096	0.0655478	−	0.997849i	$-0.479121\pi$
$967$	4.07663	0.131096	0.0655478	+	0.997849i	$0.479121\pi$
$968$	0	0
$969$	22.9442	0.737075
$970$	0	0
$971$	20.2988	0.651419	0.325709	−	0.945470i	$-0.394397\pi$
$971$	20.2988	0.651419	0.325709	+	0.945470i	$0.394397\pi$
$972$	0	0
$973$	−34.5015	−1.10607
$974$	0	0
$975$	−6.31054	−0.202099
$976$	0	0
$977$	−40.3782	−1.29181	−0.645907	−	0.763416i	$-0.723521\pi$
$977$	−40.3782	−1.29181	−0.645907	+	0.763416i	$0.723521\pi$
$978$	0	0
$979$	−15.7944	−0.504792
$980$	0	0
$981$	−60.9677	−1.94655
$982$	0	0
$983$	53.1628	1.69563	0.847815	−	0.530292i	$-0.177918\pi$
$983$	53.1628	1.69563	0.847815	+	0.530292i	$0.177918\pi$
$984$	0	0
$985$	3.88544	0.123801
$986$	0	0
$987$	−57.6941	−1.83642
$988$	0	0
$989$	−3.02511	−0.0961930
$990$	0	0
$991$	−36.6746	−1.16501	−0.582503	−	0.812829i	$-0.697927\pi$
$991$	−36.6746	−1.16501	−0.582503	+	0.812829i	$0.697927\pi$
$992$	0	0
$993$	−8.43043	−0.267532
$994$	0	0
$995$	−22.1612	−0.702556
$996$	0	0
$997$	−11.1189	−0.352140	−0.176070	−	0.984378i	$-0.556339\pi$
$997$	−11.1189	−0.352140	−0.176070	+	0.984378i	$0.556339\pi$
$998$	0	0
$999$	7.91677	0.250475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.u.1.4		4
4.3	odd	2		115.2.a.c.1.4	✓	4
5.4	even	2		9200.2.a.cl.1.1		4
8.3	odd	2		7360.2.a.cj.1.3		4
8.5	even	2		7360.2.a.cg.1.2		4
12.11	even	2		1035.2.a.o.1.1		4
20.3	even	4		575.2.b.e.24.1		8
20.7	even	4		575.2.b.e.24.8		8
20.19	odd	2		575.2.a.h.1.1		4
28.27	even	2		5635.2.a.v.1.4		4
60.59	even	2		5175.2.a.bx.1.4		4
92.91	even	2		2645.2.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.4	✓	4	4.3	odd	2
575.2.a.h.1.1		4	20.19	odd	2
575.2.b.e.24.1		8	20.3	even	4
575.2.b.e.24.8		8	20.7	even	4
1035.2.a.o.1.1		4	12.11	even	2
1840.2.a.u.1.4		4	1.1	even	1	trivial
2645.2.a.m.1.4		4	92.91	even	2
5175.2.a.bx.1.4		4	60.59	even	2
5635.2.a.v.1.4		4	28.27	even	2
7360.2.a.cg.1.2		4	8.5	even	2
7360.2.a.cj.1.3		4	8.3	odd	2
9200.2.a.cl.1.1		4	5.4	even	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +2.74252 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{5} +2.74252 q^{7} +3.56155 q^{9} +3.38705 q^{11} -2.46356 q^{13} +2.56155 q^{15} -2.64453 q^{17} -3.38705 q^{19} +7.02511 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} +9.20608 q^{29} +5.10809 q^{31} +8.67611 q^{33} +2.74252 q^{35} +5.50369 q^{37} -6.31054 q^{39} -1.20608 q^{41} -3.02511 q^{43} +3.56155 q^{45} -8.21255 q^{47} +0.521423 q^{49} -6.77410 q^{51} -10.5417 q^{53} +3.38705 q^{55} -8.67611 q^{57} -8.28259 q^{59} +0.263945 q^{61} +9.76763 q^{63} -2.46356 q^{65} +7.66964 q^{67} +2.56155 q^{69} +0.0150161 q^{71} -5.53644 q^{73} +2.56155 q^{75} +9.28906 q^{77} +8.67611 q^{79} -7.00000 q^{81} +3.52142 q^{83} -2.64453 q^{85} +23.5819 q^{87} -4.66318 q^{89} -6.75637 q^{91} +13.0846 q^{93} -3.38705 q^{95} -4.09799 q^{97} +12.0632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9	\( 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - q^{17} + 4 q^{19} + 10 q^{21} + 4 q^{23} + 4 q^{25} + 14 q^{27} + 19 q^{29} + q^{31} - 2 q^{33} + 3 q^{35} - 3 q^{37} + 13 q^{41} + 6 q^{43} + 6 q^{45} - 6 q^{47} + 9 q^{49} + 8 q^{51} + 19 q^{53} - 4 q^{55} + 2 q^{57} - 23 q^{59} + 13 q^{63} + 3 q^{67} + 2 q^{69} + 3 q^{71} - 32 q^{73} + 2 q^{75} + 18 q^{77} - 2 q^{79} - 28 q^{81} + 21 q^{83} - q^{85} + 18 q^{87} + 40 q^{91} - 8 q^{93} + 4 q^{95} - 18 q^{97} - 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - q^17 + 4 * q^19 + 10 * q^21 + 4 * q^23 + 4 * q^25 + 14 * q^27 + 19 * q^29 + q^31 - 2 * q^33 + 3 * q^35 - 3 * q^37 + 13 * q^41 + 6 * q^43 + 6 * q^45 - 6 * q^47 + 9 * q^49 + 8 * q^51 + 19 * q^53 - 4 * q^55 + 2 * q^57 - 23 * q^59 + 13 * q^63 + 3 * q^67 + 2 * q^69 + 3 * q^71 - 32 * q^73 + 2 * q^75 + 18 * q^77 - 2 * q^79 - 28 * q^81 + 21 * q^83 - q^85 + 18 * q^87 + 40 * q^91 - 8 * q^93 + 4 * q^95 - 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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