LMFDB - Embedded newform 2394.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.23607 q^{10} -0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.70820 q^{17} +1.00000 q^{19} -3.23607 q^{20} -0.763932 q^{22} -5.23607 q^{23} +5.47214 q^{25} +1.23607 q^{26} -1.00000 q^{28} -8.47214 q^{29} -2.00000 q^{31} +1.00000 q^{32} +7.70820 q^{34} +3.23607 q^{35} -10.4721 q^{37} +1.00000 q^{38} -3.23607 q^{40} -6.00000 q^{41} +8.94427 q^{43} -0.763932 q^{44} -5.23607 q^{46} -8.94427 q^{47} +1.00000 q^{49} +5.47214 q^{50} +1.23607 q^{52} -12.4721 q^{53} +2.47214 q^{55} -1.00000 q^{56} -8.47214 q^{58} -8.00000 q^{59} -0.472136 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +0.291796 q^{67} +7.70820 q^{68} +3.23607 q^{70} -10.4721 q^{71} -12.4721 q^{73} -10.4721 q^{74} +1.00000 q^{76} +0.763932 q^{77} +0.763932 q^{79} -3.23607 q^{80} -6.00000 q^{82} +10.0000 q^{83} -24.9443 q^{85} +8.94427 q^{86} -0.763932 q^{88} -6.00000 q^{89} -1.23607 q^{91} -5.23607 q^{92} -8.94427 q^{94} -3.23607 q^{95} +4.76393 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 6 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} - 2 q^{20} - 6 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{26} - 2 q^{28} - 8 q^{29} - 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} - 12 q^{37} + 2 q^{38} - 2 q^{40} - 12 q^{41} - 6 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 16 q^{53} - 4 q^{55} - 2 q^{56} - 8 q^{58} - 16 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} + 14 q^{67} + 2 q^{68} + 2 q^{70} - 12 q^{71} - 16 q^{73} - 12 q^{74} + 2 q^{76} + 6 q^{77} + 6 q^{79} - 2 q^{80} - 12 q^{82} + 20 q^{83} - 32 q^{85} - 6 q^{88} - 12 q^{89} + 2 q^{91} - 6 q^{92} - 2 q^{95} + 14 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 6 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^19 - 2 * q^20 - 6 * q^22 - 6 * q^23 + 2 * q^25 - 2 * q^26 - 2 * q^28 - 8 * q^29 - 4 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 - 12 * q^37 + 2 * q^38 - 2 * q^40 - 12 * q^41 - 6 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 16 * q^53 - 4 * q^55 - 2 * q^56 - 8 * q^58 - 16 * q^59 + 8 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^65 + 14 * q^67 + 2 * q^68 + 2 * q^70 - 12 * q^71 - 16 * q^73 - 12 * q^74 + 2 * q^76 + 6 * q^77 + 6 * q^79 - 2 * q^80 - 12 * q^82 + 20 * q^83 - 32 * q^85 - 6 * q^88 - 12 * q^89 + 2 * q^91 - 6 * q^92 - 2 * q^95 + 14 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.23607	−1.02333
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.70820	1.86951	0.934757	−	0.355288i	$-0.115617\pi$
$17$	7.70820	1.86951	0.934757	+	0.355288i	$0.115617\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.23607	−0.723607
$21$	0	0
$22$	−0.763932	−0.162871
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	1.23607	0.242413
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.70820	1.32195
$35$	3.23607	0.546995
$36$	0	0
$37$	−10.4721	−1.72161	−0.860804	−	0.508936i	$-0.830039\pi$
$37$	−10.4721	−1.72161	−0.860804	+	0.508936i	$0.830039\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−3.23607	−0.511667
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	−0.763932	−0.115167
$45$	0	0
$46$	−5.23607	−0.772016
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.47214	0.773877
$51$	0	0
$52$	1.23607	0.171412
$53$	−12.4721	−1.71318	−0.856590	−	0.515998i	$-0.827421\pi$
$53$	−12.4721	−1.71318	−0.856590	+	0.515998i	$0.827421\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−8.47214	−1.11245
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−0.472136	−0.0604508	−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136	−0.0604508	−0.0302254	+	0.999543i	$0.509623\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	0.291796	0.0356486	0.0178243	−	0.999841i	$-0.494326\pi$
$67$	0.291796	0.0356486	0.0178243	+	0.999841i	$0.494326\pi$
$68$	7.70820	0.934757
$69$	0	0
$70$	3.23607	0.386784
$71$	−10.4721	−1.24281	−0.621407	−	0.783488i	$-0.713439\pi$
$71$	−10.4721	−1.24281	−0.621407	+	0.783488i	$0.713439\pi$
$72$	0	0
$73$	−12.4721	−1.45975	−0.729877	−	0.683579i	$-0.760422\pi$
$73$	−12.4721	−1.45975	−0.729877	+	0.683579i	$0.760422\pi$
$74$	−10.4721	−1.21736
$75$	0	0
$76$	1.00000	0.114708
$77$	0.763932	0.0870581
$78$	0	0
$79$	0.763932	0.0859491	0.0429745	−	0.999076i	$-0.486317\pi$
$79$	0.763932	0.0859491	0.0429745	+	0.999076i	$0.486317\pi$
$80$	−3.23607	−0.361803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−24.9443	−2.70559
$86$	8.94427	0.964486
$87$	0	0
$88$	−0.763932	−0.0814354
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	−5.23607	−0.545898
$93$	0	0
$94$	−8.94427	−0.922531
$95$	−3.23607	−0.332014
$96$	0	0
$97$	4.76393	0.483704	0.241852	−	0.970313i	$-0.422245\pi$
$97$	4.76393	0.483704	0.241852	+	0.970313i	$0.422245\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	5.47214	0.547214
$101$	13.7082	1.36402	0.682009	−	0.731344i	$-0.261107\pi$
$101$	13.7082	1.36402	0.682009	+	0.731344i	$0.261107\pi$
$102$	0	0
$103$	−6.94427	−0.684239	−0.342120	−	0.939656i	$-0.611145\pi$
$103$	−6.94427	−0.684239	−0.342120	+	0.939656i	$0.611145\pi$
$104$	1.23607	0.121206
$105$	0	0
$106$	−12.4721	−1.21140
$107$	−5.52786	−0.534399	−0.267199	−	0.963641i	$-0.586098\pi$
$107$	−5.52786	−0.534399	−0.267199	+	0.963641i	$0.586098\pi$
$108$	0	0
$109$	18.4721	1.76931	0.884655	−	0.466246i	$-0.154394\pi$
$109$	18.4721	1.76931	0.884655	+	0.466246i	$0.154394\pi$
$110$	2.47214	0.235709
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	16.9443	1.58006
$116$	−8.47214	−0.786618
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−7.70820	−0.706610
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−0.472136	−0.0427452
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−9.70820	−0.861464	−0.430732	−	0.902480i	$-0.641745\pi$
$127$	−9.70820	−0.861464	−0.430732	+	0.902480i	$0.641745\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	16.4721	1.43918	0.719589	−	0.694401i	$-0.244330\pi$
$131$	16.4721	1.43918	0.719589	+	0.694401i	$0.244330\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0.291796	0.0252073
$135$	0	0
$136$	7.70820	0.660973
$137$	15.4164	1.31711	0.658556	−	0.752531i	$-0.271167\pi$
$137$	15.4164	1.31711	0.658556	+	0.752531i	$0.271167\pi$
$138$	0	0
$139$	21.8885	1.85656	0.928281	−	0.371879i	$-0.121287\pi$
$139$	21.8885	1.85656	0.928281	+	0.371879i	$0.121287\pi$
$140$	3.23607	0.273498
$141$	0	0
$142$	−10.4721	−0.878802
$143$	−0.944272	−0.0789640
$144$	0	0
$145$	27.4164	2.27681
$146$	−12.4721	−1.03220
$147$	0	0
$148$	−10.4721	−0.860804
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	0.763932	0.0621679	0.0310840	−	0.999517i	$-0.490104\pi$
$151$	0.763932	0.0621679	0.0310840	+	0.999517i	$0.490104\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0.763932	0.0615594
$155$	6.47214	0.519854
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	0.763932	0.0607752
$159$	0	0
$160$	−3.23607	−0.255834
$161$	5.23607	0.412660
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	10.0000	0.776151
$167$	−16.9443	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$167$	−16.9443	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	−24.9443	−1.91314
$171$	0	0
$172$	8.94427	0.681994
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	−0.763932	−0.0575835
$177$	0	0
$178$	−6.00000	−0.449719
$179$	13.8885	1.03808	0.519039	−	0.854750i	$-0.326290\pi$
$179$	13.8885	1.03808	0.519039	+	0.854750i	$0.326290\pi$
$180$	0	0
$181$	−5.23607	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$181$	−5.23607	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$182$	−1.23607	−0.0916235
$183$	0	0
$184$	−5.23607	−0.386008
$185$	33.8885	2.49154
$186$	0	0
$187$	−5.88854	−0.430613
$188$	−8.94427	−0.652328
$189$	0	0
$190$	−3.23607	−0.234769
$191$	−17.2361	−1.24716	−0.623579	−	0.781760i	$-0.714322\pi$
$191$	−17.2361	−1.24716	−0.623579	+	0.781760i	$0.714322\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	4.76393	0.342030
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.9443	0.779747	0.389874	−	0.920868i	$-0.372519\pi$
$197$	10.9443	0.779747	0.389874	+	0.920868i	$0.372519\pi$
$198$	0	0
$199$	15.4164	1.09284	0.546420	−	0.837511i	$-0.315990\pi$
$199$	15.4164	1.09284	0.546420	+	0.837511i	$0.315990\pi$
$200$	5.47214	0.386938
$201$	0	0
$202$	13.7082	0.964506
$203$	8.47214	0.594627
$204$	0	0
$205$	19.4164	1.35610
$206$	−6.94427	−0.483830
$207$	0	0
$208$	1.23607	0.0857059
$209$	−0.763932	−0.0528423
$210$	0	0
$211$	−21.5967	−1.48678	−0.743391	−	0.668857i	$-0.766784\pi$
$211$	−21.5967	−1.48678	−0.743391	+	0.668857i	$0.766784\pi$
$212$	−12.4721	−0.856590
$213$	0	0
$214$	−5.52786	−0.377877
$215$	−28.9443	−1.97398
$216$	0	0
$217$	2.00000	0.135769
$218$	18.4721	1.25109
$219$	0	0
$220$	2.47214	0.166671
$221$	9.52786	0.640913
$222$	0	0
$223$	28.4721	1.90664	0.953318	−	0.301969i	$-0.0976440\pi$
$223$	28.4721	1.90664	0.953318	+	0.301969i	$0.0976440\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−10.0000	−0.665190
$227$	−11.4164	−0.757734	−0.378867	−	0.925451i	$-0.623686\pi$
$227$	−11.4164	−0.757734	−0.378867	+	0.925451i	$0.623686\pi$
$228$	0	0
$229$	−1.05573	−0.0697645	−0.0348822	−	0.999391i	$-0.511106\pi$
$229$	−1.05573	−0.0697645	−0.0348822	+	0.999391i	$0.511106\pi$
$230$	16.9443	1.11727
$231$	0	0
$232$	−8.47214	−0.556223
$233$	−5.52786	−0.362142	−0.181071	−	0.983470i	$-0.557956\pi$
$233$	−5.52786	−0.362142	−0.181071	+	0.983470i	$0.557956\pi$
$234$	0	0
$235$	28.9443	1.88812
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−7.70820	−0.499649
$239$	−26.1803	−1.69347	−0.846733	−	0.532019i	$-0.821434\pi$
$239$	−26.1803	−1.69347	−0.846733	+	0.532019i	$0.821434\pi$
$240$	0	0
$241$	−0.763932	−0.0492092	−0.0246046	−	0.999697i	$-0.507833\pi$
$241$	−0.763932	−0.0492092	−0.0246046	+	0.999697i	$0.507833\pi$
$242$	−10.4164	−0.669592
$243$	0	0
$244$	−0.472136	−0.0302254
$245$	−3.23607	−0.206745
$246$	0	0
$247$	1.23607	0.0786491
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−1.52786	−0.0966306
$251$	1.41641	0.0894029	0.0447014	−	0.999000i	$-0.485766\pi$
$251$	1.41641	0.0894029	0.0447014	+	0.999000i	$0.485766\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−9.70820	−0.609147
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.52786	0.220062	0.110031	−	0.993928i	$-0.464905\pi$
$257$	3.52786	0.220062	0.110031	+	0.993928i	$0.464905\pi$
$258$	0	0
$259$	10.4721	0.650707
$260$	−4.00000	−0.248069
$261$	0	0
$262$	16.4721	1.01765
$263$	−7.70820	−0.475308	−0.237654	−	0.971350i	$-0.576378\pi$
$263$	−7.70820	−0.475308	−0.237654	+	0.971350i	$0.576378\pi$
$264$	0	0
$265$	40.3607	2.47934
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	0.291796	0.0178243
$269$	6.94427	0.423400	0.211700	−	0.977335i	$-0.432100\pi$
$269$	6.94427	0.423400	0.211700	+	0.977335i	$0.432100\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	7.70820	0.467379
$273$	0	0
$274$	15.4164	0.931339
$275$	−4.18034	−0.252084
$276$	0	0
$277$	−14.9443	−0.897914	−0.448957	−	0.893553i	$-0.648204\pi$
$277$	−14.9443	−0.897914	−0.448957	+	0.893553i	$0.648204\pi$
$278$	21.8885	1.31279
$279$	0	0
$280$	3.23607	0.193392
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−21.8885	−1.30114	−0.650569	−	0.759447i	$-0.725470\pi$
$283$	−21.8885	−1.30114	−0.650569	+	0.759447i	$0.725470\pi$
$284$	−10.4721	−0.621407
$285$	0	0
$286$	−0.944272	−0.0558360
$287$	6.00000	0.354169
$288$	0	0
$289$	42.4164	2.49508
$290$	27.4164	1.60995
$291$	0	0
$292$	−12.4721	−0.729877
$293$	−15.8885	−0.928219	−0.464109	−	0.885778i	$-0.653626\pi$
$293$	−15.8885	−0.928219	−0.464109	+	0.885778i	$0.653626\pi$
$294$	0	0
$295$	25.8885	1.50729
$296$	−10.4721	−0.608681
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−6.47214	−0.374293
$300$	0	0
$301$	−8.94427	−0.515539
$302$	0.763932	0.0439593
$303$	0	0
$304$	1.00000	0.0573539
$305$	1.52786	0.0874852
$306$	0	0
$307$	4.58359	0.261599	0.130800	−	0.991409i	$-0.458246\pi$
$307$	4.58359	0.261599	0.130800	+	0.991409i	$0.458246\pi$
$308$	0.763932	0.0435291
$309$	0	0
$310$	6.47214	0.367593
$311$	−16.3607	−0.927729	−0.463865	−	0.885906i	$-0.653538\pi$
$311$	−16.3607	−0.927729	−0.463865	+	0.885906i	$0.653538\pi$
$312$	0	0
$313$	6.94427	0.392513	0.196257	−	0.980553i	$-0.437121\pi$
$313$	6.94427	0.392513	0.196257	+	0.980553i	$0.437121\pi$
$314$	13.4164	0.757132
$315$	0	0
$316$	0.763932	0.0429745
$317$	27.8885	1.56638	0.783188	−	0.621785i	$-0.213592\pi$
$317$	27.8885	1.56638	0.783188	+	0.621785i	$0.213592\pi$
$318$	0	0
$319$	6.47214	0.362370
$320$	−3.23607	−0.180902
$321$	0	0
$322$	5.23607	0.291795
$323$	7.70820	0.428896
$324$	0	0
$325$	6.76393	0.375195
$326$	4.00000	0.221540
$327$	0	0
$328$	−6.00000	−0.331295
$329$	8.94427	0.493114
$330$	0	0
$331$	29.5967	1.62678	0.813392	−	0.581716i	$-0.197618\pi$
$331$	29.5967	1.62678	0.813392	+	0.581716i	$0.197618\pi$
$332$	10.0000	0.548821
$333$	0	0
$334$	−16.9443	−0.927149
$335$	−0.944272	−0.0515911
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	−11.4721	−0.624002
$339$	0	0
$340$	−24.9443	−1.35279
$341$	1.52786	0.0827385
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.94427	0.482243
$345$	0	0
$346$	2.00000	0.107521
$347$	−6.29180	−0.337761	−0.168881	−	0.985637i	$-0.554015\pi$
$347$	−6.29180	−0.337761	−0.168881	+	0.985637i	$0.554015\pi$
$348$	0	0
$349$	−24.8328	−1.32927	−0.664635	−	0.747168i	$-0.731413\pi$
$349$	−24.8328	−1.32927	−0.664635	+	0.747168i	$0.731413\pi$
$350$	−5.47214	−0.292498
$351$	0	0
$352$	−0.763932	−0.0407177
$353$	11.7082	0.623165	0.311582	−	0.950219i	$-0.399141\pi$
$353$	11.7082	0.623165	0.311582	+	0.950219i	$0.399141\pi$
$354$	0	0
$355$	33.8885	1.79862
$356$	−6.00000	−0.317999
$357$	0	0
$358$	13.8885	0.734032
$359$	11.1246	0.587135	0.293567	−	0.955938i	$-0.405158\pi$
$359$	11.1246	0.587135	0.293567	+	0.955938i	$0.405158\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.23607	−0.275202
$363$	0	0
$364$	−1.23607	−0.0647876
$365$	40.3607	2.11257
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	−5.23607	−0.272949
$369$	0	0
$370$	33.8885	1.76178
$371$	12.4721	0.647521
$372$	0	0
$373$	−4.94427	−0.256005	−0.128002	−	0.991774i	$-0.540857\pi$
$373$	−4.94427	−0.256005	−0.128002	+	0.991774i	$0.540857\pi$
$374$	−5.88854	−0.304489
$375$	0	0
$376$	−8.94427	−0.461266
$377$	−10.4721	−0.539342
$378$	0	0
$379$	−27.1246	−1.39330	−0.696649	−	0.717412i	$-0.745326\pi$
$379$	−27.1246	−1.39330	−0.696649	+	0.717412i	$0.745326\pi$
$380$	−3.23607	−0.166007
$381$	0	0
$382$	−17.2361	−0.881874
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	−18.0000	−0.916176
$387$	0	0
$388$	4.76393	0.241852
$389$	2.58359	0.130993	0.0654967	−	0.997853i	$-0.479137\pi$
$389$	2.58359	0.130993	0.0654967	+	0.997853i	$0.479137\pi$
$390$	0	0
$391$	−40.3607	−2.04113
$392$	1.00000	0.0505076
$393$	0	0
$394$	10.9443	0.551364
$395$	−2.47214	−0.124387
$396$	0	0
$397$	−25.4164	−1.27561	−0.637806	−	0.770197i	$-0.720158\pi$
$397$	−25.4164	−1.27561	−0.637806	+	0.770197i	$0.720158\pi$
$398$	15.4164	0.772755
$399$	0	0
$400$	5.47214	0.273607
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−2.47214	−0.123146
$404$	13.7082	0.682009
$405$	0	0
$406$	8.47214	0.420465
$407$	8.00000	0.396545
$408$	0	0
$409$	−37.1246	−1.83569	−0.917847	−	0.396934i	$-0.870074\pi$
$409$	−37.1246	−1.83569	−0.917847	+	0.396934i	$0.870074\pi$
$410$	19.4164	0.958908
$411$	0	0
$412$	−6.94427	−0.342120
$413$	8.00000	0.393654
$414$	0	0
$415$	−32.3607	−1.58852
$416$	1.23607	0.0606032
$417$	0	0
$418$	−0.763932	−0.0373651
$419$	30.3607	1.48322	0.741608	−	0.670833i	$-0.234063\pi$
$419$	30.3607	1.48322	0.741608	+	0.670833i	$0.234063\pi$
$420$	0	0
$421$	18.8328	0.917855	0.458928	−	0.888474i	$-0.348234\pi$
$421$	18.8328	0.917855	0.458928	+	0.888474i	$0.348234\pi$
$422$	−21.5967	−1.05131
$423$	0	0
$424$	−12.4721	−0.605700
$425$	42.1803	2.04605
$426$	0	0
$427$	0.472136	0.0228483
$428$	−5.52786	−0.267199
$429$	0	0
$430$	−28.9443	−1.39582
$431$	30.8328	1.48516	0.742582	−	0.669755i	$-0.233601\pi$
$431$	30.8328	1.48516	0.742582	+	0.669755i	$0.233601\pi$
$432$	0	0
$433$	−23.2361	−1.11665	−0.558327	−	0.829621i	$-0.688557\pi$
$433$	−23.2361	−1.11665	−0.558327	+	0.829621i	$0.688557\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	18.4721	0.884655
$437$	−5.23607	−0.250475
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	2.47214	0.117854
$441$	0	0
$442$	9.52786	0.453194
$443$	−5.70820	−0.271205	−0.135602	−	0.990763i	$-0.543297\pi$
$443$	−5.70820	−0.271205	−0.135602	+	0.990763i	$0.543297\pi$
$444$	0	0
$445$	19.4164	0.920426
$446$	28.4721	1.34819
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−20.8328	−0.983161	−0.491581	−	0.870832i	$-0.663581\pi$
$449$	−20.8328	−0.983161	−0.491581	+	0.870832i	$0.663581\pi$
$450$	0	0
$451$	4.58359	0.215833
$452$	−10.0000	−0.470360
$453$	0	0
$454$	−11.4164	−0.535799
$455$	4.00000	0.187523
$456$	0	0
$457$	−27.3050	−1.27727	−0.638636	−	0.769509i	$-0.720501\pi$
$457$	−27.3050	−1.27727	−0.638636	+	0.769509i	$0.720501\pi$
$458$	−1.05573	−0.0493309
$459$	0	0
$460$	16.9443	0.790031
$461$	−24.1803	−1.12619	−0.563095	−	0.826392i	$-0.690390\pi$
$461$	−24.1803	−1.12619	−0.563095	+	0.826392i	$0.690390\pi$
$462$	0	0
$463$	21.3050	0.990125	0.495063	−	0.868857i	$-0.335145\pi$
$463$	21.3050	0.990125	0.495063	+	0.868857i	$0.335145\pi$
$464$	−8.47214	−0.393309
$465$	0	0
$466$	−5.52786	−0.256073
$467$	−15.8885	−0.735234	−0.367617	−	0.929977i	$-0.619826\pi$
$467$	−15.8885	−0.735234	−0.367617	+	0.929977i	$0.619826\pi$
$468$	0	0
$469$	−0.291796	−0.0134739
$470$	28.9443	1.33510
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−6.83282	−0.314173
$474$	0	0
$475$	5.47214	0.251079
$476$	−7.70820	−0.353305
$477$	0	0
$478$	−26.1803	−1.19746
$479$	25.5279	1.16640	0.583199	−	0.812329i	$-0.301801\pi$
$479$	25.5279	1.16640	0.583199	+	0.812329i	$0.301801\pi$
$480$	0	0
$481$	−12.9443	−0.590208
$482$	−0.763932	−0.0347962
$483$	0	0
$484$	−10.4164	−0.473473
$485$	−15.4164	−0.700023
$486$	0	0
$487$	30.0689	1.36255	0.681276	−	0.732027i	$-0.261426\pi$
$487$	30.0689	1.36255	0.681276	+	0.732027i	$0.261426\pi$
$488$	−0.472136	−0.0213726
$489$	0	0
$490$	−3.23607	−0.146191
$491$	13.7082	0.618643	0.309321	−	0.950958i	$-0.399898\pi$
$491$	13.7082	0.618643	0.309321	+	0.950958i	$0.399898\pi$
$492$	0	0
$493$	−65.3050	−2.94119
$494$	1.23607	0.0556133
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	10.4721	0.469739
$498$	0	0
$499$	−37.3050	−1.67000	−0.834999	−	0.550251i	$-0.814532\pi$
$499$	−37.3050	−1.67000	−0.834999	+	0.550251i	$0.814532\pi$
$500$	−1.52786	−0.0683282
$501$	0	0
$502$	1.41641	0.0632174
$503$	26.4721	1.18033	0.590167	−	0.807281i	$-0.299062\pi$
$503$	26.4721	1.18033	0.590167	+	0.807281i	$0.299062\pi$
$504$	0	0
$505$	−44.3607	−1.97402
$506$	4.00000	0.177822
$507$	0	0
$508$	−9.70820	−0.430732
$509$	13.4164	0.594672	0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164	0.594672	0.297336	+	0.954773i	$0.403902\pi$
$510$	0	0
$511$	12.4721	0.551735
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.52786	0.155607
$515$	22.4721	0.990241
$516$	0	0
$517$	6.83282	0.300507
$518$	10.4721	0.460119
$519$	0	0
$520$	−4.00000	−0.175412
$521$	−26.3607	−1.15488	−0.577441	−	0.816432i	$-0.695949\pi$
$521$	−26.3607	−1.15488	−0.577441	+	0.816432i	$0.695949\pi$
$522$	0	0
$523$	5.52786	0.241717	0.120858	−	0.992670i	$-0.461435\pi$
$523$	5.52786	0.241717	0.120858	+	0.992670i	$0.461435\pi$
$524$	16.4721	0.719589
$525$	0	0
$526$	−7.70820	−0.336094
$527$	−15.4164	−0.671549
$528$	0	0
$529$	4.41641	0.192018
$530$	40.3607	1.75316
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	−7.41641	−0.321240
$534$	0	0
$535$	17.8885	0.773389
$536$	0.291796	0.0126037
$537$	0	0
$538$	6.94427	0.299389
$539$	−0.763932	−0.0329049
$540$	0	0
$541$	25.7771	1.10824	0.554122	−	0.832436i	$-0.313054\pi$
$541$	25.7771	1.10824	0.554122	+	0.832436i	$0.313054\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	7.70820	0.330487
$545$	−59.7771	−2.56057
$546$	0	0
$547$	11.1246	0.475654	0.237827	−	0.971308i	$-0.423565\pi$
$547$	11.1246	0.475654	0.237827	+	0.971308i	$0.423565\pi$
$548$	15.4164	0.658556
$549$	0	0
$550$	−4.18034	−0.178250
$551$	−8.47214	−0.360925
$552$	0	0
$553$	−0.763932	−0.0324857
$554$	−14.9443	−0.634921
$555$	0	0
$556$	21.8885	0.928281
$557$	0.111456	0.00472255	0.00236127	−	0.999997i	$-0.499248\pi$
$557$	0.111456	0.00472255	0.00236127	+	0.999997i	$0.499248\pi$
$558$	0	0
$559$	11.0557	0.467607
$560$	3.23607	0.136749
$561$	0	0
$562$	−6.00000	−0.253095
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	32.3607	1.36142
$566$	−21.8885	−0.920044
$567$	0	0
$568$	−10.4721	−0.439401
$569$	2.94427	0.123430	0.0617151	−	0.998094i	$-0.480343\pi$
$569$	2.94427	0.123430	0.0617151	+	0.998094i	$0.480343\pi$
$570$	0	0
$571$	6.47214	0.270850	0.135425	−	0.990788i	$-0.456760\pi$
$571$	6.47214	0.270850	0.135425	+	0.990788i	$0.456760\pi$
$572$	−0.944272	−0.0394820
$573$	0	0
$574$	6.00000	0.250435
$575$	−28.6525	−1.19489
$576$	0	0
$577$	21.4164	0.891577	0.445788	−	0.895138i	$-0.352923\pi$
$577$	21.4164	0.891577	0.445788	+	0.895138i	$0.352923\pi$
$578$	42.4164	1.76429
$579$	0	0
$580$	27.4164	1.13840
$581$	−10.0000	−0.414870
$582$	0	0
$583$	9.52786	0.394604
$584$	−12.4721	−0.516101
$585$	0	0
$586$	−15.8885	−0.656350
$587$	44.8328	1.85045	0.925224	−	0.379421i	$-0.123877\pi$
$587$	44.8328	1.85045	0.925224	+	0.379421i	$0.123877\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	25.8885	1.06581
$591$	0	0
$592$	−10.4721	−0.430402
$593$	3.34752	0.137466	0.0687332	−	0.997635i	$-0.478104\pi$
$593$	3.34752	0.137466	0.0687332	+	0.997635i	$0.478104\pi$
$594$	0	0
$595$	24.9443	1.02262
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−6.47214	−0.264665
$599$	39.4164	1.61051	0.805255	−	0.592928i	$-0.202028\pi$
$599$	39.4164	1.61051	0.805255	+	0.592928i	$0.202028\pi$
$600$	0	0
$601$	1.70820	0.0696791	0.0348395	−	0.999393i	$-0.488908\pi$
$601$	1.70820	0.0696791	0.0348395	+	0.999393i	$0.488908\pi$
$602$	−8.94427	−0.364541
$603$	0	0
$604$	0.763932	0.0310840
$605$	33.7082	1.37043
$606$	0	0
$607$	25.4164	1.03162	0.515810	−	0.856703i	$-0.327491\pi$
$607$	25.4164	1.03162	0.515810	+	0.856703i	$0.327491\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	1.52786	0.0618614
$611$	−11.0557	−0.447267
$612$	0	0
$613$	8.11146	0.327619	0.163809	−	0.986492i	$-0.447622\pi$
$613$	8.11146	0.327619	0.163809	+	0.986492i	$0.447622\pi$
$614$	4.58359	0.184979
$615$	0	0
$616$	0.763932	0.0307797
$617$	3.41641	0.137539	0.0687697	−	0.997633i	$-0.478093\pi$
$617$	3.41641	0.137539	0.0687697	+	0.997633i	$0.478093\pi$
$618$	0	0
$619$	−41.8885	−1.68364	−0.841821	−	0.539756i	$-0.818516\pi$
$619$	−41.8885	−1.68364	−0.841821	+	0.539756i	$0.818516\pi$
$620$	6.47214	0.259927
$621$	0	0
$622$	−16.3607	−0.656003
$623$	6.00000	0.240385
$624$	0	0
$625$	−22.4164	−0.896656
$626$	6.94427	0.277549
$627$	0	0
$628$	13.4164	0.535373
$629$	−80.7214	−3.21857
$630$	0	0
$631$	−6.47214	−0.257652	−0.128826	−	0.991667i	$-0.541121\pi$
$631$	−6.47214	−0.257652	−0.128826	+	0.991667i	$0.541121\pi$
$632$	0.763932	0.0303876
$633$	0	0
$634$	27.8885	1.10760
$635$	31.4164	1.24672
$636$	0	0
$637$	1.23607	0.0489748
$638$	6.47214	0.256234
$639$	0	0
$640$	−3.23607	−0.127917
$641$	47.8885	1.89148	0.945742	−	0.324919i	$-0.105337\pi$
$641$	47.8885	1.89148	0.945742	+	0.324919i	$0.105337\pi$
$642$	0	0
$643$	−27.0557	−1.06697	−0.533487	−	0.845808i	$-0.679119\pi$
$643$	−27.0557	−1.06697	−0.533487	+	0.845808i	$0.679119\pi$
$644$	5.23607	0.206330
$645$	0	0
$646$	7.70820	0.303275
$647$	46.2492	1.81824	0.909122	−	0.416529i	$-0.136754\pi$
$647$	46.2492	1.81824	0.909122	+	0.416529i	$0.136754\pi$
$648$	0	0
$649$	6.11146	0.239896
$650$	6.76393	0.265303
$651$	0	0
$652$	4.00000	0.156652
$653$	9.41641	0.368493	0.184246	−	0.982880i	$-0.441016\pi$
$653$	9.41641	0.368493	0.184246	+	0.982880i	$0.441016\pi$
$654$	0	0
$655$	−53.3050	−2.08280
$656$	−6.00000	−0.234261
$657$	0	0
$658$	8.94427	0.348684
$659$	−5.52786	−0.215335	−0.107668	−	0.994187i	$-0.534338\pi$
$659$	−5.52786	−0.215335	−0.107668	+	0.994187i	$0.534338\pi$
$660$	0	0
$661$	−45.2361	−1.75948	−0.879740	−	0.475456i	$-0.842283\pi$
$661$	−45.2361	−1.75948	−0.879740	+	0.475456i	$0.842283\pi$
$662$	29.5967	1.15031
$663$	0	0
$664$	10.0000	0.388075
$665$	3.23607	0.125489
$666$	0	0
$667$	44.3607	1.71765
$668$	−16.9443	−0.655594
$669$	0	0
$670$	−0.944272	−0.0364804
$671$	0.360680	0.0139239
$672$	0	0
$673$	36.2492	1.39730	0.698652	−	0.715461i	$-0.253783\pi$
$673$	36.2492	1.39730	0.698652	+	0.715461i	$0.253783\pi$
$674$	−20.4721	−0.788557
$675$	0	0
$676$	−11.4721	−0.441236
$677$	28.4721	1.09427	0.547137	−	0.837043i	$-0.315718\pi$
$677$	28.4721	1.09427	0.547137	+	0.837043i	$0.315718\pi$
$678$	0	0
$679$	−4.76393	−0.182823
$680$	−24.9443	−0.956569
$681$	0	0
$682$	1.52786	0.0585049
$683$	−10.1115	−0.386904	−0.193452	−	0.981110i	$-0.561968\pi$
$683$	−10.1115	−0.386904	−0.193452	+	0.981110i	$0.561968\pi$
$684$	0	0
$685$	−49.8885	−1.90614
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	8.94427	0.340997
$689$	−15.4164	−0.587318
$690$	0	0
$691$	−26.8328	−1.02077	−0.510384	−	0.859946i	$-0.670497\pi$
$691$	−26.8328	−1.02077	−0.510384	+	0.859946i	$0.670497\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−6.29180	−0.238833
$695$	−70.8328	−2.68684
$696$	0	0
$697$	−46.2492	−1.75181
$698$	−24.8328	−0.939936
$699$	0	0
$700$	−5.47214	−0.206827
$701$	27.5279	1.03971	0.519857	−	0.854254i	$-0.325985\pi$
$701$	27.5279	1.03971	0.519857	+	0.854254i	$0.325985\pi$
$702$	0	0
$703$	−10.4721	−0.394964
$704$	−0.763932	−0.0287918
$705$	0	0
$706$	11.7082	0.440644
$707$	−13.7082	−0.515550
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	33.8885	1.27181
$711$	0	0
$712$	−6.00000	−0.224860
$713$	10.4721	0.392185
$714$	0	0
$715$	3.05573	0.114278
$716$	13.8885	0.519039
$717$	0	0
$718$	11.1246	0.415167
$719$	6.47214	0.241370	0.120685	−	0.992691i	$-0.461491\pi$
$719$	6.47214	0.241370	0.120685	+	0.992691i	$0.461491\pi$
$720$	0	0
$721$	6.94427	0.258618
$722$	1.00000	0.0372161
$723$	0	0
$724$	−5.23607	−0.194597
$725$	−46.3607	−1.72179
$726$	0	0
$727$	−27.7771	−1.03020	−0.515098	−	0.857132i	$-0.672244\pi$
$727$	−27.7771	−1.03020	−0.515098	+	0.857132i	$0.672244\pi$
$728$	−1.23607	−0.0458117
$729$	0	0
$730$	40.3607	1.49382
$731$	68.9443	2.55000
$732$	0	0
$733$	12.8328	0.473991	0.236995	−	0.971511i	$-0.423837\pi$
$733$	12.8328	0.473991	0.236995	+	0.971511i	$0.423837\pi$
$734$	−4.00000	−0.147643
$735$	0	0
$736$	−5.23607	−0.193004
$737$	−0.222912	−0.00821108
$738$	0	0
$739$	−3.41641	−0.125675	−0.0628373	−	0.998024i	$-0.520015\pi$
$739$	−3.41641	−0.125675	−0.0628373	+	0.998024i	$0.520015\pi$
$740$	33.8885	1.24577
$741$	0	0
$742$	12.4721	0.457867
$743$	23.4164	0.859065	0.429532	−	0.903051i	$-0.358678\pi$
$743$	23.4164	0.859065	0.429532	+	0.903051i	$0.358678\pi$
$744$	0	0
$745$	19.4164	0.711362
$746$	−4.94427	−0.181023
$747$	0	0
$748$	−5.88854	−0.215306
$749$	5.52786	0.201984
$750$	0	0
$751$	23.0132	0.839762	0.419881	−	0.907579i	$-0.362072\pi$
$751$	23.0132	0.839762	0.419881	+	0.907579i	$0.362072\pi$
$752$	−8.94427	−0.326164
$753$	0	0
$754$	−10.4721	−0.381373
$755$	−2.47214	−0.0899702
$756$	0	0
$757$	35.8885	1.30439	0.652196	−	0.758051i	$-0.273848\pi$
$757$	35.8885	1.30439	0.652196	+	0.758051i	$0.273848\pi$
$758$	−27.1246	−0.985210
$759$	0	0
$760$	−3.23607	−0.117385
$761$	−28.6525	−1.03865	−0.519326	−	0.854576i	$-0.673817\pi$
$761$	−28.6525	−1.03865	−0.519326	+	0.854576i	$0.673817\pi$
$762$	0	0
$763$	−18.4721	−0.668736
$764$	−17.2361	−0.623579
$765$	0	0
$766$	8.00000	0.289052
$767$	−9.88854	−0.357055
$768$	0	0
$769$	−10.9443	−0.394661	−0.197330	−	0.980337i	$-0.563227\pi$
$769$	−10.9443	−0.394661	−0.197330	+	0.980337i	$0.563227\pi$
$770$	−2.47214	−0.0890896
$771$	0	0
$772$	−18.0000	−0.647834
$773$	45.4164	1.63351	0.816757	−	0.576981i	$-0.195769\pi$
$773$	45.4164	1.63351	0.816757	+	0.576981i	$0.195769\pi$
$774$	0	0
$775$	−10.9443	−0.393130
$776$	4.76393	0.171015
$777$	0	0
$778$	2.58359	0.0926263
$779$	−6.00000	−0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	−40.3607	−1.44329
$783$	0	0
$784$	1.00000	0.0357143
$785$	−43.4164	−1.54960
$786$	0	0
$787$	19.4164	0.692120	0.346060	−	0.938212i	$-0.387519\pi$
$787$	19.4164	0.692120	0.346060	+	0.938212i	$0.387519\pi$
$788$	10.9443	0.389874
$789$	0	0
$790$	−2.47214	−0.0879547
$791$	10.0000	0.355559
$792$	0	0
$793$	−0.583592	−0.0207240
$794$	−25.4164	−0.901995
$795$	0	0
$796$	15.4164	0.546420
$797$	−18.5836	−0.658265	−0.329132	−	0.944284i	$-0.606756\pi$
$797$	−18.5836	−0.658265	−0.329132	+	0.944284i	$0.606756\pi$
$798$	0	0
$799$	−68.9443	−2.43907
$800$	5.47214	0.193469
$801$	0	0
$802$	−30.0000	−1.05934
$803$	9.52786	0.336231
$804$	0	0
$805$	−16.9443	−0.597207
$806$	−2.47214	−0.0870773
$807$	0	0
$808$	13.7082	0.482253
$809$	8.94427	0.314464	0.157232	−	0.987562i	$-0.449743\pi$
$809$	8.94427	0.314464	0.157232	+	0.987562i	$0.449743\pi$
$810$	0	0
$811$	3.41641	0.119966	0.0599832	−	0.998199i	$-0.480895\pi$
$811$	3.41641	0.119966	0.0599832	+	0.998199i	$0.480895\pi$
$812$	8.47214	0.297314
$813$	0	0
$814$	8.00000	0.280400
$815$	−12.9443	−0.453418
$816$	0	0
$817$	8.94427	0.312920
$818$	−37.1246	−1.29803
$819$	0	0
$820$	19.4164	0.678050
$821$	10.9443	0.381958	0.190979	−	0.981594i	$-0.438834\pi$
$821$	10.9443	0.381958	0.190979	+	0.981594i	$0.438834\pi$
$822$	0	0
$823$	38.4721	1.34105	0.670527	−	0.741885i	$-0.266068\pi$
$823$	38.4721	1.34105	0.670527	+	0.741885i	$0.266068\pi$
$824$	−6.94427	−0.241915
$825$	0	0
$826$	8.00000	0.278356
$827$	−10.8328	−0.376694	−0.188347	−	0.982103i	$-0.560313\pi$
$827$	−10.8328	−0.376694	−0.188347	+	0.982103i	$0.560313\pi$
$828$	0	0
$829$	−18.7639	−0.651698	−0.325849	−	0.945422i	$-0.605650\pi$
$829$	−18.7639	−0.651698	−0.325849	+	0.945422i	$0.605650\pi$
$830$	−32.3607	−1.12326
$831$	0	0
$832$	1.23607	0.0428529
$833$	7.70820	0.267073
$834$	0	0
$835$	54.8328	1.89757
$836$	−0.763932	−0.0264211
$837$	0	0
$838$	30.3607	1.04879
$839$	−18.1115	−0.625277	−0.312638	−	0.949872i	$-0.601213\pi$
$839$	−18.1115	−0.625277	−0.312638	+	0.949872i	$0.601213\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	18.8328	0.649022
$843$	0	0
$844$	−21.5967	−0.743391
$845$	37.1246	1.27713
$846$	0	0
$847$	10.4164	0.357912
$848$	−12.4721	−0.428295
$849$	0	0
$850$	42.1803	1.44677
$851$	54.8328	1.87964
$852$	0	0
$853$	−10.9443	−0.374725	−0.187362	−	0.982291i	$-0.559994\pi$
$853$	−10.9443	−0.374725	−0.187362	+	0.982291i	$0.559994\pi$
$854$	0.472136	0.0161562
$855$	0	0
$856$	−5.52786	−0.188939
$857$	3.88854	0.132830	0.0664151	−	0.997792i	$-0.478844\pi$
$857$	3.88854	0.132830	0.0664151	+	0.997792i	$0.478844\pi$
$858$	0	0
$859$	57.8885	1.97513	0.987566	−	0.157206i	$-0.0502487\pi$
$859$	57.8885	1.97513	0.987566	+	0.157206i	$0.0502487\pi$
$860$	−28.9443	−0.986991
$861$	0	0
$862$	30.8328	1.05017
$863$	−3.63932	−0.123884	−0.0619420	−	0.998080i	$-0.519729\pi$
$863$	−3.63932	−0.123884	−0.0619420	+	0.998080i	$0.519729\pi$
$864$	0	0
$865$	−6.47214	−0.220059
$866$	−23.2361	−0.789594
$867$	0	0
$868$	2.00000	0.0678844
$869$	−0.583592	−0.0197970
$870$	0	0
$871$	0.360680	0.0122212
$872$	18.4721	0.625545
$873$	0	0
$874$	−5.23607	−0.177113
$875$	1.52786	0.0516512
$876$	0	0
$877$	7.41641	0.250434	0.125217	−	0.992129i	$-0.460037\pi$
$877$	7.41641	0.250434	0.125217	+	0.992129i	$0.460037\pi$
$878$	10.0000	0.337484
$879$	0	0
$880$	2.47214	0.0833357
$881$	5.59675	0.188559	0.0942796	−	0.995546i	$-0.469945\pi$
$881$	5.59675	0.188559	0.0942796	+	0.995546i	$0.469945\pi$
$882$	0	0
$883$	−14.4721	−0.487026	−0.243513	−	0.969898i	$-0.578300\pi$
$883$	−14.4721	−0.487026	−0.243513	+	0.969898i	$0.578300\pi$
$884$	9.52786	0.320457
$885$	0	0
$886$	−5.70820	−0.191771
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	9.70820	0.325603
$890$	19.4164	0.650839
$891$	0	0
$892$	28.4721	0.953318
$893$	−8.94427	−0.299309
$894$	0	0
$895$	−44.9443	−1.50232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−20.8328	−0.695200
$899$	16.9443	0.565123
$900$	0	0
$901$	−96.1378	−3.20281
$902$	4.58359	0.152617
$903$	0	0
$904$	−10.0000	−0.332595
$905$	16.9443	0.563247
$906$	0	0
$907$	18.7639	0.623046	0.311523	−	0.950239i	$-0.399161\pi$
$907$	18.7639	0.623046	0.311523	+	0.950239i	$0.399161\pi$
$908$	−11.4164	−0.378867
$909$	0	0
$910$	4.00000	0.132599
$911$	−38.2492	−1.26725	−0.633627	−	0.773639i	$-0.718434\pi$
$911$	−38.2492	−1.26725	−0.633627	+	0.773639i	$0.718434\pi$
$912$	0	0
$913$	−7.63932	−0.252825
$914$	−27.3050	−0.903168
$915$	0	0
$916$	−1.05573	−0.0348822
$917$	−16.4721	−0.543958
$918$	0	0
$919$	32.3607	1.06748	0.533740	−	0.845649i	$-0.320786\pi$
$919$	32.3607	1.06748	0.533740	+	0.845649i	$0.320786\pi$
$920$	16.9443	0.558636
$921$	0	0
$922$	−24.1803	−0.796337
$923$	−12.9443	−0.426066
$924$	0	0
$925$	−57.3050	−1.88418
$926$	21.3050	0.700124
$927$	0	0
$928$	−8.47214	−0.278111
$929$	−16.0689	−0.527203	−0.263601	−	0.964632i	$-0.584910\pi$
$929$	−16.0689	−0.527203	−0.263601	+	0.964632i	$0.584910\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	−5.52786	−0.181071
$933$	0	0
$934$	−15.8885	−0.519889
$935$	19.0557	0.623189
$936$	0	0
$937$	−29.0557	−0.949209	−0.474605	−	0.880199i	$-0.657409\pi$
$937$	−29.0557	−0.949209	−0.474605	+	0.880199i	$0.657409\pi$
$938$	−0.291796	−0.00952748
$939$	0	0
$940$	28.9443	0.944058
$941$	−44.4721	−1.44975	−0.724875	−	0.688880i	$-0.758103\pi$
$941$	−44.4721	−1.44975	−0.724875	+	0.688880i	$0.758103\pi$
$942$	0	0
$943$	31.4164	1.02306
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−6.83282	−0.222154
$947$	−59.0132	−1.91767	−0.958835	−	0.283964i	$-0.908350\pi$
$947$	−59.0132	−1.91767	−0.958835	+	0.283964i	$0.908350\pi$
$948$	0	0
$949$	−15.4164	−0.500438
$950$	5.47214	0.177540
$951$	0	0
$952$	−7.70820	−0.249824
$953$	−25.0557	−0.811635	−0.405817	−	0.913954i	$-0.633013\pi$
$953$	−25.0557	−0.811635	−0.405817	+	0.913954i	$0.633013\pi$
$954$	0	0
$955$	55.7771	1.80490
$956$	−26.1803	−0.846733
$957$	0	0
$958$	25.5279	0.824768
$959$	−15.4164	−0.497822
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−12.9443	−0.417340
$963$	0	0
$964$	−0.763932	−0.0246046
$965$	58.2492	1.87511
$966$	0	0
$967$	−22.8328	−0.734254	−0.367127	−	0.930171i	$-0.619659\pi$
$967$	−22.8328	−0.734254	−0.367127	+	0.930171i	$0.619659\pi$
$968$	−10.4164	−0.334796
$969$	0	0
$970$	−15.4164	−0.494991
$971$	−25.5279	−0.819228	−0.409614	−	0.912259i	$-0.634337\pi$
$971$	−25.5279	−0.819228	−0.409614	+	0.912259i	$0.634337\pi$
$972$	0	0
$973$	−21.8885	−0.701714
$974$	30.0689	0.963469
$975$	0	0
$976$	−0.472136	−0.0151127
$977$	−7.88854	−0.252377	−0.126188	−	0.992006i	$-0.540274\pi$
$977$	−7.88854	−0.252377	−0.126188	+	0.992006i	$0.540274\pi$
$978$	0	0
$979$	4.58359	0.146492
$980$	−3.23607	−0.103372
$981$	0	0
$982$	13.7082	0.437446
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	−35.4164	−1.12846
$986$	−65.3050	−2.07973
$987$	0	0
$988$	1.23607	0.0393246
$989$	−46.8328	−1.48920
$990$	0	0
$991$	24.7639	0.786652	0.393326	−	0.919399i	$-0.371324\pi$
$991$	24.7639	0.786652	0.393326	+	0.919399i	$0.371324\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	10.4721	0.332156
$995$	−49.8885	−1.58157
$996$	0	0
$997$	−61.1935	−1.93802	−0.969009	−	0.247027i	$-0.920547\pi$
$997$	−61.1935	−1.93802	−0.969009	+	0.247027i	$0.920547\pi$
$998$	−37.3050	−1.18087
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.v.1.1	yes	2
3.2	odd	2		2394.2.a.s.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.s.1.2	✓	2	3.2	odd	2
2394.2.a.v.1.1	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.23607 q^{10} -0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.70820 q^{17} +1.00000 q^{19} -3.23607 q^{20} -0.763932 q^{22} -5.23607 q^{23} +5.47214 q^{25} +1.23607 q^{26} -1.00000 q^{28} -8.47214 q^{29} -2.00000 q^{31} +1.00000 q^{32} +7.70820 q^{34} +3.23607 q^{35} -10.4721 q^{37} +1.00000 q^{38} -3.23607 q^{40} -6.00000 q^{41} +8.94427 q^{43} -0.763932 q^{44} -5.23607 q^{46} -8.94427 q^{47} +1.00000 q^{49} +5.47214 q^{50} +1.23607 q^{52} -12.4721 q^{53} +2.47214 q^{55} -1.00000 q^{56} -8.47214 q^{58} -8.00000 q^{59} -0.472136 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +0.291796 q^{67} +7.70820 q^{68} +3.23607 q^{70} -10.4721 q^{71} -12.4721 q^{73} -10.4721 q^{74} +1.00000 q^{76} +0.763932 q^{77} +0.763932 q^{79} -3.23607 q^{80} -6.00000 q^{82} +10.0000 q^{83} -24.9443 q^{85} +8.94427 q^{86} -0.763932 q^{88} -6.00000 q^{89} -1.23607 q^{91} -5.23607 q^{92} -8.94427 q^{94} -3.23607 q^{95} +4.76393 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 6 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{19} - 2 q^{20} - 6 q^{22} - 6 q^{23} + 2 q^{25} - 2 q^{26} - 2 q^{28} - 8 q^{29} - 4 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{35} - 12 q^{37} + 2 q^{38} - 2 q^{40} - 12 q^{41} - 6 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} - 2 q^{52} - 16 q^{53} - 4 q^{55} - 2 q^{56} - 8 q^{58} - 16 q^{59} + 8 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} + 14 q^{67} + 2 q^{68} + 2 q^{70} - 12 q^{71} - 16 q^{73} - 12 q^{74} + 2 q^{76} + 6 q^{77} + 6 q^{79} - 2 q^{80} - 12 q^{82} + 20 q^{83} - 32 q^{85} - 6 q^{88} - 12 q^{89} + 2 q^{91} - 6 q^{92} - 2 q^{95} + 14 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 6 * q^11 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^19 - 2 * q^20 - 6 * q^22 - 6 * q^23 + 2 * q^25 - 2 * q^26 - 2 * q^28 - 8 * q^29 - 4 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^35 - 12 * q^37 + 2 * q^38 - 2 * q^40 - 12 * q^41 - 6 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 - 2 * q^52 - 16 * q^53 - 4 * q^55 - 2 * q^56 - 8 * q^58 - 16 * q^59 + 8 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^65 + 14 * q^67 + 2 * q^68 + 2 * q^70 - 12 * q^71 - 16 * q^73 - 12 * q^74 + 2 * q^76 + 6 * q^77 + 6 * q^79 - 2 * q^80 - 12 * q^82 + 20 * q^83 - 32 * q^85 - 6 * q^88 - 12 * q^89 + 2 * q^91 - 6 * q^92 - 2 * q^95 + 14 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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