LMFDB - Embedded newform 2842.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -2.82843 q^{13} +1.00000 q^{16} -2.82843 q^{17} +3.00000 q^{18} -5.65685 q^{19} -2.82843 q^{20} -4.00000 q^{22} +3.00000 q^{25} +2.82843 q^{26} -1.00000 q^{29} +2.82843 q^{31} -1.00000 q^{32} +2.82843 q^{34} -3.00000 q^{36} +2.00000 q^{37} +5.65685 q^{38} +2.82843 q^{40} +2.82843 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.48528 q^{45} +2.82843 q^{47} -3.00000 q^{50} -2.82843 q^{52} +6.00000 q^{53} -11.3137 q^{55} +1.00000 q^{58} -14.1421 q^{59} -2.82843 q^{62} +1.00000 q^{64} +8.00000 q^{65} +4.00000 q^{67} -2.82843 q^{68} -8.00000 q^{71} +3.00000 q^{72} -8.48528 q^{73} -2.00000 q^{74} -5.65685 q^{76} -2.82843 q^{80} +9.00000 q^{81} -2.82843 q^{82} +14.1421 q^{83} +8.00000 q^{85} +4.00000 q^{86} -4.00000 q^{88} +2.82843 q^{89} -8.48528 q^{90} -2.82843 q^{94} +16.0000 q^{95} +8.48528 q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} + 8 q^{11} + 2 q^{16} + 6 q^{18} - 8 q^{22} + 6 q^{25} - 2 q^{29} - 2 q^{32} - 6 q^{36} + 4 q^{37} - 8 q^{43} + 8 q^{44} - 6 q^{50} + 12 q^{53} + 2 q^{58} + 2 q^{64} + 16 q^{65} + 8 q^{67} - 16 q^{71} + 6 q^{72} - 4 q^{74} + 18 q^{81} + 16 q^{85} + 8 q^{86} - 8 q^{88} + 32 q^{95} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 + 8 * q^11 + 2 * q^16 + 6 * q^18 - 8 * q^22 + 6 * q^25 - 2 * q^29 - 2 * q^32 - 6 * q^36 + 4 * q^37 - 8 * q^43 + 8 * q^44 - 6 * q^50 + 12 * q^53 + 2 * q^58 + 2 * q^64 + 16 * q^65 + 8 * q^67 - 16 * q^71 + 6 * q^72 - 4 * q^74 + 18 * q^81 + 16 * q^85 + 8 * q^86 - 8 * q^88 + 32 * q^95 - 24 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	2.82843	0.894427
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	3.00000	0.707107
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	−2.82843	−0.632456
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	2.82843	0.554700
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.82843	0.485071
$35$	0	0
$36$	−3.00000	−0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	5.65685	0.917663
$39$	0	0
$40$	2.82843	0.447214
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	8.48528	1.26491
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	−2.82843	−0.392232
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−11.3137	−1.52554
$56$	0	0
$57$	0	0
$58$	1.00000	0.131306
$59$	−14.1421	−1.84115	−0.920575	−	0.390567i	$-0.872279\pi$
$59$	−14.1421	−1.84115	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−2.82843	−0.359211
$63$	0	0
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.82843	−0.342997
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	3.00000	0.353553
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−5.65685	−0.648886
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.82843	−0.316228
$81$	9.00000	1.00000
$82$	−2.82843	−0.312348
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	4.00000	0.431331
$87$	0	0
$88$	−4.00000	−0.426401
$89$	2.82843	0.299813	0.149906	−	0.988700i	$-0.452103\pi$
$89$	2.82843	0.299813	0.149906	+	0.988700i	$0.452103\pi$
$90$	−8.48528	−0.894427
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−2.82843	−0.291730
$95$	16.0000	1.64157
$96$	0	0
$97$	8.48528	0.861550	0.430775	−	0.902459i	$-0.358240\pi$
$97$	8.48528	0.861550	0.430775	+	0.902459i	$0.358240\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	3.00000	0.300000
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	0	0
$103$	−5.65685	−0.557386	−0.278693	−	0.960380i	$-0.589901\pi$
$103$	−5.65685	−0.557386	−0.278693	+	0.960380i	$0.589901\pi$
$104$	2.82843	0.277350
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	11.3137	1.07872
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	8.48528	0.784465
$118$	14.1421	1.30189
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	2.82843	0.254000
$125$	5.65685	0.505964
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.00000	−0.701646
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.82843	0.242536
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−11.3137	−0.946100
$144$	−3.00000	−0.250000
$145$	2.82843	0.234888
$146$	8.48528	0.702247
$147$	0	0
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	5.65685	0.458831
$153$	8.48528	0.685994
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−5.65685	−0.451466	−0.225733	−	0.974189i	$-0.572478\pi$
$157$	−5.65685	−0.451466	−0.225733	+	0.974189i	$0.572478\pi$
$158$	0	0
$159$	0	0
$160$	2.82843	0.223607
$161$	0	0
$162$	−9.00000	−0.707107
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	2.82843	0.220863
$165$	0	0
$166$	−14.1421	−1.09764
$167$	−11.3137	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	−8.00000	−0.613572
$171$	16.9706	1.29777
$172$	−4.00000	−0.304997
$173$	19.7990	1.50529	0.752645	−	0.658427i	$-0.228778\pi$
$173$	19.7990	1.50529	0.752645	+	0.658427i	$0.228778\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	−2.82843	−0.212000
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	8.48528	0.632456
$181$	19.7990	1.47165	0.735824	−	0.677173i	$-0.236795\pi$
$181$	19.7990	1.47165	0.735824	+	0.677173i	$0.236795\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−11.3137	−0.827340
$188$	2.82843	0.206284
$189$	0	0
$190$	−16.0000	−1.16076
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−8.48528	−0.609208
$195$	0	0
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	12.0000	0.852803
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−11.3137	−0.796030
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	5.65685	0.394132
$207$	0	0
$208$	−2.82843	−0.196116
$209$	−22.6274	−1.56517
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−11.3137	−0.762770
$221$	8.00000	0.538138
$222$	0	0
$223$	−16.9706	−1.13643	−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706	−1.13643	−0.568216	+	0.822879i	$0.692366\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	−2.00000	−0.133038
$227$	19.7990	1.31411	0.657053	−	0.753845i	$-0.271803\pi$
$227$	19.7990	1.31411	0.657053	+	0.753845i	$0.271803\pi$
$228$	0	0
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	1.00000	0.0656532
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	−8.48528	−0.554700
$235$	−8.00000	−0.521862
$236$	−14.1421	−0.920575
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−22.6274	−1.45756	−0.728780	−	0.684748i	$-0.759912\pi$
$241$	−22.6274	−1.45756	−0.728780	+	0.684748i	$0.759912\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	−2.82843	−0.179605
$249$	0	0
$250$	−5.65685	−0.357771
$251$	−28.2843	−1.78529	−0.892644	−	0.450763i	$-0.851152\pi$
$251$	−28.2843	−1.78529	−0.892644	+	0.450763i	$0.851152\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	28.2843	1.76432	0.882162	−	0.470946i	$-0.156087\pi$
$257$	28.2843	1.76432	0.882162	+	0.470946i	$0.156087\pi$
$258$	0	0
$259$	0	0
$260$	8.00000	0.496139
$261$	3.00000	0.185695
$262$	5.65685	0.349482
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−16.9706	−1.04249
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	−11.3137	−0.689809	−0.344904	−	0.938638i	$-0.612089\pi$
$269$	−11.3137	−0.689809	−0.344904	+	0.938638i	$0.612089\pi$
$270$	0	0
$271$	−14.1421	−0.859074	−0.429537	−	0.903049i	$-0.641323\pi$
$271$	−14.1421	−0.859074	−0.429537	+	0.903049i	$0.641323\pi$
$272$	−2.82843	−0.171499
$273$	0	0
$274$	−22.0000	−1.32907
$275$	12.0000	0.723627
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	−2.82843	−0.169638
$279$	−8.48528	−0.508001
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	11.3137	0.668994
$287$	0	0
$288$	3.00000	0.176777
$289$	−9.00000	−0.529412
$290$	−2.82843	−0.166091
$291$	0	0
$292$	−8.48528	−0.496564
$293$	5.65685	0.330477	0.165238	−	0.986254i	$-0.447161\pi$
$293$	5.65685	0.330477	0.165238	+	0.986254i	$0.447161\pi$
$294$	0	0
$295$	40.0000	2.32889
$296$	−2.00000	−0.116248
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	−5.65685	−0.324443
$305$	0	0
$306$	−8.48528	−0.485071
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	19.7990	1.12270	0.561349	−	0.827579i	$-0.310283\pi$
$311$	19.7990	1.12270	0.561349	+	0.827579i	$0.310283\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	5.65685	0.319235
$315$	0	0
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	−2.82843	−0.158114
$321$	0	0
$322$	0	0
$323$	16.0000	0.890264
$324$	9.00000	0.500000
$325$	−8.48528	−0.470679
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−2.82843	−0.156174
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	14.1421	0.776151
$333$	−6.00000	−0.328798
$334$	11.3137	0.619059
$335$	−11.3137	−0.618134
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	8.00000	0.433861
$341$	11.3137	0.612672
$342$	−16.9706	−0.917663
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	−19.7990	−1.06440
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	36.7696	1.96823	0.984115	−	0.177535i	$-0.0568122\pi$
$349$	36.7696	1.96823	0.984115	+	0.177535i	$0.0568122\pi$
$350$	0	0
$351$	0	0
$352$	−4.00000	−0.213201
$353$	22.6274	1.20434	0.602168	−	0.798369i	$-0.294304\pi$
$353$	22.6274	1.20434	0.602168	+	0.798369i	$0.294304\pi$
$354$	0	0
$355$	22.6274	1.20094
$356$	2.82843	0.149906
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	−8.48528	−0.447214
$361$	13.0000	0.684211
$362$	−19.7990	−1.04061
$363$	0	0
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	−25.4558	−1.32878	−0.664392	−	0.747384i	$-0.731309\pi$
$367$	−25.4558	−1.32878	−0.664392	+	0.747384i	$0.731309\pi$
$368$	0	0
$369$	−8.48528	−0.441726
$370$	5.65685	0.294086
$371$	0	0
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	11.3137	0.585018
$375$	0	0
$376$	−2.82843	−0.145865
$377$	2.82843	0.145671
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	16.0000	0.820783
$381$	0	0
$382$	0	0
$383$	−33.9411	−1.73431	−0.867155	−	0.498038i	$-0.834054\pi$
$383$	−33.9411	−1.73431	−0.867155	+	0.498038i	$0.834054\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	12.0000	0.609994
$388$	8.48528	0.430775
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−12.0000	−0.603023
$397$	−25.4558	−1.27759	−0.638796	−	0.769376i	$-0.720567\pi$
$397$	−25.4558	−1.27759	−0.638796	+	0.769376i	$0.720567\pi$
$398$	−16.9706	−0.850657
$399$	0	0
$400$	3.00000	0.150000
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	11.3137	0.562878
$405$	−25.4558	−1.26491
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	31.1127	1.53842	0.769212	−	0.638994i	$-0.220649\pi$
$409$	31.1127	1.53842	0.769212	+	0.638994i	$0.220649\pi$
$410$	8.00000	0.395092
$411$	0	0
$412$	−5.65685	−0.278693
$413$	0	0
$414$	0	0
$415$	−40.0000	−1.96352
$416$	2.82843	0.138675
$417$	0	0
$418$	22.6274	1.10674
$419$	25.4558	1.24360	0.621800	−	0.783176i	$-0.286402\pi$
$419$	25.4558	1.24360	0.621800	+	0.783176i	$0.286402\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−12.0000	−0.584151
$423$	−8.48528	−0.412568
$424$	−6.00000	−0.291386
$425$	−8.48528	−0.411597
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	−11.3137	−0.545595
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	25.4558	1.22333	0.611665	−	0.791117i	$-0.290500\pi$
$433$	25.4558	1.22333	0.611665	+	0.791117i	$0.290500\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	11.3137	0.539974	0.269987	−	0.962864i	$-0.412981\pi$
$439$	11.3137	0.539974	0.269987	+	0.962864i	$0.412981\pi$
$440$	11.3137	0.539360
$441$	0	0
$442$	−8.00000	−0.380521
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	16.9706	0.803579
$447$	0	0
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	9.00000	0.424264
$451$	11.3137	0.532742
$452$	2.00000	0.0940721
$453$	0	0
$454$	−19.7990	−0.929213
$455$	0	0
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	16.9706	0.792982
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−28.2843	−1.30884	−0.654420	−	0.756131i	$-0.727087\pi$
$467$	−28.2843	−1.30884	−0.654420	+	0.756131i	$0.727087\pi$
$468$	8.48528	0.392232
$469$	0	0
$470$	8.00000	0.369012
$471$	0	0
$472$	14.1421	0.650945
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−16.9706	−0.778663
$476$	0	0
$477$	−18.0000	−0.824163
$478$	0	0
$479$	−8.48528	−0.387702	−0.193851	−	0.981031i	$-0.562098\pi$
$479$	−8.48528	−0.387702	−0.193851	+	0.981031i	$0.562098\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	22.6274	1.03065
$483$	0	0
$484$	5.00000	0.227273
$485$	−24.0000	−1.08978
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	−16.0000	−0.719874
$495$	33.9411	1.52554
$496$	2.82843	0.127000
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	28.2843	1.26239
$503$	−25.4558	−1.13502	−0.567510	−	0.823367i	$-0.692093\pi$
$503$	−25.4558	−1.13502	−0.567510	+	0.823367i	$0.692093\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	14.1421	0.626839	0.313420	−	0.949615i	$-0.398525\pi$
$509$	14.1421	0.626839	0.313420	+	0.949615i	$0.398525\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−28.2843	−1.24757
$515$	16.0000	0.705044
$516$	0	0
$517$	11.3137	0.497576
$518$	0	0
$519$	0	0
$520$	−8.00000	−0.350823
$521$	−28.2843	−1.23916	−0.619578	−	0.784935i	$-0.712696\pi$
$521$	−28.2843	−1.23916	−0.619578	+	0.784935i	$0.712696\pi$
$522$	−3.00000	−0.131306
$523$	2.82843	0.123678	0.0618392	−	0.998086i	$-0.480303\pi$
$523$	2.82843	0.123678	0.0618392	+	0.998086i	$0.480303\pi$
$524$	−5.65685	−0.247121
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−23.0000	−1.00000
$530$	16.9706	0.737154
$531$	42.4264	1.84115
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	−33.9411	−1.46740
$536$	−4.00000	−0.172774
$537$	0	0
$538$	11.3137	0.487769
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	14.1421	0.607457
$543$	0	0
$544$	2.82843	0.121268
$545$	−5.65685	−0.242313
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	22.0000	0.939793
$549$	0	0
$550$	−12.0000	−0.511682
$551$	5.65685	0.240990
$552$	0	0
$553$	0	0
$554$	26.0000	1.10463
$555$	0	0
$556$	2.82843	0.119952
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	8.48528	0.359211
$559$	11.3137	0.478519
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	−5.65685	−0.237986
$566$	−8.48528	−0.356663
$567$	0	0
$568$	8.00000	0.335673
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	−11.3137	−0.473050
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.00000	−0.125000
$577$	−8.48528	−0.353247	−0.176623	−	0.984278i	$-0.556517\pi$
$577$	−8.48528	−0.353247	−0.176623	+	0.984278i	$0.556517\pi$
$578$	9.00000	0.374351
$579$	0	0
$580$	2.82843	0.117444
$581$	0	0
$582$	0	0
$583$	24.0000	0.993978
$584$	8.48528	0.351123
$585$	−24.0000	−0.992278
$586$	−5.65685	−0.233682
$587$	42.4264	1.75113	0.875563	−	0.483105i	$-0.160491\pi$
$587$	42.4264	1.75113	0.875563	+	0.483105i	$0.160491\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	−40.0000	−1.64677
$591$	0	0
$592$	2.00000	0.0821995
$593$	22.6274	0.929197	0.464598	−	0.885522i	$-0.346199\pi$
$593$	22.6274	0.929197	0.464598	+	0.885522i	$0.346199\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−19.7990	−0.807618	−0.403809	−	0.914843i	$-0.632314\pi$
$601$	−19.7990	−0.807618	−0.403809	+	0.914843i	$0.632314\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	16.0000	0.651031
$605$	−14.1421	−0.574960
$606$	0	0
$607$	42.4264	1.72203	0.861017	−	0.508576i	$-0.169828\pi$
$607$	42.4264	1.72203	0.861017	+	0.508576i	$0.169828\pi$
$608$	5.65685	0.229416
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	8.48528	0.342997
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	−11.3137	−0.456584
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	39.5980	1.59158	0.795789	−	0.605575i	$-0.207057\pi$
$619$	39.5980	1.59158	0.795789	+	0.605575i	$0.207057\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−19.7990	−0.793867
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	−5.65685	−0.225733
$629$	−5.65685	−0.225554
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	0	0
$634$	2.00000	0.0794301
$635$	22.6274	0.897942
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	24.0000	0.949425
$640$	2.82843	0.111803
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	14.1421	0.557711	0.278856	−	0.960333i	$-0.410045\pi$
$643$	14.1421	0.557711	0.278856	+	0.960333i	$0.410045\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−5.65685	−0.222394	−0.111197	−	0.993798i	$-0.535468\pi$
$647$	−5.65685	−0.222394	−0.111197	+	0.993798i	$0.535468\pi$
$648$	−9.00000	−0.353553
$649$	−56.5685	−2.22051
$650$	8.48528	0.332820
$651$	0	0
$652$	12.0000	0.469956
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	2.82843	0.110432
$657$	25.4558	0.993127
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−14.1421	−0.550065	−0.275033	−	0.961435i	$-0.588689\pi$
$661$	−14.1421	−0.550065	−0.275033	+	0.961435i	$0.588689\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−14.1421	−0.548821
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	−11.3137	−0.437741
$669$	0	0
$670$	11.3137	0.437087
$671$	0	0
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−5.00000	−0.192308
$677$	−50.9117	−1.95670	−0.978348	−	0.206969i	$-0.933640\pi$
$677$	−50.9117	−1.95670	−0.978348	+	0.206969i	$0.933640\pi$
$678$	0	0
$679$	0	0
$680$	−8.00000	−0.306786
$681$	0	0
$682$	−11.3137	−0.433224
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	16.9706	0.648886
$685$	−62.2254	−2.37751
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−16.9706	−0.646527
$690$	0	0
$691$	14.1421	0.537992	0.268996	−	0.963141i	$-0.413308\pi$
$691$	14.1421	0.537992	0.268996	+	0.963141i	$0.413308\pi$
$692$	19.7990	0.752645
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−8.00000	−0.303022
$698$	−36.7696	−1.39175
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	4.00000	0.150756
$705$	0	0
$706$	−22.6274	−0.851594
$707$	0	0
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−22.6274	−0.849192
$711$	0	0
$712$	−2.82843	−0.106000
$713$	0	0
$714$	0	0
$715$	32.0000	1.19673
$716$	4.00000	0.149487
$717$	0	0
$718$	16.0000	0.597115
$719$	−39.5980	−1.47676	−0.738378	−	0.674387i	$-0.764408\pi$
$719$	−39.5980	−1.47676	−0.738378	+	0.674387i	$0.764408\pi$
$720$	8.48528	0.316228
$721$	0	0
$722$	−13.0000	−0.483810
$723$	0	0
$724$	19.7990	0.735824
$725$	−3.00000	−0.111417
$726$	0	0
$727$	8.48528	0.314702	0.157351	−	0.987543i	$-0.449705\pi$
$727$	8.48528	0.314702	0.157351	+	0.987543i	$0.449705\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−24.0000	−0.888280
$731$	11.3137	0.418453
$732$	0	0
$733$	22.6274	0.835763	0.417881	−	0.908502i	$-0.362773\pi$
$733$	22.6274	0.835763	0.417881	+	0.908502i	$0.362773\pi$
$734$	25.4558	0.939592
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	8.48528	0.312348
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−5.65685	−0.207950
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	28.2843	1.03626
$746$	26.0000	0.951928
$747$	−42.4264	−1.55230
$748$	−11.3137	−0.413670
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	2.82843	0.103142
$753$	0	0
$754$	−2.82843	−0.103005
$755$	−45.2548	−1.64699
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	−16.0000	−0.580381
$761$	11.3137	0.410122	0.205061	−	0.978749i	$-0.434261\pi$
$761$	11.3137	0.410122	0.205061	+	0.978749i	$0.434261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−24.0000	−0.867722
$766$	33.9411	1.22634
$767$	40.0000	1.44432
$768$	0	0
$769$	8.48528	0.305987	0.152994	−	0.988227i	$-0.451109\pi$
$769$	8.48528	0.305987	0.152994	+	0.988227i	$0.451109\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	28.2843	1.01731	0.508657	−	0.860969i	$-0.330142\pi$
$773$	28.2843	1.01731	0.508657	+	0.860969i	$0.330142\pi$
$774$	−12.0000	−0.431331
$775$	8.48528	0.304800
$776$	−8.48528	−0.304604
$777$	0	0
$778$	14.0000	0.501924
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.0000	0.571064
$786$	0	0
$787$	−2.82843	−0.100823	−0.0504113	−	0.998729i	$-0.516053\pi$
$787$	−2.82843	−0.100823	−0.0504113	+	0.998729i	$0.516053\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	0	0
$791$	0	0
$792$	12.0000	0.426401
$793$	0	0
$794$	25.4558	0.903394
$795$	0	0
$796$	16.9706	0.601506
$797$	−50.9117	−1.80338	−0.901692	−	0.432378i	$-0.857674\pi$
$797$	−50.9117	−1.80338	−0.901692	+	0.432378i	$0.857674\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−3.00000	−0.106066
$801$	−8.48528	−0.299813
$802$	34.0000	1.20058
$803$	−33.9411	−1.19776
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−11.3137	−0.398015
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	25.4558	0.894427
$811$	8.48528	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$811$	8.48528	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$812$	0	0
$813$	0	0
$814$	−8.00000	−0.280400
$815$	−33.9411	−1.18891
$816$	0	0
$817$	22.6274	0.791633
$818$	−31.1127	−1.08783
$819$	0	0
$820$	−8.00000	−0.279372
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	5.65685	0.197066
$825$	0	0
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	45.2548	1.57177	0.785883	−	0.618376i	$-0.212209\pi$
$829$	45.2548	1.57177	0.785883	+	0.618376i	$0.212209\pi$
$830$	40.0000	1.38842
$831$	0	0
$832$	−2.82843	−0.0980581
$833$	0	0
$834$	0	0
$835$	32.0000	1.10741
$836$	−22.6274	−0.782586
$837$	0	0
$838$	−25.4558	−0.879358
$839$	31.1127	1.07413	0.537065	−	0.843541i	$-0.319533\pi$
$839$	31.1127	1.07413	0.537065	+	0.843541i	$0.319533\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.00000	0.0689246
$843$	0	0
$844$	12.0000	0.413057
$845$	14.1421	0.486504
$846$	8.48528	0.291730
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	8.48528	0.291043
$851$	0	0
$852$	0	0
$853$	22.6274	0.774748	0.387374	−	0.921923i	$-0.373382\pi$
$853$	22.6274	0.774748	0.387374	+	0.921923i	$0.373382\pi$
$854$	0	0
$855$	−48.0000	−1.64157
$856$	−12.0000	−0.410152
$857$	11.3137	0.386469	0.193234	−	0.981153i	$-0.438102\pi$
$857$	11.3137	0.386469	0.193234	+	0.981153i	$0.438102\pi$
$858$	0	0
$859$	33.9411	1.15806	0.579028	−	0.815308i	$-0.303432\pi$
$859$	33.9411	1.15806	0.579028	+	0.815308i	$0.303432\pi$
$860$	11.3137	0.385794
$861$	0	0
$862$	−32.0000	−1.08992
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−56.0000	−1.90406
$866$	−25.4558	−0.865025
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−11.3137	−0.383350
$872$	−2.00000	−0.0677285
$873$	−25.4558	−0.861550
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−11.3137	−0.381819
$879$	0	0
$880$	−11.3137	−0.381385
$881$	−36.7696	−1.23880	−0.619399	−	0.785076i	$-0.712624\pi$
$881$	−36.7696	−1.23880	−0.619399	+	0.785076i	$0.712624\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−28.0000	−0.940678
$887$	14.1421	0.474846	0.237423	−	0.971406i	$-0.423697\pi$
$887$	14.1421	0.474846	0.237423	+	0.971406i	$0.423697\pi$
$888$	0	0
$889$	0	0
$890$	8.00000	0.268161
$891$	36.0000	1.20605
$892$	−16.9706	−0.568216
$893$	−16.0000	−0.535420
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	0	0
$898$	14.0000	0.467186
$899$	−2.82843	−0.0943333
$900$	−9.00000	−0.300000
$901$	−16.9706	−0.565371
$902$	−11.3137	−0.376705
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	−56.0000	−1.86150
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	19.7990	0.657053
$909$	−33.9411	−1.12576
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	56.5685	1.87215
$914$	14.0000	0.463079
$915$	0	0
$916$	−16.9706	−0.560723
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.6274	0.744791
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	16.9706	0.557386
$928$	1.00000	0.0328266
$929$	22.6274	0.742381	0.371191	−	0.928557i	$-0.378950\pi$
$929$	22.6274	0.742381	0.371191	+	0.928557i	$0.378950\pi$
$930$	0	0
$931$	0	0
$932$	22.0000	0.720634
$933$	0	0
$934$	28.2843	0.925490
$935$	32.0000	1.04651
$936$	−8.48528	−0.277350
$937$	33.9411	1.10881	0.554404	−	0.832248i	$-0.312946\pi$
$937$	33.9411	1.10881	0.554404	+	0.832248i	$0.312946\pi$
$938$	0	0
$939$	0	0
$940$	−8.00000	−0.260931
$941$	19.7990	0.645429	0.322714	−	0.946496i	$-0.395405\pi$
$941$	19.7990	0.645429	0.322714	+	0.946496i	$0.395405\pi$
$942$	0	0
$943$	0	0
$944$	−14.1421	−0.460287
$945$	0	0
$946$	16.0000	0.520205
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	16.9706	0.550598
$951$	0	0
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	0	0
$957$	0	0
$958$	8.48528	0.274147
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	5.65685	0.182384
$963$	−36.0000	−1.16008
$964$	−22.6274	−0.728780
$965$	39.5980	1.27470
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	24.0000	0.770594
$971$	16.9706	0.544611	0.272306	−	0.962211i	$-0.412214\pi$
$971$	16.9706	0.544611	0.272306	+	0.962211i	$0.412214\pi$
$972$	0	0
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	11.3137	0.361588
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−36.0000	−1.14881
$983$	−14.1421	−0.451064	−0.225532	−	0.974236i	$-0.572412\pi$
$983$	−14.1421	−0.451064	−0.225532	+	0.974236i	$0.572412\pi$
$984$	0	0
$985$	−62.2254	−1.98267
$986$	−2.82843	−0.0900755
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	−33.9411	−1.07872
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−2.82843	−0.0898027
$993$	0	0
$994$	0	0
$995$	−48.0000	−1.52170
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.g.1.1	✓	2
7.6	odd	2	inner	2842.2.a.g.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.g.1.1	✓	2	1.1	even	1	trivial
2842.2.a.g.1.2	yes	2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} +2.82843 q^{10} +4.00000 q^{11} -2.82843 q^{13} +1.00000 q^{16} -2.82843 q^{17} +3.00000 q^{18} -5.65685 q^{19} -2.82843 q^{20} -4.00000 q^{22} +3.00000 q^{25} +2.82843 q^{26} -1.00000 q^{29} +2.82843 q^{31} -1.00000 q^{32} +2.82843 q^{34} -3.00000 q^{36} +2.00000 q^{37} +5.65685 q^{38} +2.82843 q^{40} +2.82843 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.48528 q^{45} +2.82843 q^{47} -3.00000 q^{50} -2.82843 q^{52} +6.00000 q^{53} -11.3137 q^{55} +1.00000 q^{58} -14.1421 q^{59} -2.82843 q^{62} +1.00000 q^{64} +8.00000 q^{65} +4.00000 q^{67} -2.82843 q^{68} -8.00000 q^{71} +3.00000 q^{72} -8.48528 q^{73} -2.00000 q^{74} -5.65685 q^{76} -2.82843 q^{80} +9.00000 q^{81} -2.82843 q^{82} +14.1421 q^{83} +8.00000 q^{85} +4.00000 q^{86} -4.00000 q^{88} +2.82843 q^{89} -8.48528 q^{90} -2.82843 q^{94} +16.0000 q^{95} +8.48528 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} + 8 q^{11} + 2 q^{16} + 6 q^{18} - 8 q^{22} + 6 q^{25} - 2 q^{29} - 2 q^{32} - 6 q^{36} + 4 q^{37} - 8 q^{43} + 8 q^{44} - 6 q^{50} + 12 q^{53} + 2 q^{58} + 2 q^{64} + 16 q^{65} + 8 q^{67} - 16 q^{71} + 6 q^{72} - 4 q^{74} + 18 q^{81} + 16 q^{85} + 8 q^{86} - 8 q^{88} + 32 q^{95} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 6 * q^9 + 8 * q^11 + 2 * q^16 + 6 * q^18 - 8 * q^22 + 6 * q^25 - 2 * q^29 - 2 * q^32 - 6 * q^36 + 4 * q^37 - 8 * q^43 + 8 * q^44 - 6 * q^50 + 12 * q^53 + 2 * q^58 + 2 * q^64 + 16 * q^65 + 8 * q^67 - 16 * q^71 + 6 * q^72 - 4 * q^74 + 18 * q^81 + 16 * q^85 + 8 * q^86 - 8 * q^88 + 32 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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