LMFDB - Embedded newform 6028.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$2$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} -3.61803 q^{5} -2.38197 q^{7} -2.61803 q^{9} +O(q^{10})$	$q+0.618034 q^{3} -3.61803 q^{5} -2.38197 q^{7} -2.61803 q^{9} +1.00000 q^{11} -4.23607 q^{13} -2.23607 q^{15} -6.23607 q^{17} -2.61803 q^{19} -1.47214 q^{21} -5.85410 q^{23} +8.09017 q^{25} -3.47214 q^{27} -2.76393 q^{29} -0.527864 q^{31} +0.618034 q^{33} +8.61803 q^{35} +0.854102 q^{37} -2.61803 q^{39} -3.00000 q^{41} -10.7082 q^{43} +9.47214 q^{45} +5.94427 q^{47} -1.32624 q^{49} -3.85410 q^{51} -3.76393 q^{53} -3.61803 q^{55} -1.61803 q^{57} +5.09017 q^{59} -14.2361 q^{61} +6.23607 q^{63} +15.3262 q^{65} +5.32624 q^{67} -3.61803 q^{69} -3.14590 q^{71} +15.9443 q^{73} +5.00000 q^{75} -2.38197 q^{77} -14.7082 q^{79} +5.70820 q^{81} -4.56231 q^{83} +22.5623 q^{85} -1.70820 q^{87} +7.09017 q^{89} +10.0902 q^{91} -0.326238 q^{93} +9.47214 q^{95} -3.94427 q^{97} -2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9	$ 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 10 q^{31} - q^{33} + 15 q^{35} - 5 q^{37} - 3 q^{39} - 6 q^{41} - 8 q^{43} + 10 q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 12 q^{53} - 5 q^{55} - q^{57} - q^{59} - 24 q^{61} + 8 q^{63} + 15 q^{65} - 5 q^{67} - 5 q^{69} - 13 q^{71} + 14 q^{73} + 10 q^{75} - 7 q^{77} - 16 q^{79} - 2 q^{81} + 11 q^{83} + 25 q^{85} + 10 q^{87} + 3 q^{89} + 9 q^{91} + 15 q^{93} + 10 q^{95} + 10 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^17 - 3 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 10 * q^31 - q^33 + 15 * q^35 - 5 * q^37 - 3 * q^39 - 6 * q^41 - 8 * q^43 + 10 * q^45 - 6 * q^47 + 13 * q^49 - q^51 - 12 * q^53 - 5 * q^55 - q^57 - q^59 - 24 * q^61 + 8 * q^63 + 15 * q^65 - 5 * q^67 - 5 * q^69 - 13 * q^71 + 14 * q^73 + 10 * q^75 - 7 * q^77 - 16 * q^79 - 2 * q^81 + 11 * q^83 + 25 * q^85 + 10 * q^87 + 3 * q^89 + 9 * q^91 + 15 * q^93 + 10 * q^95 + 10 * q^97 - 3 * 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0	0
$7$	−2.38197	−0.900299	−0.450149	−	0.892953i	$-0.648629\pi$
$7$	−2.38197	−0.900299	−0.450149	+	0.892953i	$0.648629\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	0	0
$15$	−2.23607	−0.577350
$16$	0	0
$17$	−6.23607	−1.51247	−0.756234	−	0.654301i	$-0.772963\pi$
$17$	−6.23607	−1.51247	−0.756234	+	0.654301i	$0.772963\pi$
$18$	0	0
$19$	−2.61803	−0.600618	−0.300309	−	0.953842i	$-0.597090\pi$
$19$	−2.61803	−0.600618	−0.300309	+	0.953842i	$0.597090\pi$
$20$	0	0
$21$	−1.47214	−0.321246
$22$	0	0
$23$	−5.85410	−1.22066	−0.610332	−	0.792145i	$-0.708964\pi$
$23$	−5.85410	−1.22066	−0.610332	+	0.792145i	$0.708964\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	−0.527864	−0.0948072	−0.0474036	−	0.998876i	$-0.515095\pi$
$31$	−0.527864	−0.0948072	−0.0474036	+	0.998876i	$0.515095\pi$
$32$	0	0
$33$	0.618034	0.107586
$34$	0	0
$35$	8.61803	1.45671
$36$	0	0
$37$	0.854102	0.140413	0.0702067	−	0.997532i	$-0.477634\pi$
$37$	0.854102	0.140413	0.0702067	+	0.997532i	$0.477634\pi$
$38$	0	0
$39$	−2.61803	−0.419221
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−10.7082	−1.63299	−0.816493	−	0.577355i	$-0.804085\pi$
$43$	−10.7082	−1.63299	−0.816493	+	0.577355i	$0.804085\pi$
$44$	0	0
$45$	9.47214	1.41202
$46$	0	0
$47$	5.94427	0.867061	0.433531	−	0.901139i	$-0.357268\pi$
$47$	5.94427	0.867061	0.433531	+	0.901139i	$0.357268\pi$
$48$	0	0
$49$	−1.32624	−0.189463
$50$	0	0
$51$	−3.85410	−0.539682
$52$	0	0
$53$	−3.76393	−0.517016	−0.258508	−	0.966009i	$-0.583231\pi$
$53$	−3.76393	−0.517016	−0.258508	+	0.966009i	$0.583231\pi$
$54$	0	0
$55$	−3.61803	−0.487856
$56$	0	0
$57$	−1.61803	−0.214314
$58$	0	0
$59$	5.09017	0.662684	0.331342	−	0.943511i	$-0.392499\pi$
$59$	5.09017	0.662684	0.331342	+	0.943511i	$0.392499\pi$
$60$	0	0
$61$	−14.2361	−1.82274	−0.911371	−	0.411586i	$-0.864975\pi$
$61$	−14.2361	−1.82274	−0.911371	+	0.411586i	$0.864975\pi$
$62$	0	0
$63$	6.23607	0.785671
$64$	0	0
$65$	15.3262	1.90099
$66$	0	0
$67$	5.32624	0.650704	0.325352	−	0.945593i	$-0.394517\pi$
$67$	5.32624	0.650704	0.325352	+	0.945593i	$0.394517\pi$
$68$	0	0
$69$	−3.61803	−0.435560
$70$	0	0
$71$	−3.14590	−0.373349	−0.186675	−	0.982422i	$-0.559771\pi$
$71$	−3.14590	−0.373349	−0.186675	+	0.982422i	$0.559771\pi$
$72$	0	0
$73$	15.9443	1.86614	0.933068	−	0.359700i	$-0.117121\pi$
$73$	15.9443	1.86614	0.933068	+	0.359700i	$0.117121\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	−2.38197	−0.271450
$78$	0	0
$79$	−14.7082	−1.65480	−0.827401	−	0.561611i	$-0.810182\pi$
$79$	−14.7082	−1.65480	−0.827401	+	0.561611i	$0.810182\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−4.56231	−0.500778	−0.250389	−	0.968145i	$-0.580559\pi$
$83$	−4.56231	−0.500778	−0.250389	+	0.968145i	$0.580559\pi$
$84$	0	0
$85$	22.5623	2.44723
$86$	0	0
$87$	−1.70820	−0.183139
$88$	0	0
$89$	7.09017	0.751557	0.375778	−	0.926710i	$-0.377375\pi$
$89$	7.09017	0.751557	0.375778	+	0.926710i	$0.377375\pi$
$90$	0	0
$91$	10.0902	1.05774
$92$	0	0
$93$	−0.326238	−0.0338293
$94$	0	0
$95$	9.47214	0.971821
$96$	0	0
$97$	−3.94427	−0.400480	−0.200240	−	0.979747i	$-0.564172\pi$
$97$	−3.94427	−0.400480	−0.200240	+	0.979747i	$0.564172\pi$
$98$	0	0
$99$	−2.61803	−0.263122
$100$	0	0
$101$	−9.14590	−0.910051	−0.455025	−	0.890478i	$-0.650370\pi$
$101$	−9.14590	−0.910051	−0.455025	+	0.890478i	$0.650370\pi$
$102$	0	0
$103$	−0.0557281	−0.00549105	−0.00274553	−	0.999996i	$-0.500874\pi$
$103$	−0.0557281	−0.00549105	−0.00274553	+	0.999996i	$0.500874\pi$
$104$	0	0
$105$	5.32624	0.519788
$106$	0	0
$107$	−6.56231	−0.634402	−0.317201	−	0.948358i	$-0.602743\pi$
$107$	−6.56231	−0.634402	−0.317201	+	0.948358i	$0.602743\pi$
$108$	0	0
$109$	2.23607	0.214176	0.107088	−	0.994250i	$-0.465847\pi$
$109$	2.23607	0.214176	0.107088	+	0.994250i	$0.465847\pi$
$110$	0	0
$111$	0.527864	0.0501026
$112$	0	0
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	0	0
$115$	21.1803	1.97508
$116$	0	0
$117$	11.0902	1.02529
$118$	0	0
$119$	14.8541	1.36167
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.85410	−0.167179
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−8.52786	−0.756726	−0.378363	−	0.925657i	$-0.623513\pi$
$127$	−8.52786	−0.756726	−0.378363	+	0.925657i	$0.623513\pi$
$128$	0	0
$129$	−6.61803	−0.582685
$130$	0	0
$131$	2.29180	0.200235	0.100118	−	0.994976i	$-0.468078\pi$
$131$	2.29180	0.200235	0.100118	+	0.994976i	$0.468078\pi$
$132$	0	0
$133$	6.23607	0.540736
$134$	0	0
$135$	12.5623	1.08119
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	1.29180	0.109569	0.0547844	−	0.998498i	$-0.482553\pi$
$139$	1.29180	0.109569	0.0547844	+	0.998498i	$0.482553\pi$
$140$	0	0
$141$	3.67376	0.309387
$142$	0	0
$143$	−4.23607	−0.354238
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	−0.819660	−0.0676044
$148$	0	0
$149$	9.32624	0.764035	0.382018	−	0.924155i	$-0.375229\pi$
$149$	9.32624	0.764035	0.382018	+	0.924155i	$0.375229\pi$
$150$	0	0
$151$	−10.3262	−0.840337	−0.420169	−	0.907446i	$-0.638029\pi$
$151$	−10.3262	−0.840337	−0.420169	+	0.907446i	$0.638029\pi$
$152$	0	0
$153$	16.3262	1.31990
$154$	0	0
$155$	1.90983	0.153401
$156$	0	0
$157$	1.14590	0.0914526	0.0457263	−	0.998954i	$-0.485440\pi$
$157$	1.14590	0.0914526	0.0457263	+	0.998954i	$0.485440\pi$
$158$	0	0
$159$	−2.32624	−0.184483
$160$	0	0
$161$	13.9443	1.09896
$162$	0	0
$163$	−15.4164	−1.20751	−0.603753	−	0.797171i	$-0.706329\pi$
$163$	−15.4164	−1.20751	−0.603753	+	0.797171i	$0.706329\pi$
$164$	0	0
$165$	−2.23607	−0.174078
$166$	0	0
$167$	−11.0902	−0.858183	−0.429092	−	0.903261i	$-0.641166\pi$
$167$	−11.0902	−0.858183	−0.429092	+	0.903261i	$0.641166\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	0	0
$171$	6.85410	0.524146
$172$	0	0
$173$	0.326238	0.0248034	0.0124017	−	0.999923i	$-0.496052\pi$
$173$	0.326238	0.0248034	0.0124017	+	0.999923i	$0.496052\pi$
$174$	0	0
$175$	−19.2705	−1.45671
$176$	0	0
$177$	3.14590	0.236460
$178$	0	0
$179$	−6.94427	−0.519039	−0.259520	−	0.965738i	$-0.583564\pi$
$179$	−6.94427	−0.519039	−0.259520	+	0.965738i	$0.583564\pi$
$180$	0	0
$181$	−17.1803	−1.27700	−0.638502	−	0.769620i	$-0.720446\pi$
$181$	−17.1803	−1.27700	−0.638502	+	0.769620i	$0.720446\pi$
$182$	0	0
$183$	−8.79837	−0.650395
$184$	0	0
$185$	−3.09017	−0.227194
$186$	0	0
$187$	−6.23607	−0.456026
$188$	0	0
$189$	8.27051	0.601591
$190$	0	0
$191$	−0.708204	−0.0512438	−0.0256219	−	0.999672i	$-0.508157\pi$
$191$	−0.708204	−0.0512438	−0.0256219	+	0.999672i	$0.508157\pi$
$192$	0	0
$193$	−1.70820	−0.122959	−0.0614796	−	0.998108i	$-0.519582\pi$
$193$	−1.70820	−0.122959	−0.0614796	+	0.998108i	$0.519582\pi$
$194$	0	0
$195$	9.47214	0.678314
$196$	0	0
$197$	7.23607	0.515548	0.257774	−	0.966205i	$-0.417011\pi$
$197$	7.23607	0.515548	0.257774	+	0.966205i	$0.417011\pi$
$198$	0	0
$199$	10.2361	0.725616	0.362808	−	0.931864i	$-0.381818\pi$
$199$	10.2361	0.725616	0.362808	+	0.931864i	$0.381818\pi$
$200$	0	0
$201$	3.29180	0.232185
$202$	0	0
$203$	6.58359	0.462078
$204$	0	0
$205$	10.8541	0.758083
$206$	0	0
$207$	15.3262	1.06525
$208$	0	0
$209$	−2.61803	−0.181093
$210$	0	0
$211$	−1.09017	−0.0750504	−0.0375252	−	0.999296i	$-0.511947\pi$
$211$	−1.09017	−0.0750504	−0.0375252	+	0.999296i	$0.511947\pi$
$212$	0	0
$213$	−1.94427	−0.133219
$214$	0	0
$215$	38.7426	2.64223
$216$	0	0
$217$	1.25735	0.0853548
$218$	0	0
$219$	9.85410	0.665879
$220$	0	0
$221$	26.4164	1.77696
$222$	0	0
$223$	−6.88854	−0.461291	−0.230646	−	0.973038i	$-0.574084\pi$
$223$	−6.88854	−0.461291	−0.230646	+	0.973038i	$0.574084\pi$
$224$	0	0
$225$	−21.1803	−1.41202
$226$	0	0
$227$	28.9787	1.92338	0.961692	−	0.274131i	$-0.0883902\pi$
$227$	28.9787	1.92338	0.961692	+	0.274131i	$0.0883902\pi$
$228$	0	0
$229$	10.9443	0.723218	0.361609	−	0.932330i	$-0.382228\pi$
$229$	10.9443	0.723218	0.361609	+	0.932330i	$0.382228\pi$
$230$	0	0
$231$	−1.47214	−0.0968594
$232$	0	0
$233$	−15.3820	−1.00771	−0.503853	−	0.863789i	$-0.668085\pi$
$233$	−15.3820	−1.00771	−0.503853	+	0.863789i	$0.668085\pi$
$234$	0	0
$235$	−21.5066	−1.40293
$236$	0	0
$237$	−9.09017	−0.590470
$238$	0	0
$239$	−1.00000	−0.0646846	−0.0323423	−	0.999477i	$-0.510297\pi$
$239$	−1.00000	−0.0646846	−0.0323423	+	0.999477i	$0.510297\pi$
$240$	0	0
$241$	−12.7639	−0.822197	−0.411099	−	0.911591i	$-0.634855\pi$
$241$	−12.7639	−0.822197	−0.411099	+	0.911591i	$0.634855\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	4.79837	0.306557
$246$	0	0
$247$	11.0902	0.705651
$248$	0	0
$249$	−2.81966	−0.178689
$250$	0	0
$251$	3.79837	0.239751	0.119876	−	0.992789i	$-0.461750\pi$
$251$	3.79837	0.239751	0.119876	+	0.992789i	$0.461750\pi$
$252$	0	0
$253$	−5.85410	−0.368044
$254$	0	0
$255$	13.9443	0.873224
$256$	0	0
$257$	−0.381966	−0.0238264	−0.0119132	−	0.999929i	$-0.503792\pi$
$257$	−0.381966	−0.0238264	−0.0119132	+	0.999929i	$0.503792\pi$
$258$	0	0
$259$	−2.03444	−0.126414
$260$	0	0
$261$	7.23607	0.447901
$262$	0	0
$263$	−7.09017	−0.437199	−0.218599	−	0.975815i	$-0.570149\pi$
$263$	−7.09017	−0.437199	−0.218599	+	0.975815i	$0.570149\pi$
$264$	0	0
$265$	13.6180	0.836549
$266$	0	0
$267$	4.38197	0.268172
$268$	0	0
$269$	5.94427	0.362429	0.181214	−	0.983444i	$-0.441997\pi$
$269$	5.94427	0.362429	0.181214	+	0.983444i	$0.441997\pi$
$270$	0	0
$271$	−4.61803	−0.280526	−0.140263	−	0.990114i	$-0.544795\pi$
$271$	−4.61803	−0.280526	−0.140263	+	0.990114i	$0.544795\pi$
$272$	0	0
$273$	6.23607	0.377424
$274$	0	0
$275$	8.09017	0.487856
$276$	0	0
$277$	21.6525	1.30097	0.650486	−	0.759519i	$-0.274565\pi$
$277$	21.6525	1.30097	0.650486	+	0.759519i	$0.274565\pi$
$278$	0	0
$279$	1.38197	0.0827361
$280$	0	0
$281$	−5.76393	−0.343847	−0.171924	−	0.985110i	$-0.554998\pi$
$281$	−5.76393	−0.343847	−0.171924	+	0.985110i	$0.554998\pi$
$282$	0	0
$283$	−15.4721	−0.919723	−0.459862	−	0.887991i	$-0.652101\pi$
$283$	−15.4721	−0.919723	−0.459862	+	0.887991i	$0.652101\pi$
$284$	0	0
$285$	5.85410	0.346767
$286$	0	0
$287$	7.14590	0.421809
$288$	0	0
$289$	21.8885	1.28756
$290$	0	0
$291$	−2.43769	−0.142900
$292$	0	0
$293$	8.70820	0.508739	0.254369	−	0.967107i	$-0.418132\pi$
$293$	8.70820	0.508739	0.254369	+	0.967107i	$0.418132\pi$
$294$	0	0
$295$	−18.4164	−1.07224
$296$	0	0
$297$	−3.47214	−0.201474
$298$	0	0
$299$	24.7984	1.43413
$300$	0	0
$301$	25.5066	1.47017
$302$	0	0
$303$	−5.65248	−0.324726
$304$	0	0
$305$	51.5066	2.94926
$306$	0	0
$307$	4.61803	0.263565	0.131783	−	0.991279i	$-0.457930\pi$
$307$	4.61803	0.263565	0.131783	+	0.991279i	$0.457930\pi$
$308$	0	0
$309$	−0.0344419	−0.00195933
$310$	0	0
$311$	−29.4164	−1.66805	−0.834026	−	0.551726i	$-0.813970\pi$
$311$	−29.4164	−1.66805	−0.834026	+	0.551726i	$0.813970\pi$
$312$	0	0
$313$	21.2705	1.20228	0.601140	−	0.799144i	$-0.294713\pi$
$313$	21.2705	1.20228	0.601140	+	0.799144i	$0.294713\pi$
$314$	0	0
$315$	−22.5623	−1.27124
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−2.76393	−0.154750
$320$	0	0
$321$	−4.05573	−0.226369
$322$	0	0
$323$	16.3262	0.908416
$324$	0	0
$325$	−34.2705	−1.90099
$326$	0	0
$327$	1.38197	0.0764229
$328$	0	0
$329$	−14.1591	−0.780614
$330$	0	0
$331$	−17.0344	−0.936298	−0.468149	−	0.883650i	$-0.655079\pi$
$331$	−17.0344	−0.936298	−0.468149	+	0.883650i	$0.655079\pi$
$332$	0	0
$333$	−2.23607	−0.122536
$334$	0	0
$335$	−19.2705	−1.05286
$336$	0	0
$337$	−6.76393	−0.368455	−0.184227	−	0.982884i	$-0.558978\pi$
$337$	−6.76393	−0.368455	−0.184227	+	0.982884i	$0.558978\pi$
$338$	0	0
$339$	4.29180	0.233098
$340$	0	0
$341$	−0.527864	−0.0285854
$342$	0	0
$343$	19.8328	1.07087
$344$	0	0
$345$	13.0902	0.704751
$346$	0	0
$347$	1.29180	0.0693472	0.0346736	−	0.999399i	$-0.488961\pi$
$347$	1.29180	0.0693472	0.0346736	+	0.999399i	$0.488961\pi$
$348$	0	0
$349$	−21.2148	−1.13560	−0.567801	−	0.823166i	$-0.692206\pi$
$349$	−21.2148	−1.13560	−0.567801	+	0.823166i	$0.692206\pi$
$350$	0	0
$351$	14.7082	0.785066
$352$	0	0
$353$	20.1246	1.07113	0.535563	−	0.844496i	$-0.320100\pi$
$353$	20.1246	1.07113	0.535563	+	0.844496i	$0.320100\pi$
$354$	0	0
$355$	11.3820	0.604092
$356$	0	0
$357$	9.18034	0.485875
$358$	0	0
$359$	−16.9443	−0.894284	−0.447142	−	0.894463i	$-0.647558\pi$
$359$	−16.9443	−0.894284	−0.447142	+	0.894463i	$0.647558\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	0	0
$363$	0.618034	0.0324384
$364$	0	0
$365$	−57.6869	−3.01947
$366$	0	0
$367$	7.70820	0.402365	0.201182	−	0.979554i	$-0.435522\pi$
$367$	7.70820	0.402365	0.201182	+	0.979554i	$0.435522\pi$
$368$	0	0
$369$	7.85410	0.408868
$370$	0	0
$371$	8.96556	0.465469
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	0	0
$375$	−6.90983	−0.356822
$376$	0	0
$377$	11.7082	0.603003
$378$	0	0
$379$	−24.7082	−1.26918	−0.634588	−	0.772851i	$-0.718830\pi$
$379$	−24.7082	−1.26918	−0.634588	+	0.772851i	$0.718830\pi$
$380$	0	0
$381$	−5.27051	−0.270016
$382$	0	0
$383$	28.7082	1.46692	0.733460	−	0.679732i	$-0.237904\pi$
$383$	28.7082	1.46692	0.733460	+	0.679732i	$0.237904\pi$
$384$	0	0
$385$	8.61803	0.439216
$386$	0	0
$387$	28.0344	1.42507
$388$	0	0
$389$	−32.5410	−1.64990	−0.824948	−	0.565209i	$-0.808795\pi$
$389$	−32.5410	−1.64990	−0.824948	+	0.565209i	$0.808795\pi$
$390$	0	0
$391$	36.5066	1.84622
$392$	0	0
$393$	1.41641	0.0714483
$394$	0	0
$395$	53.2148	2.67753
$396$	0	0
$397$	−7.50658	−0.376744	−0.188372	−	0.982098i	$-0.560321\pi$
$397$	−7.50658	−0.376744	−0.188372	+	0.982098i	$0.560321\pi$
$398$	0	0
$399$	3.85410	0.192946
$400$	0	0
$401$	−36.5066	−1.82305	−0.911526	−	0.411243i	$-0.865095\pi$
$401$	−36.5066	−1.82305	−0.911526	+	0.411243i	$0.865095\pi$
$402$	0	0
$403$	2.23607	0.111386
$404$	0	0
$405$	−20.6525	−1.02623
$406$	0	0
$407$	0.854102	0.0423363
$408$	0	0
$409$	−1.67376	−0.0827622	−0.0413811	−	0.999143i	$-0.513176\pi$
$409$	−1.67376	−0.0827622	−0.0413811	+	0.999143i	$0.513176\pi$
$410$	0	0
$411$	0.618034	0.0304854
$412$	0	0
$413$	−12.1246	−0.596613
$414$	0	0
$415$	16.5066	0.810276
$416$	0	0
$417$	0.798374	0.0390965
$418$	0	0
$419$	6.18034	0.301929	0.150965	−	0.988539i	$-0.451762\pi$
$419$	6.18034	0.301929	0.150965	+	0.988539i	$0.451762\pi$
$420$	0	0
$421$	14.7426	0.718513	0.359256	−	0.933239i	$-0.383030\pi$
$421$	14.7426	0.718513	0.359256	+	0.933239i	$0.383030\pi$
$422$	0	0
$423$	−15.5623	−0.756665
$424$	0	0
$425$	−50.4508	−2.44723
$426$	0	0
$427$	33.9098	1.64101
$428$	0	0
$429$	−2.61803	−0.126400
$430$	0	0
$431$	6.65248	0.320438	0.160219	−	0.987081i	$-0.448780\pi$
$431$	6.65248	0.320438	0.160219	+	0.987081i	$0.448780\pi$
$432$	0	0
$433$	9.05573	0.435191	0.217595	−	0.976039i	$-0.430179\pi$
$433$	9.05573	0.435191	0.217595	+	0.976039i	$0.430179\pi$
$434$	0	0
$435$	6.18034	0.296325
$436$	0	0
$437$	15.3262	0.733153
$438$	0	0
$439$	−16.7639	−0.800099	−0.400049	−	0.916494i	$-0.631007\pi$
$439$	−16.7639	−0.800099	−0.400049	+	0.916494i	$0.631007\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	21.0902	1.00202	0.501012	−	0.865440i	$-0.332961\pi$
$443$	21.0902	1.00202	0.501012	+	0.865440i	$0.332961\pi$
$444$	0	0
$445$	−25.6525	−1.21604
$446$	0	0
$447$	5.76393	0.272625
$448$	0	0
$449$	−12.1803	−0.574826	−0.287413	−	0.957807i	$-0.592795\pi$
$449$	−12.1803	−0.574826	−0.287413	+	0.957807i	$0.592795\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	0	0
$453$	−6.38197	−0.299851
$454$	0	0
$455$	−36.5066	−1.71145
$456$	0	0
$457$	−13.8197	−0.646456	−0.323228	−	0.946321i	$-0.604768\pi$
$457$	−13.8197	−0.646456	−0.323228	+	0.946321i	$0.604768\pi$
$458$	0	0
$459$	21.6525	1.01065
$460$	0	0
$461$	−24.2918	−1.13138	−0.565691	−	0.824617i	$-0.691390\pi$
$461$	−24.2918	−1.13138	−0.565691	+	0.824617i	$0.691390\pi$
$462$	0	0
$463$	−40.0689	−1.86216	−0.931079	−	0.364816i	$-0.881132\pi$
$463$	−40.0689	−1.86216	−0.931079	+	0.364816i	$0.881132\pi$
$464$	0	0
$465$	1.18034	0.0547370
$466$	0	0
$467$	−34.0902	−1.57750	−0.788752	−	0.614711i	$-0.789273\pi$
$467$	−34.0902	−1.57750	−0.788752	+	0.614711i	$0.789273\pi$
$468$	0	0
$469$	−12.6869	−0.585827
$470$	0	0
$471$	0.708204	0.0326323
$472$	0	0
$473$	−10.7082	−0.492364
$474$	0	0
$475$	−21.1803	−0.971821
$476$	0	0
$477$	9.85410	0.451188
$478$	0	0
$479$	30.8541	1.40976	0.704880	−	0.709327i	$-0.251001\pi$
$479$	30.8541	1.40976	0.704880	+	0.709327i	$0.251001\pi$
$480$	0	0
$481$	−3.61803	−0.164968
$482$	0	0
$483$	8.61803	0.392134
$484$	0	0
$485$	14.2705	0.647990
$486$	0	0
$487$	12.2918	0.556994	0.278497	−	0.960437i	$-0.410164\pi$
$487$	12.2918	0.556994	0.278497	+	0.960437i	$0.410164\pi$
$488$	0	0
$489$	−9.52786	−0.430865
$490$	0	0
$491$	−8.96556	−0.404610	−0.202305	−	0.979323i	$-0.564843\pi$
$491$	−8.96556	−0.404610	−0.202305	+	0.979323i	$0.564843\pi$
$492$	0	0
$493$	17.2361	0.776273
$494$	0	0
$495$	9.47214	0.425741
$496$	0	0
$497$	7.49342	0.336126
$498$	0	0
$499$	9.41641	0.421536	0.210768	−	0.977536i	$-0.432403\pi$
$499$	9.41641	0.421536	0.210768	+	0.977536i	$0.432403\pi$
$500$	0	0
$501$	−6.85410	−0.306219
$502$	0	0
$503$	0.437694	0.0195158	0.00975791	−	0.999952i	$-0.496894\pi$
$503$	0.437694	0.0195158	0.00975791	+	0.999952i	$0.496894\pi$
$504$	0	0
$505$	33.0902	1.47249
$506$	0	0
$507$	3.05573	0.135710
$508$	0	0
$509$	−43.9787	−1.94932	−0.974661	−	0.223687i	$-0.928191\pi$
$509$	−43.9787	−1.94932	−0.974661	+	0.223687i	$0.928191\pi$
$510$	0	0
$511$	−37.9787	−1.68008
$512$	0	0
$513$	9.09017	0.401341
$514$	0	0
$515$	0.201626	0.00888471
$516$	0	0
$517$	5.94427	0.261429
$518$	0	0
$519$	0.201626	0.00885040
$520$	0	0
$521$	−37.3607	−1.63680	−0.818401	−	0.574648i	$-0.805139\pi$
$521$	−37.3607	−1.63680	−0.818401	+	0.574648i	$0.805139\pi$
$522$	0	0
$523$	−3.61803	−0.158206	−0.0791028	−	0.996866i	$-0.525206\pi$
$523$	−3.61803	−0.158206	−0.0791028	+	0.996866i	$0.525206\pi$
$524$	0	0
$525$	−11.9098	−0.519788
$526$	0	0
$527$	3.29180	0.143393
$528$	0	0
$529$	11.2705	0.490022
$530$	0	0
$531$	−13.3262	−0.578309
$532$	0	0
$533$	12.7082	0.550453
$534$	0	0
$535$	23.7426	1.02648
$536$	0	0
$537$	−4.29180	−0.185205
$538$	0	0
$539$	−1.32624	−0.0571251
$540$	0	0
$541$	−42.2361	−1.81587	−0.907935	−	0.419111i	$-0.862342\pi$
$541$	−42.2361	−1.81587	−0.907935	+	0.419111i	$0.862342\pi$
$542$	0	0
$543$	−10.6180	−0.455663
$544$	0	0
$545$	−8.09017	−0.346545
$546$	0	0
$547$	27.2705	1.16600	0.583001	−	0.812471i	$-0.301878\pi$
$547$	27.2705	1.16600	0.583001	+	0.812471i	$0.301878\pi$
$548$	0	0
$549$	37.2705	1.59067
$550$	0	0
$551$	7.23607	0.308267
$552$	0	0
$553$	35.0344	1.48982
$554$	0	0
$555$	−1.90983	−0.0810678
$556$	0	0
$557$	−43.6525	−1.84961	−0.924807	−	0.380436i	$-0.875774\pi$
$557$	−43.6525	−1.84961	−0.924807	+	0.380436i	$0.875774\pi$
$558$	0	0
$559$	45.3607	1.91855
$560$	0	0
$561$	−3.85410	−0.162720
$562$	0	0
$563$	12.5967	0.530890	0.265445	−	0.964126i	$-0.414481\pi$
$563$	12.5967	0.530890	0.265445	+	0.964126i	$0.414481\pi$
$564$	0	0
$565$	−25.1246	−1.05700
$566$	0	0
$567$	−13.5967	−0.571010
$568$	0	0
$569$	30.0902	1.26145	0.630723	−	0.776008i	$-0.282759\pi$
$569$	30.0902	1.26145	0.630723	+	0.776008i	$0.282759\pi$
$570$	0	0
$571$	−18.2918	−0.765488	−0.382744	−	0.923854i	$-0.625021\pi$
$571$	−18.2918	−0.765488	−0.382744	+	0.923854i	$0.625021\pi$
$572$	0	0
$573$	−0.437694	−0.0182849
$574$	0	0
$575$	−47.3607	−1.97508
$576$	0	0
$577$	−5.67376	−0.236202	−0.118101	−	0.993002i	$-0.537681\pi$
$577$	−5.67376	−0.236202	−0.118101	+	0.993002i	$0.537681\pi$
$578$	0	0
$579$	−1.05573	−0.0438746
$580$	0	0
$581$	10.8673	0.450850
$582$	0	0
$583$	−3.76393	−0.155886
$584$	0	0
$585$	−40.1246	−1.65895
$586$	0	0
$587$	−20.1246	−0.830632	−0.415316	−	0.909677i	$-0.636329\pi$
$587$	−20.1246	−0.830632	−0.415316	+	0.909677i	$0.636329\pi$
$588$	0	0
$589$	1.38197	0.0569429
$590$	0	0
$591$	4.47214	0.183959
$592$	0	0
$593$	−0.270510	−0.0111085	−0.00555425	−	0.999985i	$-0.501768\pi$
$593$	−0.270510	−0.0111085	−0.00555425	+	0.999985i	$0.501768\pi$
$594$	0	0
$595$	−53.7426	−2.20323
$596$	0	0
$597$	6.32624	0.258916
$598$	0	0
$599$	1.58359	0.0647038	0.0323519	−	0.999477i	$-0.489700\pi$
$599$	1.58359	0.0647038	0.0323519	+	0.999477i	$0.489700\pi$
$600$	0	0
$601$	−39.1246	−1.59593	−0.797963	−	0.602706i	$-0.794089\pi$
$601$	−39.1246	−1.59593	−0.797963	+	0.602706i	$0.794089\pi$
$602$	0	0
$603$	−13.9443	−0.567855
$604$	0	0
$605$	−3.61803	−0.147094
$606$	0	0
$607$	−45.1033	−1.83069	−0.915344	−	0.402673i	$-0.868081\pi$
$607$	−45.1033	−1.83069	−0.915344	+	0.402673i	$0.868081\pi$
$608$	0	0
$609$	4.06888	0.164879
$610$	0	0
$611$	−25.1803	−1.01869
$612$	0	0
$613$	0.180340	0.00728386	0.00364193	−	0.999993i	$-0.498841\pi$
$613$	0.180340	0.00728386	0.00364193	+	0.999993i	$0.498841\pi$
$614$	0	0
$615$	6.70820	0.270501
$616$	0	0
$617$	17.5279	0.705645	0.352823	−	0.935690i	$-0.385222\pi$
$617$	17.5279	0.705645	0.352823	+	0.935690i	$0.385222\pi$
$618$	0	0
$619$	42.8885	1.72384	0.861918	−	0.507048i	$-0.169263\pi$
$619$	42.8885	1.72384	0.861918	+	0.507048i	$0.169263\pi$
$620$	0	0
$621$	20.3262	0.815664
$622$	0	0
$623$	−16.8885	−0.676625
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.61803	−0.0646181
$628$	0	0
$629$	−5.32624	−0.212371
$630$	0	0
$631$	−26.2361	−1.04444	−0.522221	−	0.852810i	$-0.674896\pi$
$631$	−26.2361	−1.04444	−0.522221	+	0.852810i	$0.674896\pi$
$632$	0	0
$633$	−0.673762	−0.0267796
$634$	0	0
$635$	30.8541	1.22441
$636$	0	0
$637$	5.61803	0.222595
$638$	0	0
$639$	8.23607	0.325814
$640$	0	0
$641$	33.3607	1.31767	0.658834	−	0.752289i	$-0.271050\pi$
$641$	33.3607	1.31767	0.658834	+	0.752289i	$0.271050\pi$
$642$	0	0
$643$	35.7082	1.40819	0.704097	−	0.710104i	$-0.251352\pi$
$643$	35.7082	1.40819	0.704097	+	0.710104i	$0.251352\pi$
$644$	0	0
$645$	23.9443	0.942805
$646$	0	0
$647$	7.58359	0.298142	0.149071	−	0.988827i	$-0.452372\pi$
$647$	7.58359	0.298142	0.149071	+	0.988827i	$0.452372\pi$
$648$	0	0
$649$	5.09017	0.199807
$650$	0	0
$651$	0.777088	0.0304565
$652$	0	0
$653$	−13.4721	−0.527205	−0.263603	−	0.964631i	$-0.584911\pi$
$653$	−13.4721	−0.527205	−0.263603	+	0.964631i	$0.584911\pi$
$654$	0	0
$655$	−8.29180	−0.323987
$656$	0	0
$657$	−41.7426	−1.62854
$658$	0	0
$659$	16.9098	0.658713	0.329357	−	0.944206i	$-0.393168\pi$
$659$	16.9098	0.658713	0.329357	+	0.944206i	$0.393168\pi$
$660$	0	0
$661$	−21.9230	−0.852705	−0.426353	−	0.904557i	$-0.640202\pi$
$661$	−21.9230	−0.852705	−0.426353	+	0.904557i	$0.640202\pi$
$662$	0	0
$663$	16.3262	0.634059
$664$	0	0
$665$	−22.5623	−0.874929
$666$	0	0
$667$	16.1803	0.626505
$668$	0	0
$669$	−4.25735	−0.164599
$670$	0	0
$671$	−14.2361	−0.549577
$672$	0	0
$673$	−3.47214	−0.133841	−0.0669205	−	0.997758i	$-0.521317\pi$
$673$	−3.47214	−0.133841	−0.0669205	+	0.997758i	$0.521317\pi$
$674$	0	0
$675$	−28.0902	−1.08119
$676$	0	0
$677$	38.4721	1.47860	0.739302	−	0.673374i	$-0.235156\pi$
$677$	38.4721	1.47860	0.739302	+	0.673374i	$0.235156\pi$
$678$	0	0
$679$	9.39512	0.360552
$680$	0	0
$681$	17.9098	0.686306
$682$	0	0
$683$	−9.81966	−0.375739	−0.187869	−	0.982194i	$-0.560158\pi$
$683$	−9.81966	−0.375739	−0.187869	+	0.982194i	$0.560158\pi$
$684$	0	0
$685$	−3.61803	−0.138238
$686$	0	0
$687$	6.76393	0.258060
$688$	0	0
$689$	15.9443	0.607428
$690$	0	0
$691$	−11.2705	−0.428750	−0.214375	−	0.976751i	$-0.568772\pi$
$691$	−11.2705	−0.428750	−0.214375	+	0.976751i	$0.568772\pi$
$692$	0	0
$693$	6.23607	0.236889
$694$	0	0
$695$	−4.67376	−0.177286
$696$	0	0
$697$	18.7082	0.708624
$698$	0	0
$699$	−9.50658	−0.359572
$700$	0	0
$701$	−40.8885	−1.54434	−0.772169	−	0.635417i	$-0.780828\pi$
$701$	−40.8885	−1.54434	−0.772169	+	0.635417i	$0.780828\pi$
$702$	0	0
$703$	−2.23607	−0.0843349
$704$	0	0
$705$	−13.2918	−0.500598
$706$	0	0
$707$	21.7852	0.819317
$708$	0	0
$709$	12.5623	0.471787	0.235894	−	0.971779i	$-0.424198\pi$
$709$	12.5623	0.471787	0.235894	+	0.971779i	$0.424198\pi$
$710$	0	0
$711$	38.5066	1.44411
$712$	0	0
$713$	3.09017	0.115728
$714$	0	0
$715$	15.3262	0.573169
$716$	0	0
$717$	−0.618034	−0.0230809
$718$	0	0
$719$	−33.7984	−1.26047	−0.630233	−	0.776406i	$-0.717041\pi$
$719$	−33.7984	−1.26047	−0.630233	+	0.776406i	$0.717041\pi$
$720$	0	0
$721$	0.132742	0.00494359
$722$	0	0
$723$	−7.88854	−0.293378
$724$	0	0
$725$	−22.3607	−0.830455
$726$	0	0
$727$	1.56231	0.0579427	0.0289714	−	0.999580i	$-0.490777\pi$
$727$	1.56231	0.0579427	0.0289714	+	0.999580i	$0.490777\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	66.7771	2.46984
$732$	0	0
$733$	38.2148	1.41150	0.705748	−	0.708463i	$-0.250611\pi$
$733$	38.2148	1.41150	0.705748	+	0.708463i	$0.250611\pi$
$734$	0	0
$735$	2.96556	0.109386
$736$	0	0
$737$	5.32624	0.196194
$738$	0	0
$739$	−40.5623	−1.49211	−0.746054	−	0.665885i	$-0.768054\pi$
$739$	−40.5623	−1.49211	−0.746054	+	0.665885i	$0.768054\pi$
$740$	0	0
$741$	6.85410	0.251792
$742$	0	0
$743$	52.8328	1.93825	0.969124	−	0.246574i	$-0.0793048\pi$
$743$	52.8328	1.93825	0.969124	+	0.246574i	$0.0793048\pi$
$744$	0	0
$745$	−33.7426	−1.23623
$746$	0	0
$747$	11.9443	0.437018
$748$	0	0
$749$	15.6312	0.571151
$750$	0	0
$751$	−9.36068	−0.341576	−0.170788	−	0.985308i	$-0.554631\pi$
$751$	−9.36068	−0.341576	−0.170788	+	0.985308i	$0.554631\pi$
$752$	0	0
$753$	2.34752	0.0855485
$754$	0	0
$755$	37.3607	1.35969
$756$	0	0
$757$	−12.7295	−0.462661	−0.231331	−	0.972875i	$-0.574308\pi$
$757$	−12.7295	−0.462661	−0.231331	+	0.972875i	$0.574308\pi$
$758$	0	0
$759$	−3.61803	−0.131326
$760$	0	0
$761$	−1.18034	−0.0427873	−0.0213936	−	0.999771i	$-0.506810\pi$
$761$	−1.18034	−0.0427873	−0.0213936	+	0.999771i	$0.506810\pi$
$762$	0	0
$763$	−5.32624	−0.192823
$764$	0	0
$765$	−59.0689	−2.13564
$766$	0	0
$767$	−21.5623	−0.778570
$768$	0	0
$769$	20.7082	0.746757	0.373378	−	0.927679i	$-0.378199\pi$
$769$	20.7082	0.746757	0.373378	+	0.927679i	$0.378199\pi$
$770$	0	0
$771$	−0.236068	−0.00850178
$772$	0	0
$773$	−18.7426	−0.674126	−0.337063	−	0.941482i	$-0.609434\pi$
$773$	−18.7426	−0.674126	−0.337063	+	0.941482i	$0.609434\pi$
$774$	0	0
$775$	−4.27051	−0.153401
$776$	0	0
$777$	−1.25735	−0.0451073
$778$	0	0
$779$	7.85410	0.281402
$780$	0	0
$781$	−3.14590	−0.112569
$782$	0	0
$783$	9.59675	0.342960
$784$	0	0
$785$	−4.14590	−0.147973
$786$	0	0
$787$	−20.4164	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$787$	−20.4164	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$788$	0	0
$789$	−4.38197	−0.156002
$790$	0	0
$791$	−16.5410	−0.588131
$792$	0	0
$793$	60.3050	2.14149
$794$	0	0
$795$	8.41641	0.298499
$796$	0	0
$797$	−39.3050	−1.39225	−0.696126	−	0.717919i	$-0.745095\pi$
$797$	−39.3050	−1.39225	−0.696126	+	0.717919i	$0.745095\pi$
$798$	0	0
$799$	−37.0689	−1.31140
$800$	0	0
$801$	−18.5623	−0.655867
$802$	0	0
$803$	15.9443	0.562661
$804$	0	0
$805$	−50.4508	−1.77816
$806$	0	0
$807$	3.67376	0.129323
$808$	0	0
$809$	48.1033	1.69122	0.845611	−	0.533799i	$-0.179236\pi$
$809$	48.1033	1.69122	0.845611	+	0.533799i	$0.179236\pi$
$810$	0	0
$811$	−46.7426	−1.64136	−0.820678	−	0.571391i	$-0.806404\pi$
$811$	−46.7426	−1.64136	−0.820678	+	0.571391i	$0.806404\pi$
$812$	0	0
$813$	−2.85410	−0.100098
$814$	0	0
$815$	55.7771	1.95379
$816$	0	0
$817$	28.0344	0.980801
$818$	0	0
$819$	−26.4164	−0.923064
$820$	0	0
$821$	−15.8754	−0.554055	−0.277027	−	0.960862i	$-0.589349\pi$
$821$	−15.8754	−0.554055	−0.277027	+	0.960862i	$0.589349\pi$
$822$	0	0
$823$	3.20163	0.111602	0.0558008	−	0.998442i	$-0.482229\pi$
$823$	3.20163	0.111602	0.0558008	+	0.998442i	$0.482229\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	−14.5066	−0.504443	−0.252222	−	0.967670i	$-0.581161\pi$
$827$	−14.5066	−0.504443	−0.252222	+	0.967670i	$0.581161\pi$
$828$	0	0
$829$	−46.7771	−1.62464	−0.812318	−	0.583215i	$-0.801794\pi$
$829$	−46.7771	−1.62464	−0.812318	+	0.583215i	$0.801794\pi$
$830$	0	0
$831$	13.3820	0.464215
$832$	0	0
$833$	8.27051	0.286556
$834$	0	0
$835$	40.1246	1.38857
$836$	0	0
$837$	1.83282	0.0633514
$838$	0	0
$839$	32.3050	1.11529	0.557645	−	0.830079i	$-0.311705\pi$
$839$	32.3050	1.11529	0.557645	+	0.830079i	$0.311705\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	−3.56231	−0.122692
$844$	0	0
$845$	−17.8885	−0.615385
$846$	0	0
$847$	−2.38197	−0.0818453
$848$	0	0
$849$	−9.56231	−0.328177
$850$	0	0
$851$	−5.00000	−0.171398
$852$	0	0
$853$	16.3050	0.558271	0.279135	−	0.960252i	$-0.409952\pi$
$853$	16.3050	0.558271	0.279135	+	0.960252i	$0.409952\pi$
$854$	0	0
$855$	−24.7984	−0.848086
$856$	0	0
$857$	−18.1246	−0.619125	−0.309562	−	0.950879i	$-0.600183\pi$
$857$	−18.1246	−0.619125	−0.309562	+	0.950879i	$0.600183\pi$
$858$	0	0
$859$	−12.4377	−0.424369	−0.212184	−	0.977230i	$-0.568058\pi$
$859$	−12.4377	−0.424369	−0.212184	+	0.977230i	$0.568058\pi$
$860$	0	0
$861$	4.41641	0.150511
$862$	0	0
$863$	−26.0689	−0.887395	−0.443698	−	0.896177i	$-0.646334\pi$
$863$	−26.0689	−0.887395	−0.443698	+	0.896177i	$0.646334\pi$
$864$	0	0
$865$	−1.18034	−0.0401328
$866$	0	0
$867$	13.5279	0.459430
$868$	0	0
$869$	−14.7082	−0.498942
$870$	0	0
$871$	−22.5623	−0.764495
$872$	0	0
$873$	10.3262	0.349490
$874$	0	0
$875$	26.6312	0.900299
$876$	0	0
$877$	−48.1803	−1.62693	−0.813467	−	0.581611i	$-0.802423\pi$
$877$	−48.1803	−1.62693	−0.813467	+	0.581611i	$0.802423\pi$
$878$	0	0
$879$	5.38197	0.181529
$880$	0	0
$881$	10.4377	0.351655	0.175827	−	0.984421i	$-0.443740\pi$
$881$	10.4377	0.351655	0.175827	+	0.984421i	$0.443740\pi$
$882$	0	0
$883$	−2.34752	−0.0790005	−0.0395002	−	0.999220i	$-0.512577\pi$
$883$	−2.34752	−0.0790005	−0.0395002	+	0.999220i	$0.512577\pi$
$884$	0	0
$885$	−11.3820	−0.382601
$886$	0	0
$887$	−1.97871	−0.0664387	−0.0332194	−	0.999448i	$-0.510576\pi$
$887$	−1.97871	−0.0664387	−0.0332194	+	0.999448i	$0.510576\pi$
$888$	0	0
$889$	20.3131	0.681279
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	−15.5623	−0.520773
$894$	0	0
$895$	25.1246	0.839823
$896$	0	0
$897$	15.3262	0.511728
$898$	0	0
$899$	1.45898	0.0486597
$900$	0	0
$901$	23.4721	0.781970
$902$	0	0
$903$	15.7639	0.524591
$904$	0	0
$905$	62.1591	2.06624
$906$	0	0
$907$	38.2148	1.26890	0.634451	−	0.772963i	$-0.281226\pi$
$907$	38.2148	1.26890	0.634451	+	0.772963i	$0.281226\pi$
$908$	0	0
$909$	23.9443	0.794181
$910$	0	0
$911$	49.3050	1.63355	0.816773	−	0.576959i	$-0.195761\pi$
$911$	49.3050	1.63355	0.816773	+	0.576959i	$0.195761\pi$
$912$	0	0
$913$	−4.56231	−0.150990
$914$	0	0
$915$	31.8328	1.05236
$916$	0	0
$917$	−5.45898	−0.180271
$918$	0	0
$919$	21.8885	0.722036	0.361018	−	0.932559i	$-0.382429\pi$
$919$	21.8885	0.722036	0.361018	+	0.932559i	$0.382429\pi$
$920$	0	0
$921$	2.85410	0.0940459
$922$	0	0
$923$	13.3262	0.438638
$924$	0	0
$925$	6.90983	0.227194
$926$	0	0
$927$	0.145898	0.00479192
$928$	0	0
$929$	58.3951	1.91588	0.957941	−	0.286966i	$-0.0926465\pi$
$929$	58.3951	1.91588	0.957941	+	0.286966i	$0.0926465\pi$
$930$	0	0
$931$	3.47214	0.113795
$932$	0	0
$933$	−18.1803	−0.595198
$934$	0	0
$935$	22.5623	0.737866
$936$	0	0
$937$	16.4721	0.538121	0.269061	−	0.963123i	$-0.413287\pi$
$937$	16.4721	0.538121	0.269061	+	0.963123i	$0.413287\pi$
$938$	0	0
$939$	13.1459	0.429000
$940$	0	0
$941$	42.5623	1.38749	0.693746	−	0.720220i	$-0.255959\pi$
$941$	42.5623	1.38749	0.693746	+	0.720220i	$0.255959\pi$
$942$	0	0
$943$	17.5623	0.571907
$944$	0	0
$945$	−29.9230	−0.973395
$946$	0	0
$947$	−35.4164	−1.15088	−0.575439	−	0.817844i	$-0.695169\pi$
$947$	−35.4164	−1.15088	−0.575439	+	0.817844i	$0.695169\pi$
$948$	0	0
$949$	−67.5410	−2.19247
$950$	0	0
$951$	11.1246	0.360740
$952$	0	0
$953$	22.5410	0.730175	0.365088	−	0.930973i	$-0.381039\pi$
$953$	22.5410	0.730175	0.365088	+	0.930973i	$0.381039\pi$
$954$	0	0
$955$	2.56231	0.0829143
$956$	0	0
$957$	−1.70820	−0.0552184
$958$	0	0
$959$	−2.38197	−0.0769177
$960$	0	0
$961$	−30.7214	−0.991012
$962$	0	0
$963$	17.1803	0.553629
$964$	0	0
$965$	6.18034	0.198952
$966$	0	0
$967$	−39.8885	−1.28273	−0.641365	−	0.767236i	$-0.721631\pi$
$967$	−39.8885	−1.28273	−0.641365	+	0.767236i	$0.721631\pi$
$968$	0	0
$969$	10.0902	0.324143
$970$	0	0
$971$	−4.05573	−0.130155	−0.0650773	−	0.997880i	$-0.520729\pi$
$971$	−4.05573	−0.130155	−0.0650773	+	0.997880i	$0.520729\pi$
$972$	0	0
$973$	−3.07701	−0.0986446
$974$	0	0
$975$	−21.1803	−0.678314
$976$	0	0
$977$	15.0000	0.479893	0.239946	−	0.970786i	$-0.422870\pi$
$977$	15.0000	0.479893	0.239946	+	0.970786i	$0.422870\pi$
$978$	0	0
$979$	7.09017	0.226603
$980$	0	0
$981$	−5.85410	−0.186907
$982$	0	0
$983$	−12.8197	−0.408884	−0.204442	−	0.978879i	$-0.565538\pi$
$983$	−12.8197	−0.408884	−0.204442	+	0.978879i	$0.565538\pi$
$984$	0	0
$985$	−26.1803	−0.834175
$986$	0	0
$987$	−8.75078	−0.278540
$988$	0	0
$989$	62.6869	1.99333
$990$	0	0
$991$	−54.5967	−1.73432	−0.867161	−	0.498027i	$-0.834058\pi$
$991$	−54.5967	−1.73432	−0.867161	+	0.498027i	$0.834058\pi$
$992$	0	0
$993$	−10.5279	−0.334092
$994$	0	0
$995$	−37.0344	−1.17407
$996$	0	0
$997$	11.0689	0.350555	0.175278	−	0.984519i	$-0.443918\pi$
$997$	11.0689	0.350555	0.175278	+	0.984519i	$0.443918\pi$
$998$	0	0
$999$	−2.96556	−0.0938261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.a.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.a.1.2	✓	2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} -3.61803 q^{5} -2.38197 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q+0.618034 q^{3} -3.61803 q^{5} -2.38197 q^{7} -2.61803 q^{9} +1.00000 q^{11} -4.23607 q^{13} -2.23607 q^{15} -6.23607 q^{17} -2.61803 q^{19} -1.47214 q^{21} -5.85410 q^{23} +8.09017 q^{25} -3.47214 q^{27} -2.76393 q^{29} -0.527864 q^{31} +0.618034 q^{33} +8.61803 q^{35} +0.854102 q^{37} -2.61803 q^{39} -3.00000 q^{41} -10.7082 q^{43} +9.47214 q^{45} +5.94427 q^{47} -1.32624 q^{49} -3.85410 q^{51} -3.76393 q^{53} -3.61803 q^{55} -1.61803 q^{57} +5.09017 q^{59} -14.2361 q^{61} +6.23607 q^{63} +15.3262 q^{65} +5.32624 q^{67} -3.61803 q^{69} -3.14590 q^{71} +15.9443 q^{73} +5.00000 q^{75} -2.38197 q^{77} -14.7082 q^{79} +5.70820 q^{81} -4.56231 q^{83} +22.5623 q^{85} -1.70820 q^{87} +7.09017 q^{89} +10.0902 q^{91} -0.326238 q^{93} +9.47214 q^{95} -3.94427 q^{97} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9	\( 2 q - q^{3} - 5 q^{5} - 7 q^{7} - 3 q^{9} + 2 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} + 6 q^{21} - 5 q^{23} + 5 q^{25} + 2 q^{27} - 10 q^{29} - 10 q^{31} - q^{33} + 15 q^{35} - 5 q^{37} - 3 q^{39} - 6 q^{41} - 8 q^{43} + 10 q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 12 q^{53} - 5 q^{55} - q^{57} - q^{59} - 24 q^{61} + 8 q^{63} + 15 q^{65} - 5 q^{67} - 5 q^{69} - 13 q^{71} + 14 q^{73} + 10 q^{75} - 7 q^{77} - 16 q^{79} - 2 q^{81} + 11 q^{83} + 25 q^{85} + 10 q^{87} + 3 q^{89} + 9 q^{91} + 15 q^{93} + 10 q^{95} + 10 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - 5 * q^5 - 7 * q^7 - 3 * q^9 + 2 * q^11 - 4 * q^13 - 8 * q^17 - 3 * q^19 + 6 * q^21 - 5 * q^23 + 5 * q^25 + 2 * q^27 - 10 * q^29 - 10 * q^31 - q^33 + 15 * q^35 - 5 * q^37 - 3 * q^39 - 6 * q^41 - 8 * q^43 + 10 * q^45 - 6 * q^47 + 13 * q^49 - q^51 - 12 * q^53 - 5 * q^55 - q^57 - q^59 - 24 * q^61 + 8 * q^63 + 15 * q^65 - 5 * q^67 - 5 * q^69 - 13 * q^71 + 14 * q^73 + 10 * q^75 - 7 * q^77 - 16 * q^79 - 2 * q^81 + 11 * q^83 + 25 * q^85 + 10 * q^87 + 3 * q^89 + 9 * q^91 + 15 * q^93 + 10 * q^95 + 10 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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