LMFDB - Embedded newform 6384.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
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:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +5.23607 q^{13} +1.23607 q^{15} +2.47214 q^{17} +1.00000 q^{19} -1.00000 q^{21} +5.70820 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -7.70820 q^{31} +2.00000 q^{33} -1.23607 q^{35} +0.763932 q^{37} +5.23607 q^{39} -6.94427 q^{41} -1.52786 q^{43} +1.23607 q^{45} -2.76393 q^{47} +1.00000 q^{49} +2.47214 q^{51} +8.47214 q^{53} +2.47214 q^{55} +1.00000 q^{57} +12.9443 q^{59} +12.4721 q^{61} -1.00000 q^{63} +6.47214 q^{65} -10.9443 q^{67} +5.70820 q^{69} -14.4721 q^{71} +6.00000 q^{73} -3.47214 q^{75} -2.00000 q^{77} -12.1803 q^{79} +1.00000 q^{81} -8.94427 q^{83} +3.05573 q^{85} +8.47214 q^{87} -2.94427 q^{89} -5.23607 q^{91} -7.70820 q^{93} +1.23607 q^{95} +10.4721 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{31} + 4 q^{33} + 2 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 4 q^{65} - 4 q^{67} - 2 q^{69} - 20 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} - 2 q^{79} + 2 q^{81} + 24 q^{85} + 8 q^{87} + 12 q^{89} - 6 q^{91} - 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^31 + 4 * q^33 + 2 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 12 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 + 16 * q^61 - 2 * q^63 + 4 * q^65 - 4 * q^67 - 2 * q^69 - 20 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 - 2 * q^79 + 2 * q^81 + 24 * q^85 + 8 * q^87 + 12 * q^89 - 6 * q^91 - 2 * q^93 - 2 * q^95 + 12 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0	0
$15$	1.23607	0.319151
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	−7.70820	−1.38443	−0.692217	−	0.721689i	$-0.743366\pi$
$31$	−7.70820	−1.38443	−0.692217	+	0.721689i	$0.743366\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−1.23607	−0.208934
$36$	0	0
$37$	0.763932	0.125590	0.0627948	−	0.998026i	$-0.479999\pi$
$37$	0.763932	0.125590	0.0627948	+	0.998026i	$0.479999\pi$
$38$	0	0
$39$	5.23607	0.838442
$40$	0	0
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	1.23607	0.184262
$46$	0	0
$47$	−2.76393	−0.403161	−0.201580	−	0.979472i	$-0.564608\pi$
$47$	−2.76393	−0.403161	−0.201580	+	0.979472i	$0.564608\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	12.9443	1.68520	0.842600	−	0.538539i	$-0.181024\pi$
$59$	12.9443	1.68520	0.842600	+	0.538539i	$0.181024\pi$
$60$	0	0
$61$	12.4721	1.59689	0.798447	−	0.602066i	$-0.205655\pi$
$61$	12.4721	1.59689	0.798447	+	0.602066i	$0.205655\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	6.47214	0.802770
$66$	0	0
$67$	−10.9443	−1.33706	−0.668528	−	0.743687i	$-0.733075\pi$
$67$	−10.9443	−1.33706	−0.668528	+	0.743687i	$0.733075\pi$
$68$	0	0
$69$	5.70820	0.687187
$70$	0	0
$71$	−14.4721	−1.71753	−0.858763	−	0.512373i	$-0.828767\pi$
$71$	−14.4721	−1.71753	−0.858763	+	0.512373i	$0.828767\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−12.1803	−1.37040	−0.685198	−	0.728357i	$-0.740284\pi$
$79$	−12.1803	−1.37040	−0.685198	+	0.728357i	$0.740284\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.94427	−0.981761	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427	−0.981761	−0.490881	+	0.871227i	$0.663325\pi$
$84$	0	0
$85$	3.05573	0.331440
$86$	0	0
$87$	8.47214	0.908308
$88$	0	0
$89$	−2.94427	−0.312092	−0.156046	−	0.987750i	$-0.549875\pi$
$89$	−2.94427	−0.312092	−0.156046	+	0.987750i	$0.549875\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	0	0
$93$	−7.70820	−0.799304
$94$	0	0
$95$	1.23607	0.126818
$96$	0	0
$97$	10.4721	1.06328	0.531642	−	0.846969i	$-0.321575\pi$
$97$	10.4721	1.06328	0.531642	+	0.846969i	$0.321575\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−19.7082	−1.96104	−0.980520	−	0.196420i	$-0.937068\pi$
$101$	−19.7082	−1.96104	−0.980520	+	0.196420i	$0.937068\pi$
$102$	0	0
$103$	12.6525	1.24669	0.623343	−	0.781949i	$-0.285774\pi$
$103$	12.6525	1.24669	0.623343	+	0.781949i	$0.285774\pi$
$104$	0	0
$105$	−1.23607	−0.120628
$106$	0	0
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−1.70820	−0.163616	−0.0818081	−	0.996648i	$-0.526069\pi$
$109$	−1.70820	−0.163616	−0.0818081	+	0.996648i	$0.526069\pi$
$110$	0	0
$111$	0.763932	0.0725092
$112$	0	0
$113$	12.4721	1.17328	0.586640	−	0.809848i	$-0.300450\pi$
$113$	12.4721	1.17328	0.586640	+	0.809848i	$0.300450\pi$
$114$	0	0
$115$	7.05573	0.657950
$116$	0	0
$117$	5.23607	0.484075
$118$	0	0
$119$	−2.47214	−0.226620
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.94427	−0.626144
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−21.7082	−1.92629	−0.963146	−	0.268980i	$-0.913313\pi$
$127$	−21.7082	−1.92629	−0.963146	+	0.268980i	$0.913313\pi$
$128$	0	0
$129$	−1.52786	−0.134521
$130$	0	0
$131$	3.05573	0.266980	0.133490	−	0.991050i	$-0.457382\pi$
$131$	3.05573	0.266980	0.133490	+	0.991050i	$0.457382\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	1.23607	0.106384
$136$	0	0
$137$	−9.41641	−0.804498	−0.402249	−	0.915530i	$-0.631771\pi$
$137$	−9.41641	−0.804498	−0.402249	+	0.915530i	$0.631771\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−2.76393	−0.232765
$142$	0	0
$143$	10.4721	0.875724
$144$	0	0
$145$	10.4721	0.869664
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	12.1803	0.997852	0.498926	−	0.866644i	$-0.333728\pi$
$149$	12.1803	0.997852	0.498926	+	0.866644i	$0.333728\pi$
$150$	0	0
$151$	13.7082	1.11556	0.557779	−	0.829990i	$-0.311654\pi$
$151$	13.7082	1.11556	0.557779	+	0.829990i	$0.311654\pi$
$152$	0	0
$153$	2.47214	0.199860
$154$	0	0
$155$	−9.52786	−0.765296
$156$	0	0
$157$	−4.47214	−0.356915	−0.178458	−	0.983948i	$-0.557111\pi$
$157$	−4.47214	−0.356915	−0.178458	+	0.983948i	$0.557111\pi$
$158$	0	0
$159$	8.47214	0.671884
$160$	0	0
$161$	−5.70820	−0.449869
$162$	0	0
$163$	20.9443	1.64048	0.820241	−	0.572018i	$-0.193839\pi$
$163$	20.9443	1.64048	0.820241	+	0.572018i	$0.193839\pi$
$164$	0	0
$165$	2.47214	0.192456
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	11.8885	0.903869	0.451935	−	0.892051i	$-0.350734\pi$
$173$	11.8885	0.903869	0.451935	+	0.892051i	$0.350734\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	12.9443	0.972951
$178$	0	0
$179$	15.4164	1.15228	0.576138	−	0.817352i	$-0.304559\pi$
$179$	15.4164	1.15228	0.576138	+	0.817352i	$0.304559\pi$
$180$	0	0
$181$	−9.23607	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$181$	−9.23607	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$182$	0	0
$183$	12.4721	0.921967
$184$	0	0
$185$	0.944272	0.0694243
$186$	0	0
$187$	4.94427	0.361561
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	3.81966	0.276381	0.138190	−	0.990406i	$-0.455871\pi$
$191$	3.81966	0.276381	0.138190	+	0.990406i	$0.455871\pi$
$192$	0	0
$193$	−0.472136	−0.0339851	−0.0169925	−	0.999856i	$-0.505409\pi$
$193$	−0.472136	−0.0339851	−0.0169925	+	0.999856i	$0.505409\pi$
$194$	0	0
$195$	6.47214	0.463479
$196$	0	0
$197$	2.29180	0.163284	0.0816419	−	0.996662i	$-0.473984\pi$
$197$	2.29180	0.163284	0.0816419	+	0.996662i	$0.473984\pi$
$198$	0	0
$199$	5.52786	0.391860	0.195930	−	0.980618i	$-0.437227\pi$
$199$	5.52786	0.391860	0.195930	+	0.980618i	$0.437227\pi$
$200$	0	0
$201$	−10.9443	−0.771949
$202$	0	0
$203$	−8.47214	−0.594627
$204$	0	0
$205$	−8.58359	−0.599504
$206$	0	0
$207$	5.70820	0.396748
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	25.4164	1.74974	0.874869	−	0.484360i	$-0.160947\pi$
$211$	25.4164	1.74974	0.874869	+	0.484360i	$0.160947\pi$
$212$	0	0
$213$	−14.4721	−0.991614
$214$	0	0
$215$	−1.88854	−0.128798
$216$	0	0
$217$	7.70820	0.523267
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	12.9443	0.870726
$222$	0	0
$223$	−3.70820	−0.248320	−0.124160	−	0.992262i	$-0.539624\pi$
$223$	−3.70820	−0.248320	−0.124160	+	0.992262i	$0.539624\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	7.05573	0.468305	0.234153	−	0.972200i	$-0.424768\pi$
$227$	7.05573	0.468305	0.234153	+	0.972200i	$0.424768\pi$
$228$	0	0
$229$	−23.8885	−1.57860	−0.789300	−	0.614008i	$-0.789556\pi$
$229$	−23.8885	−1.57860	−0.789300	+	0.614008i	$0.789556\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−3.41641	−0.222862
$236$	0	0
$237$	−12.1803	−0.791198
$238$	0	0
$239$	−22.6525	−1.46527	−0.732633	−	0.680623i	$-0.761709\pi$
$239$	−22.6525	−1.46527	−0.732633	+	0.680623i	$0.761709\pi$
$240$	0	0
$241$	−16.3607	−1.05388	−0.526942	−	0.849901i	$-0.676662\pi$
$241$	−16.3607	−1.05388	−0.526942	+	0.849901i	$0.676662\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.23607	0.0789695
$246$	0	0
$247$	5.23607	0.333163
$248$	0	0
$249$	−8.94427	−0.566820
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	11.4164	0.717743
$254$	0	0
$255$	3.05573	0.191357
$256$	0	0
$257$	−19.8885	−1.24061	−0.620307	−	0.784359i	$-0.712992\pi$
$257$	−19.8885	−1.24061	−0.620307	+	0.784359i	$0.712992\pi$
$258$	0	0
$259$	−0.763932	−0.0474684
$260$	0	0
$261$	8.47214	0.524412
$262$	0	0
$263$	4.18034	0.257771	0.128885	−	0.991659i	$-0.458860\pi$
$263$	4.18034	0.257771	0.128885	+	0.991659i	$0.458860\pi$
$264$	0	0
$265$	10.4721	0.643298
$266$	0	0
$267$	−2.94427	−0.180187
$268$	0	0
$269$	28.4721	1.73598	0.867988	−	0.496584i	$-0.165413\pi$
$269$	28.4721	1.73598	0.867988	+	0.496584i	$0.165413\pi$
$270$	0	0
$271$	−12.9443	−0.786309	−0.393154	−	0.919473i	$-0.628616\pi$
$271$	−12.9443	−0.786309	−0.393154	+	0.919473i	$0.628616\pi$
$272$	0	0
$273$	−5.23607	−0.316901
$274$	0	0
$275$	−6.94427	−0.418755
$276$	0	0
$277$	23.8885	1.43532	0.717662	−	0.696392i	$-0.245212\pi$
$277$	23.8885	1.43532	0.717662	+	0.696392i	$0.245212\pi$
$278$	0	0
$279$	−7.70820	−0.461478
$280$	0	0
$281$	−5.41641	−0.323116	−0.161558	−	0.986863i	$-0.551652\pi$
$281$	−5.41641	−0.323116	−0.161558	+	0.986863i	$0.551652\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	1.23607	0.0732183
$286$	0	0
$287$	6.94427	0.409907
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	10.4721	0.613887
$292$	0	0
$293$	−8.47214	−0.494947	−0.247474	−	0.968895i	$-0.579600\pi$
$293$	−8.47214	−0.494947	−0.247474	+	0.968895i	$0.579600\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	29.8885	1.72850
$300$	0	0
$301$	1.52786	0.0880646
$302$	0	0
$303$	−19.7082	−1.13221
$304$	0	0
$305$	15.4164	0.882741
$306$	0	0
$307$	−5.52786	−0.315492	−0.157746	−	0.987480i	$-0.550423\pi$
$307$	−5.52786	−0.315492	−0.157746	+	0.987480i	$0.550423\pi$
$308$	0	0
$309$	12.6525	0.719774
$310$	0	0
$311$	17.2361	0.977368	0.488684	−	0.872461i	$-0.337477\pi$
$311$	17.2361	0.977368	0.488684	+	0.872461i	$0.337477\pi$
$312$	0	0
$313$	30.0000	1.69570	0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000	1.69570	0.847850	+	0.530236i	$0.177897\pi$
$314$	0	0
$315$	−1.23607	−0.0696445
$316$	0	0
$317$	12.4721	0.700505	0.350252	−	0.936655i	$-0.386096\pi$
$317$	12.4721	0.700505	0.350252	+	0.936655i	$0.386096\pi$
$318$	0	0
$319$	16.9443	0.948697
$320$	0	0
$321$	8.94427	0.499221
$322$	0	0
$323$	2.47214	0.137553
$324$	0	0
$325$	−18.1803	−1.00846
$326$	0	0
$327$	−1.70820	−0.0944639
$328$	0	0
$329$	2.76393	0.152381
$330$	0	0
$331$	18.9443	1.04127	0.520636	−	0.853779i	$-0.325695\pi$
$331$	18.9443	1.04127	0.520636	+	0.853779i	$0.325695\pi$
$332$	0	0
$333$	0.763932	0.0418632
$334$	0	0
$335$	−13.5279	−0.739106
$336$	0	0
$337$	1.05573	0.0575092	0.0287546	−	0.999587i	$-0.490846\pi$
$337$	1.05573	0.0575092	0.0287546	+	0.999587i	$0.490846\pi$
$338$	0	0
$339$	12.4721	0.677393
$340$	0	0
$341$	−15.4164	−0.834845
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.05573	0.379868
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	−22.3607	−1.19694	−0.598470	−	0.801145i	$-0.704224\pi$
$349$	−22.3607	−1.19694	−0.598470	+	0.801145i	$0.704224\pi$
$350$	0	0
$351$	5.23607	0.279481
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	−17.8885	−0.949425
$356$	0	0
$357$	−2.47214	−0.130839
$358$	0	0
$359$	−14.2918	−0.754292	−0.377146	−	0.926154i	$-0.623095\pi$
$359$	−14.2918	−0.754292	−0.377146	+	0.926154i	$0.623095\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	7.41641	0.388193
$366$	0	0
$367$	19.0557	0.994701	0.497350	−	0.867550i	$-0.334306\pi$
$367$	19.0557	0.994701	0.497350	+	0.867550i	$0.334306\pi$
$368$	0	0
$369$	−6.94427	−0.361504
$370$	0	0
$371$	−8.47214	−0.439851
$372$	0	0
$373$	1.70820	0.0884474	0.0442237	−	0.999022i	$-0.485919\pi$
$373$	1.70820	0.0884474	0.0442237	+	0.999022i	$0.485919\pi$
$374$	0	0
$375$	−10.4721	−0.540779
$376$	0	0
$377$	44.3607	2.28469
$378$	0	0
$379$	10.9443	0.562169	0.281085	−	0.959683i	$-0.409306\pi$
$379$	10.9443	0.562169	0.281085	+	0.959683i	$0.409306\pi$
$380$	0	0
$381$	−21.7082	−1.11214
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	0	0
$387$	−1.52786	−0.0776657
$388$	0	0
$389$	−7.23607	−0.366883	−0.183442	−	0.983031i	$-0.558724\pi$
$389$	−7.23607	−0.366883	−0.183442	+	0.983031i	$0.558724\pi$
$390$	0	0
$391$	14.1115	0.713647
$392$	0	0
$393$	3.05573	0.154141
$394$	0	0
$395$	−15.0557	−0.757536
$396$	0	0
$397$	−6.94427	−0.348523	−0.174262	−	0.984699i	$-0.555754\pi$
$397$	−6.94427	−0.348523	−0.174262	+	0.984699i	$0.555754\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−20.4721	−1.02233	−0.511165	−	0.859483i	$-0.670786\pi$
$401$	−20.4721	−1.02233	−0.511165	+	0.859483i	$0.670786\pi$
$402$	0	0
$403$	−40.3607	−2.01051
$404$	0	0
$405$	1.23607	0.0614207
$406$	0	0
$407$	1.52786	0.0757334
$408$	0	0
$409$	5.52786	0.273335	0.136668	−	0.990617i	$-0.456361\pi$
$409$	5.52786	0.273335	0.136668	+	0.990617i	$0.456361\pi$
$410$	0	0
$411$	−9.41641	−0.464477
$412$	0	0
$413$	−12.9443	−0.636946
$414$	0	0
$415$	−11.0557	−0.542704
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	20.3607	0.994684	0.497342	−	0.867554i	$-0.334309\pi$
$419$	20.3607	0.994684	0.497342	+	0.867554i	$0.334309\pi$
$420$	0	0
$421$	−11.2361	−0.547612	−0.273806	−	0.961785i	$-0.588283\pi$
$421$	−11.2361	−0.547612	−0.273806	+	0.961785i	$0.588283\pi$
$422$	0	0
$423$	−2.76393	−0.134387
$424$	0	0
$425$	−8.58359	−0.416365
$426$	0	0
$427$	−12.4721	−0.603569
$428$	0	0
$429$	10.4721	0.505599
$430$	0	0
$431$	9.52786	0.458941	0.229471	−	0.973316i	$-0.426301\pi$
$431$	9.52786	0.458941	0.229471	+	0.973316i	$0.426301\pi$
$432$	0	0
$433$	−13.5279	−0.650108	−0.325054	−	0.945696i	$-0.605382\pi$
$433$	−13.5279	−0.650108	−0.325054	+	0.945696i	$0.605382\pi$
$434$	0	0
$435$	10.4721	0.502100
$436$	0	0
$437$	5.70820	0.273060
$438$	0	0
$439$	−12.6525	−0.603870	−0.301935	−	0.953329i	$-0.597633\pi$
$439$	−12.6525	−0.603870	−0.301935	+	0.953329i	$0.597633\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	11.8885	0.564842	0.282421	−	0.959291i	$-0.408863\pi$
$443$	11.8885	0.564842	0.282421	+	0.959291i	$0.408863\pi$
$444$	0	0
$445$	−3.63932	−0.172520
$446$	0	0
$447$	12.1803	0.576110
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−13.8885	−0.653986
$452$	0	0
$453$	13.7082	0.644068
$454$	0	0
$455$	−6.47214	−0.303418
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	2.47214	0.115389
$460$	0	0
$461$	−25.2361	−1.17536	−0.587680	−	0.809093i	$-0.699959\pi$
$461$	−25.2361	−1.17536	−0.587680	+	0.809093i	$0.699959\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−9.52786	−0.441844
$466$	0	0
$467$	−29.8885	−1.38308	−0.691538	−	0.722340i	$-0.743067\pi$
$467$	−29.8885	−1.38308	−0.691538	+	0.722340i	$0.743067\pi$
$468$	0	0
$469$	10.9443	0.505360
$470$	0	0
$471$	−4.47214	−0.206065
$472$	0	0
$473$	−3.05573	−0.140503
$474$	0	0
$475$	−3.47214	−0.159313
$476$	0	0
$477$	8.47214	0.387912
$478$	0	0
$479$	−1.23607	−0.0564774	−0.0282387	−	0.999601i	$-0.508990\pi$
$479$	−1.23607	−0.0564774	−0.0282387	+	0.999601i	$0.508990\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	−5.70820	−0.259732
$484$	0	0
$485$	12.9443	0.587769
$486$	0	0
$487$	−19.2361	−0.871669	−0.435835	−	0.900027i	$-0.643547\pi$
$487$	−19.2361	−0.871669	−0.435835	+	0.900027i	$0.643547\pi$
$488$	0	0
$489$	20.9443	0.947133
$490$	0	0
$491$	−23.5279	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$491$	−23.5279	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$492$	0	0
$493$	20.9443	0.943283
$494$	0	0
$495$	2.47214	0.111114
$496$	0	0
$497$	14.4721	0.649164
$498$	0	0
$499$	−8.36068	−0.374275	−0.187138	−	0.982334i	$-0.559921\pi$
$499$	−8.36068	−0.374275	−0.187138	+	0.982334i	$0.559921\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	11.1246	0.496022	0.248011	−	0.968757i	$-0.420223\pi$
$503$	11.1246	0.496022	0.248011	+	0.968757i	$0.420223\pi$
$504$	0	0
$505$	−24.3607	−1.08404
$506$	0	0
$507$	14.4164	0.640255
$508$	0	0
$509$	−24.4721	−1.08471	−0.542354	−	0.840150i	$-0.682467\pi$
$509$	−24.4721	−1.08471	−0.542354	+	0.840150i	$0.682467\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	15.6393	0.689151
$516$	0	0
$517$	−5.52786	−0.243115
$518$	0	0
$519$	11.8885	0.521849
$520$	0	0
$521$	−28.8328	−1.26319	−0.631594	−	0.775299i	$-0.717599\pi$
$521$	−28.8328	−1.26319	−0.631594	+	0.775299i	$0.717599\pi$
$522$	0	0
$523$	2.47214	0.108099	0.0540495	−	0.998538i	$-0.482787\pi$
$523$	2.47214	0.108099	0.0540495	+	0.998538i	$0.482787\pi$
$524$	0	0
$525$	3.47214	0.151536
$526$	0	0
$527$	−19.0557	−0.830081
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	12.9443	0.561734
$532$	0	0
$533$	−36.3607	−1.57496
$534$	0	0
$535$	11.0557	0.477981
$536$	0	0
$537$	15.4164	0.665267
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	24.8328	1.06765	0.533823	−	0.845596i	$-0.320755\pi$
$541$	24.8328	1.06765	0.533823	+	0.845596i	$0.320755\pi$
$542$	0	0
$543$	−9.23607	−0.396358
$544$	0	0
$545$	−2.11146	−0.0904448
$546$	0	0
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	0	0
$549$	12.4721	0.532298
$550$	0	0
$551$	8.47214	0.360925
$552$	0	0
$553$	12.1803	0.517961
$554$	0	0
$555$	0.944272	0.0400821
$556$	0	0
$557$	−36.5410	−1.54829	−0.774146	−	0.633007i	$-0.781821\pi$
$557$	−36.5410	−1.54829	−0.774146	+	0.633007i	$0.781821\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	4.94427	0.208747
$562$	0	0
$563$	9.88854	0.416752	0.208376	−	0.978049i	$-0.433182\pi$
$563$	9.88854	0.416752	0.208376	+	0.978049i	$0.433182\pi$
$564$	0	0
$565$	15.4164	0.648573
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	13.0557	0.547325	0.273662	−	0.961826i	$-0.411765\pi$
$569$	13.0557	0.547325	0.273662	+	0.961826i	$0.411765\pi$
$570$	0	0
$571$	26.4721	1.10782	0.553912	−	0.832575i	$-0.313134\pi$
$571$	26.4721	1.10782	0.553912	+	0.832575i	$0.313134\pi$
$572$	0	0
$573$	3.81966	0.159569
$574$	0	0
$575$	−19.8197	−0.826537
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	−0.472136	−0.0196213
$580$	0	0
$581$	8.94427	0.371071
$582$	0	0
$583$	16.9443	0.701760
$584$	0	0
$585$	6.47214	0.267590
$586$	0	0
$587$	−19.4164	−0.801401	−0.400700	−	0.916209i	$-0.631233\pi$
$587$	−19.4164	−0.801401	−0.400700	+	0.916209i	$0.631233\pi$
$588$	0	0
$589$	−7.70820	−0.317611
$590$	0	0
$591$	2.29180	0.0942719
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	−3.05573	−0.125273
$596$	0	0
$597$	5.52786	0.226240
$598$	0	0
$599$	−32.3607	−1.32222	−0.661111	−	0.750288i	$-0.729915\pi$
$599$	−32.3607	−1.32222	−0.661111	+	0.750288i	$0.729915\pi$
$600$	0	0
$601$	−12.9443	−0.528008	−0.264004	−	0.964522i	$-0.585043\pi$
$601$	−12.9443	−0.528008	−0.264004	+	0.964522i	$0.585043\pi$
$602$	0	0
$603$	−10.9443	−0.445685
$604$	0	0
$605$	−8.65248	−0.351773
$606$	0	0
$607$	42.5410	1.72669	0.863343	−	0.504617i	$-0.168366\pi$
$607$	42.5410	1.72669	0.863343	+	0.504617i	$0.168366\pi$
$608$	0	0
$609$	−8.47214	−0.343308
$610$	0	0
$611$	−14.4721	−0.585480
$612$	0	0
$613$	−14.3607	−0.580022	−0.290011	−	0.957023i	$-0.593659\pi$
$613$	−14.3607	−0.580022	−0.290011	+	0.957023i	$0.593659\pi$
$614$	0	0
$615$	−8.58359	−0.346124
$616$	0	0
$617$	26.9443	1.08474	0.542368	−	0.840141i	$-0.317528\pi$
$617$	26.9443	1.08474	0.542368	+	0.840141i	$0.317528\pi$
$618$	0	0
$619$	26.8328	1.07850	0.539251	−	0.842145i	$-0.318707\pi$
$619$	26.8328	1.07850	0.539251	+	0.842145i	$0.318707\pi$
$620$	0	0
$621$	5.70820	0.229062
$622$	0	0
$623$	2.94427	0.117960
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	1.88854	0.0753012
$630$	0	0
$631$	−32.3607	−1.28826	−0.644129	−	0.764917i	$-0.722780\pi$
$631$	−32.3607	−1.28826	−0.644129	+	0.764917i	$0.722780\pi$
$632$	0	0
$633$	25.4164	1.01021
$634$	0	0
$635$	−26.8328	−1.06483
$636$	0	0
$637$	5.23607	0.207461
$638$	0	0
$639$	−14.4721	−0.572509
$640$	0	0
$641$	−29.0557	−1.14763	−0.573816	−	0.818984i	$-0.694538\pi$
$641$	−29.0557	−1.14763	−0.573816	+	0.818984i	$0.694538\pi$
$642$	0	0
$643$	48.7214	1.92138	0.960691	−	0.277618i	$-0.0895451\pi$
$643$	48.7214	1.92138	0.960691	+	0.277618i	$0.0895451\pi$
$644$	0	0
$645$	−1.88854	−0.0743613
$646$	0	0
$647$	4.87539	0.191671	0.0958356	−	0.995397i	$-0.469448\pi$
$647$	4.87539	0.191671	0.0958356	+	0.995397i	$0.469448\pi$
$648$	0	0
$649$	25.8885	1.01621
$650$	0	0
$651$	7.70820	0.302108
$652$	0	0
$653$	−2.87539	−0.112523	−0.0562613	−	0.998416i	$-0.517918\pi$
$653$	−2.87539	−0.112523	−0.0562613	+	0.998416i	$0.517918\pi$
$654$	0	0
$655$	3.77709	0.147583
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−42.4721	−1.65448	−0.827240	−	0.561849i	$-0.810090\pi$
$659$	−42.4721	−1.65448	−0.827240	+	0.561849i	$0.810090\pi$
$660$	0	0
$661$	37.0132	1.43964	0.719822	−	0.694158i	$-0.244223\pi$
$661$	37.0132	1.43964	0.719822	+	0.694158i	$0.244223\pi$
$662$	0	0
$663$	12.9443	0.502714
$664$	0	0
$665$	−1.23607	−0.0479327
$666$	0	0
$667$	48.3607	1.87253
$668$	0	0
$669$	−3.70820	−0.143367
$670$	0	0
$671$	24.9443	0.962963
$672$	0	0
$673$	−36.4721	−1.40590	−0.702949	−	0.711240i	$-0.748134\pi$
$673$	−36.4721	−1.40590	−0.702949	+	0.711240i	$0.748134\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	−46.3607	−1.78179	−0.890893	−	0.454214i	$-0.849920\pi$
$677$	−46.3607	−1.78179	−0.890893	+	0.454214i	$0.849920\pi$
$678$	0	0
$679$	−10.4721	−0.401884
$680$	0	0
$681$	7.05573	0.270376
$682$	0	0
$683$	−40.3607	−1.54436	−0.772179	−	0.635405i	$-0.780833\pi$
$683$	−40.3607	−1.54436	−0.772179	+	0.635405i	$0.780833\pi$
$684$	0	0
$685$	−11.6393	−0.444716
$686$	0	0
$687$	−23.8885	−0.911405
$688$	0	0
$689$	44.3607	1.69001
$690$	0	0
$691$	13.8885	0.528345	0.264173	−	0.964475i	$-0.414901\pi$
$691$	13.8885	0.528345	0.264173	+	0.964475i	$0.414901\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	19.7771	0.750188
$696$	0	0
$697$	−17.1672	−0.650253
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	−39.2361	−1.48193	−0.740963	−	0.671546i	$-0.765631\pi$
$701$	−39.2361	−1.48193	−0.740963	+	0.671546i	$0.765631\pi$
$702$	0	0
$703$	0.763932	0.0288122
$704$	0	0
$705$	−3.41641	−0.128669
$706$	0	0
$707$	19.7082	0.741203
$708$	0	0
$709$	10.5836	0.397475	0.198738	−	0.980053i	$-0.436316\pi$
$709$	10.5836	0.397475	0.198738	+	0.980053i	$0.436316\pi$
$710$	0	0
$711$	−12.1803	−0.456798
$712$	0	0
$713$	−44.0000	−1.64781
$714$	0	0
$715$	12.9443	0.484088
$716$	0	0
$717$	−22.6525	−0.845972
$718$	0	0
$719$	−18.5410	−0.691463	−0.345732	−	0.938333i	$-0.612369\pi$
$719$	−18.5410	−0.691463	−0.345732	+	0.938333i	$0.612369\pi$
$720$	0	0
$721$	−12.6525	−0.471203
$722$	0	0
$723$	−16.3607	−0.608460
$724$	0	0
$725$	−29.4164	−1.09250
$726$	0	0
$727$	10.4721	0.388390	0.194195	−	0.980963i	$-0.437791\pi$
$727$	10.4721	0.388390	0.194195	+	0.980963i	$0.437791\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.77709	−0.139701
$732$	0	0
$733$	−19.3050	−0.713045	−0.356522	−	0.934287i	$-0.616038\pi$
$733$	−19.3050	−0.713045	−0.356522	+	0.934287i	$0.616038\pi$
$734$	0	0
$735$	1.23607	0.0455931
$736$	0	0
$737$	−21.8885	−0.806275
$738$	0	0
$739$	−32.9443	−1.21187	−0.605937	−	0.795512i	$-0.707202\pi$
$739$	−32.9443	−1.21187	−0.605937	+	0.795512i	$0.707202\pi$
$740$	0	0
$741$	5.23607	0.192352
$742$	0	0
$743$	35.0557	1.28607	0.643035	−	0.765837i	$-0.277675\pi$
$743$	35.0557	1.28607	0.643035	+	0.765837i	$0.277675\pi$
$744$	0	0
$745$	15.0557	0.551599
$746$	0	0
$747$	−8.94427	−0.327254
$748$	0	0
$749$	−8.94427	−0.326817
$750$	0	0
$751$	53.4853	1.95171	0.975853	−	0.218428i	$-0.0700930\pi$
$751$	53.4853	1.95171	0.975853	+	0.218428i	$0.0700930\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16.9443	0.616665
$756$	0	0
$757$	39.3050	1.42856	0.714281	−	0.699859i	$-0.246754\pi$
$757$	39.3050	1.42856	0.714281	+	0.699859i	$0.246754\pi$
$758$	0	0
$759$	11.4164	0.414389
$760$	0	0
$761$	16.3607	0.593074	0.296537	−	0.955021i	$-0.404168\pi$
$761$	16.3607	0.593074	0.296537	+	0.955021i	$0.404168\pi$
$762$	0	0
$763$	1.70820	0.0618411
$764$	0	0
$765$	3.05573	0.110480
$766$	0	0
$767$	67.7771	2.44729
$768$	0	0
$769$	23.8885	0.861443	0.430721	−	0.902485i	$-0.358259\pi$
$769$	23.8885	0.861443	0.430721	+	0.902485i	$0.358259\pi$
$770$	0	0
$771$	−19.8885	−0.716268
$772$	0	0
$773$	31.5279	1.13398	0.566989	−	0.823725i	$-0.308108\pi$
$773$	31.5279	1.13398	0.566989	+	0.823725i	$0.308108\pi$
$774$	0	0
$775$	26.7639	0.961389
$776$	0	0
$777$	−0.763932	−0.0274059
$778$	0	0
$779$	−6.94427	−0.248804
$780$	0	0
$781$	−28.9443	−1.03571
$782$	0	0
$783$	8.47214	0.302769
$784$	0	0
$785$	−5.52786	−0.197298
$786$	0	0
$787$	38.8328	1.38424	0.692120	−	0.721782i	$-0.256677\pi$
$787$	38.8328	1.38424	0.692120	+	0.721782i	$0.256677\pi$
$788$	0	0
$789$	4.18034	0.148824
$790$	0	0
$791$	−12.4721	−0.443458
$792$	0	0
$793$	65.3050	2.31905
$794$	0	0
$795$	10.4721	0.371408
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	−6.83282	−0.241728
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−7.05573	−0.248682
$806$	0	0
$807$	28.4721	1.00227
$808$	0	0
$809$	33.4164	1.17486	0.587429	−	0.809276i	$-0.300140\pi$
$809$	33.4164	1.17486	0.587429	+	0.809276i	$0.300140\pi$
$810$	0	0
$811$	31.4164	1.10318	0.551590	−	0.834116i	$-0.314021\pi$
$811$	31.4164	1.10318	0.551590	+	0.834116i	$0.314021\pi$
$812$	0	0
$813$	−12.9443	−0.453975
$814$	0	0
$815$	25.8885	0.906836
$816$	0	0
$817$	−1.52786	−0.0534532
$818$	0	0
$819$	−5.23607	−0.182963
$820$	0	0
$821$	−22.2918	−0.777989	−0.388995	−	0.921240i	$-0.627178\pi$
$821$	−22.2918	−0.777989	−0.388995	+	0.921240i	$0.627178\pi$
$822$	0	0
$823$	−51.7771	−1.80484	−0.902418	−	0.430862i	$-0.858210\pi$
$823$	−51.7771	−1.80484	−0.902418	+	0.430862i	$0.858210\pi$
$824$	0	0
$825$	−6.94427	−0.241769
$826$	0	0
$827$	2.47214	0.0859646	0.0429823	−	0.999076i	$-0.486314\pi$
$827$	2.47214	0.0859646	0.0429823	+	0.999076i	$0.486314\pi$
$828$	0	0
$829$	13.8197	0.479977	0.239988	−	0.970776i	$-0.422856\pi$
$829$	13.8197	0.479977	0.239988	+	0.970776i	$0.422856\pi$
$830$	0	0
$831$	23.8885	0.828684
$832$	0	0
$833$	2.47214	0.0856544
$834$	0	0
$835$	−9.88854	−0.342207
$836$	0	0
$837$	−7.70820	−0.266435
$838$	0	0
$839$	−41.8885	−1.44615	−0.723077	−	0.690768i	$-0.757273\pi$
$839$	−41.8885	−1.44615	−0.723077	+	0.690768i	$0.757273\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	−5.41641	−0.186551
$844$	0	0
$845$	17.8197	0.613015
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	4.36068	0.149482
$852$	0	0
$853$	−37.4164	−1.28111	−0.640557	−	0.767911i	$-0.721296\pi$
$853$	−37.4164	−1.28111	−0.640557	+	0.767911i	$0.721296\pi$
$854$	0	0
$855$	1.23607	0.0422726
$856$	0	0
$857$	11.8885	0.406105	0.203052	−	0.979168i	$-0.434914\pi$
$857$	11.8885	0.406105	0.203052	+	0.979168i	$0.434914\pi$
$858$	0	0
$859$	−53.8885	−1.83865	−0.919327	−	0.393495i	$-0.871266\pi$
$859$	−53.8885	−1.83865	−0.919327	+	0.393495i	$0.871266\pi$
$860$	0	0
$861$	6.94427	0.236660
$862$	0	0
$863$	−44.9443	−1.52992	−0.764960	−	0.644077i	$-0.777241\pi$
$863$	−44.9443	−1.52992	−0.764960	+	0.644077i	$0.777241\pi$
$864$	0	0
$865$	14.6950	0.499647
$866$	0	0
$867$	−10.8885	−0.369794
$868$	0	0
$869$	−24.3607	−0.826379
$870$	0	0
$871$	−57.3050	−1.94170
$872$	0	0
$873$	10.4721	0.354428
$874$	0	0
$875$	10.4721	0.354023
$876$	0	0
$877$	−2.29180	−0.0773885	−0.0386942	−	0.999251i	$-0.512320\pi$
$877$	−2.29180	−0.0773885	−0.0386942	+	0.999251i	$0.512320\pi$
$878$	0	0
$879$	−8.47214	−0.285758
$880$	0	0
$881$	−36.9443	−1.24468	−0.622342	−	0.782745i	$-0.713819\pi$
$881$	−36.9443	−1.24468	−0.622342	+	0.782745i	$0.713819\pi$
$882$	0	0
$883$	0.360680	0.0121378	0.00606892	−	0.999982i	$-0.498068\pi$
$883$	0.360680	0.0121378	0.00606892	+	0.999982i	$0.498068\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	43.1935	1.45030	0.725148	−	0.688593i	$-0.241771\pi$
$887$	43.1935	1.45030	0.725148	+	0.688593i	$0.241771\pi$
$888$	0	0
$889$	21.7082	0.728070
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−2.76393	−0.0924915
$894$	0	0
$895$	19.0557	0.636963
$896$	0	0
$897$	29.8885	0.997949
$898$	0	0
$899$	−65.3050	−2.17804
$900$	0	0
$901$	20.9443	0.697755
$902$	0	0
$903$	1.52786	0.0508441
$904$	0	0
$905$	−11.4164	−0.379494
$906$	0	0
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	0	0
$909$	−19.7082	−0.653680
$910$	0	0
$911$	22.4721	0.744535	0.372268	−	0.928125i	$-0.378580\pi$
$911$	22.4721	0.744535	0.372268	+	0.928125i	$0.378580\pi$
$912$	0	0
$913$	−17.8885	−0.592024
$914$	0	0
$915$	15.4164	0.509651
$916$	0	0
$917$	−3.05573	−0.100909
$918$	0	0
$919$	−22.8328	−0.753185	−0.376593	−	0.926379i	$-0.622904\pi$
$919$	−22.8328	−0.753185	−0.376593	+	0.926379i	$0.622904\pi$
$920$	0	0
$921$	−5.52786	−0.182149
$922$	0	0
$923$	−75.7771	−2.49423
$924$	0	0
$925$	−2.65248	−0.0872129
$926$	0	0
$927$	12.6525	0.415562
$928$	0	0
$929$	−39.4164	−1.29321	−0.646605	−	0.762825i	$-0.723812\pi$
$929$	−39.4164	−1.29321	−0.646605	+	0.762825i	$0.723812\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	17.2361	0.564284
$934$	0	0
$935$	6.11146	0.199866
$936$	0	0
$937$	−33.7771	−1.10345	−0.551725	−	0.834026i	$-0.686030\pi$
$937$	−33.7771	−1.10345	−0.551725	+	0.834026i	$0.686030\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−16.4721	−0.536976	−0.268488	−	0.963283i	$-0.586524\pi$
$941$	−16.4721	−0.536976	−0.268488	+	0.963283i	$0.586524\pi$
$942$	0	0
$943$	−39.6393	−1.29083
$944$	0	0
$945$	−1.23607	−0.0402093
$946$	0	0
$947$	−32.2492	−1.04796	−0.523979	−	0.851731i	$-0.675553\pi$
$947$	−32.2492	−1.04796	−0.523979	+	0.851731i	$0.675553\pi$
$948$	0	0
$949$	31.4164	1.01982
$950$	0	0
$951$	12.4721	0.404437
$952$	0	0
$953$	18.5836	0.601982	0.300991	−	0.953627i	$-0.402683\pi$
$953$	18.5836	0.601982	0.300991	+	0.953627i	$0.402683\pi$
$954$	0	0
$955$	4.72136	0.152780
$956$	0	0
$957$	16.9443	0.547731
$958$	0	0
$959$	9.41641	0.304072
$960$	0	0
$961$	28.4164	0.916658
$962$	0	0
$963$	8.94427	0.288225
$964$	0	0
$965$	−0.583592	−0.0187865
$966$	0	0
$967$	19.4164	0.624390	0.312195	−	0.950018i	$-0.398936\pi$
$967$	19.4164	0.624390	0.312195	+	0.950018i	$0.398936\pi$
$968$	0	0
$969$	2.47214	0.0794164
$970$	0	0
$971$	44.7214	1.43518	0.717588	−	0.696467i	$-0.245246\pi$
$971$	44.7214	1.43518	0.717588	+	0.696467i	$0.245246\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	−18.1803	−0.582237
$976$	0	0
$977$	5.41641	0.173286	0.0866431	−	0.996239i	$-0.472386\pi$
$977$	5.41641	0.173286	0.0866431	+	0.996239i	$0.472386\pi$
$978$	0	0
$979$	−5.88854	−0.188199
$980$	0	0
$981$	−1.70820	−0.0545388
$982$	0	0
$983$	−17.3050	−0.551942	−0.275971	−	0.961166i	$-0.588999\pi$
$983$	−17.3050	−0.551942	−0.275971	+	0.961166i	$0.588999\pi$
$984$	0	0
$985$	2.83282	0.0902610
$986$	0	0
$987$	2.76393	0.0879769
$988$	0	0
$989$	−8.72136	−0.277323
$990$	0	0
$991$	−49.7082	−1.57903	−0.789517	−	0.613729i	$-0.789669\pi$
$991$	−49.7082	−1.57903	−0.789517	+	0.613729i	$0.789669\pi$
$992$	0	0
$993$	18.9443	0.601178
$994$	0	0
$995$	6.83282	0.216615
$996$	0	0
$997$	48.4721	1.53513	0.767564	−	0.640972i	$-0.221469\pi$
$997$	48.4721	1.53513	0.767564	+	0.640972i	$0.221469\pi$
$998$	0	0
$999$	0.763932	0.0241697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bn.1.2		2
4.3	odd	2		798.2.a.j.1.2	✓	2
12.11	even	2		2394.2.a.z.1.1		2
28.27	even	2		5586.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.a.j.1.2	✓	2	4.3	odd	2
2394.2.a.z.1.1		2	12.11	even	2
5586.2.a.be.1.1		2	28.27	even	2
6384.2.a.bn.1.2		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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +5.23607 q^{13} +1.23607 q^{15} +2.47214 q^{17} +1.00000 q^{19} -1.00000 q^{21} +5.70820 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -7.70820 q^{31} +2.00000 q^{33} -1.23607 q^{35} +0.763932 q^{37} +5.23607 q^{39} -6.94427 q^{41} -1.52786 q^{43} +1.23607 q^{45} -2.76393 q^{47} +1.00000 q^{49} +2.47214 q^{51} +8.47214 q^{53} +2.47214 q^{55} +1.00000 q^{57} +12.9443 q^{59} +12.4721 q^{61} -1.00000 q^{63} +6.47214 q^{65} -10.9443 q^{67} +5.70820 q^{69} -14.4721 q^{71} +6.00000 q^{73} -3.47214 q^{75} -2.00000 q^{77} -12.1803 q^{79} +1.00000 q^{81} -8.94427 q^{83} +3.05573 q^{85} +8.47214 q^{87} -2.94427 q^{89} -5.23607 q^{91} -7.70820 q^{93} +1.23607 q^{95} +10.4721 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{31} + 4 q^{33} + 2 q^{35} + 6 q^{37} + 6 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{45} - 10 q^{47} + 2 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 4 q^{65} - 4 q^{67} - 2 q^{69} - 20 q^{71} + 12 q^{73} + 2 q^{75} - 4 q^{77} - 2 q^{79} + 2 q^{81} + 24 q^{85} + 8 q^{87} + 12 q^{89} - 6 q^{91} - 2 q^{93} - 2 q^{95} + 12 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 + 6 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^31 + 4 * q^33 + 2 * q^35 + 6 * q^37 + 6 * q^39 + 4 * q^41 - 12 * q^43 - 2 * q^45 - 10 * q^47 + 2 * q^49 - 4 * q^51 + 8 * q^53 - 4 * q^55 + 2 * q^57 + 8 * q^59 + 16 * q^61 - 2 * q^63 + 4 * q^65 - 4 * q^67 - 2 * q^69 - 20 * q^71 + 12 * q^73 + 2 * q^75 - 4 * q^77 - 2 * q^79 + 2 * q^81 + 24 * q^85 + 8 * q^87 + 12 * q^89 - 6 * q^91 - 2 * q^93 - 2 * q^95 + 12 * q^97 + 4 * q^99

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