LMFDB - Embedded newform 644.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [644,2,Mod(1,644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(644, 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("644.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 644 = 2^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	644.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:	$5.14236589017$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6963152.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 $ x^5 - 2x^4 - 10x^3 + 10x^2 + 29x + 10
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			1.1
Root			$3.11181$ of defining polynomial
Character	$\chi$	$=$	644.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +O(q^{10})$	$q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +0.612946 q^{11} +1.84681 q^{13} -3.75608 q^{15} +4.26501 q^{17} -4.70034 q^{19} +2.11181 q^{21} -1.00000 q^{23} -1.83657 q^{25} +3.25271 q^{27} +10.2406 q^{29} +10.2181 q^{31} -1.29443 q^{33} -1.77860 q^{35} -2.47671 q^{37} -3.90011 q^{39} +6.37682 q^{41} +10.9240 q^{43} +2.59632 q^{45} +0.544914 q^{47} +1.00000 q^{49} -9.00689 q^{51} -7.61984 q^{53} +1.09019 q^{55} +9.92623 q^{57} -2.74172 q^{59} +15.1454 q^{61} -1.45975 q^{63} +3.28473 q^{65} -5.27937 q^{67} +2.11181 q^{69} -2.21583 q^{71} +5.62764 q^{73} +3.87849 q^{75} -0.612946 q^{77} -2.46755 q^{79} -11.2484 q^{81} +5.84347 q^{83} +7.58575 q^{85} -21.6262 q^{87} +0.00746465 q^{89} -1.84681 q^{91} -21.5786 q^{93} -8.36003 q^{95} -7.58815 q^{97} +0.894750 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9	$ 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 + 2 * q^11 + 13 * q^13 + 4 * q^15 + 4 * q^17 + 12 * q^19 - 3 * q^21 - 5 * q^23 + 19 * q^25 + 15 * q^27 + 13 * q^29 - 3 * q^31 + 24 * q^33 - 2 * q^35 - 4 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - 16 * q^45 + 5 * q^47 + 5 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 12 * q^59 + 20 * q^61 - 10 * q^63 - 12 * q^65 - 12 * q^67 - 3 * q^69 + 9 * q^71 - 9 * q^73 + 35 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 - 28 * q^83 + 16 * q^85 - 15 * q^87 + 32 * q^89 - 13 * q^91 - 15 * q^93 - 36 * q^95 + 4 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.11181	−1.21926	−0.609628	−	0.792688i	$-0.708681\pi$
$3$	−2.11181	−1.21926	−0.609628	+	0.792688i	$0.708681\pi$
$4$	0	0
$5$	1.77860	0.795415	0.397708	−	0.917512i	$-0.369806\pi$
$5$	1.77860	0.795415	0.397708	+	0.917512i	$0.369806\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.45975	0.486584
$10$	0	0
$11$	0.612946	0.184810	0.0924051	−	0.995721i	$-0.470545\pi$
$11$	0.612946	0.184810	0.0924051	+	0.995721i	$0.470545\pi$
$12$	0	0
$13$	1.84681	0.512212	0.256106	−	0.966649i	$-0.417560\pi$
$13$	1.84681	0.512212	0.256106	+	0.966649i	$0.417560\pi$
$14$	0	0
$15$	−3.75608	−0.969815
$16$	0	0
$17$	4.26501	1.03442	0.517208	−	0.855860i	$-0.326971\pi$
$17$	4.26501	1.03442	0.517208	+	0.855860i	$0.326971\pi$
$18$	0	0
$19$	−4.70034	−1.07833	−0.539166	−	0.842200i	$-0.681260\pi$
$19$	−4.70034	−1.07833	−0.539166	+	0.842200i	$0.681260\pi$
$20$	0	0
$21$	2.11181	0.460835
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.83657	−0.367314
$26$	0	0
$27$	3.25271	0.625985
$28$	0	0
$29$	10.2406	1.90163	0.950814	−	0.309761i	$-0.100249\pi$
$29$	10.2406	1.90163	0.950814	+	0.309761i	$0.100249\pi$
$30$	0	0
$31$	10.2181	1.83522	0.917609	−	0.397485i	$-0.130117\pi$
$31$	10.2181	1.83522	0.917609	+	0.397485i	$0.130117\pi$
$32$	0	0
$33$	−1.29443	−0.225331
$34$	0	0
$35$	−1.77860	−0.300639
$36$	0	0
$37$	−2.47671	−0.407169	−0.203584	−	0.979057i	$-0.565259\pi$
$37$	−2.47671	−0.407169	−0.203584	+	0.979057i	$0.565259\pi$
$38$	0	0
$39$	−3.90011	−0.624517
$40$	0	0
$41$	6.37682	0.995892	0.497946	−	0.867208i	$-0.334088\pi$
$41$	6.37682	0.995892	0.497946	+	0.867208i	$0.334088\pi$
$42$	0	0
$43$	10.9240	1.66589	0.832944	−	0.553357i	$-0.186653\pi$
$43$	10.9240	1.66589	0.832944	+	0.553357i	$0.186653\pi$
$44$	0	0
$45$	2.59632	0.387037
$46$	0	0
$47$	0.544914	0.0794838	0.0397419	−	0.999210i	$-0.487346\pi$
$47$	0.544914	0.0794838	0.0397419	+	0.999210i	$0.487346\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.00689	−1.26122
$52$	0	0
$53$	−7.61984	−1.04667	−0.523333	−	0.852128i	$-0.675312\pi$
$53$	−7.61984	−1.04667	−0.523333	+	0.852128i	$0.675312\pi$
$54$	0	0
$55$	1.09019	0.147001
$56$	0	0
$57$	9.92623	1.31476
$58$	0	0
$59$	−2.74172	−0.356941	−0.178471	−	0.983945i	$-0.557115\pi$
$59$	−2.74172	−0.356941	−0.178471	+	0.983945i	$0.557115\pi$
$60$	0	0
$61$	15.1454	1.93916	0.969582	−	0.244766i	$-0.0787111\pi$
$61$	15.1454	1.93916	0.969582	+	0.244766i	$0.0787111\pi$
$62$	0	0
$63$	−1.45975	−0.183912
$64$	0	0
$65$	3.28473	0.407421
$66$	0	0
$67$	−5.27937	−0.644977	−0.322489	−	0.946573i	$-0.604519\pi$
$67$	−5.27937	−0.644977	−0.322489	+	0.946573i	$0.604519\pi$
$68$	0	0
$69$	2.11181	0.254232
$70$	0	0
$71$	−2.21583	−0.262970	−0.131485	−	0.991318i	$-0.541975\pi$
$71$	−2.21583	−0.262970	−0.131485	+	0.991318i	$0.541975\pi$
$72$	0	0
$73$	5.62764	0.658665	0.329333	−	0.944214i	$-0.393176\pi$
$73$	5.62764	0.658665	0.329333	+	0.944214i	$0.393176\pi$
$74$	0	0
$75$	3.87849	0.447850
$76$	0	0
$77$	−0.612946	−0.0698517
$78$	0	0
$79$	−2.46755	−0.277621	−0.138810	−	0.990319i	$-0.544328\pi$
$79$	−2.46755	−0.277621	−0.138810	+	0.990319i	$0.544328\pi$
$80$	0	0
$81$	−11.2484	−1.24982
$82$	0	0
$83$	5.84347	0.641404	0.320702	−	0.947180i	$-0.396081\pi$
$83$	5.84347	0.641404	0.320702	+	0.947180i	$0.396081\pi$
$84$	0	0
$85$	7.58575	0.822790
$86$	0	0
$87$	−21.6262	−2.31857
$88$	0	0
$89$	0.00746465	0.000791251 0	0.000395626	−	1.00000i	$-0.499874\pi$
$89$	0.00746465	0.000791251 0	0.000395626		1.00000i	$0.499874\pi$
$90$	0	0
$91$	−1.84681	−0.193598
$92$	0	0
$93$	−21.5786	−2.23760
$94$	0	0
$95$	−8.36003	−0.857721
$96$	0	0
$97$	−7.58815	−0.770460	−0.385230	−	0.922821i	$-0.625878\pi$
$97$	−7.58815	−0.770460	−0.385230	+	0.922821i	$0.625878\pi$
$98$	0	0
$99$	0.894750	0.0899257
$100$	0	0
$101$	11.5904	1.15329	0.576643	−	0.816996i	$-0.304362\pi$
$101$	11.5904	1.15329	0.576643	+	0.816996i	$0.304362\pi$
$102$	0	0
$103$	−0.666419	−0.0656642	−0.0328321	−	0.999461i	$-0.510453\pi$
$103$	−0.666419	−0.0656642	−0.0328321	+	0.999461i	$0.510453\pi$
$104$	0	0
$105$	3.75608	0.366555
$106$	0	0
$107$	6.92396	0.669365	0.334682	−	0.942331i	$-0.391371\pi$
$107$	6.92396	0.669365	0.334682	+	0.942331i	$0.391371\pi$
$108$	0	0
$109$	4.06263	0.389130	0.194565	−	0.980890i	$-0.437671\pi$
$109$	4.06263	0.389130	0.194565	+	0.980890i	$0.437671\pi$
$110$	0	0
$111$	5.23035	0.496443
$112$	0	0
$113$	−12.5926	−1.18462	−0.592308	−	0.805711i	$-0.701783\pi$
$113$	−12.5926	−1.18462	−0.592308	+	0.805711i	$0.701783\pi$
$114$	0	0
$115$	−1.77860	−0.165856
$116$	0	0
$117$	2.69588	0.249234
$118$	0	0
$119$	−4.26501	−0.390973
$120$	0	0
$121$	−10.6243	−0.965845
$122$	0	0
$123$	−13.4666	−1.21425
$124$	0	0
$125$	−12.1595	−1.08758
$126$	0	0
$127$	−12.8535	−1.14057	−0.570283	−	0.821448i	$-0.693167\pi$
$127$	−12.8535	−1.14057	−0.570283	+	0.821448i	$0.693167\pi$
$128$	0	0
$129$	−23.0694	−2.03114
$130$	0	0
$131$	−14.6948	−1.28389	−0.641944	−	0.766752i	$-0.721872\pi$
$131$	−14.6948	−1.28389	−0.641944	+	0.766752i	$0.721872\pi$
$132$	0	0
$133$	4.70034	0.407571
$134$	0	0
$135$	5.78529	0.497918
$136$	0	0
$137$	−19.2303	−1.64296	−0.821480	−	0.570238i	$-0.806851\pi$
$137$	−19.2303	−1.64296	−0.821480	+	0.570238i	$0.806851\pi$
$138$	0	0
$139$	−6.14573	−0.521274	−0.260637	−	0.965437i	$-0.583933\pi$
$139$	−6.14573	−0.521274	−0.260637	+	0.965437i	$0.583933\pi$
$140$	0	0
$141$	−1.15076	−0.0969111
$142$	0	0
$143$	1.13199	0.0946620
$144$	0	0
$145$	18.2139	1.51258
$146$	0	0
$147$	−2.11181	−0.174179
$148$	0	0
$149$	−6.66642	−0.546134	−0.273067	−	0.961995i	$-0.588038\pi$
$149$	−6.66642	−0.546134	−0.273067	+	0.961995i	$0.588038\pi$
$150$	0	0
$151$	−1.07270	−0.0872950	−0.0436475	−	0.999047i	$-0.513898\pi$
$151$	−1.07270	−0.0872950	−0.0436475	+	0.999047i	$0.513898\pi$
$152$	0	0
$153$	6.22585	0.503330
$154$	0	0
$155$	18.1739	1.45976
$156$	0	0
$157$	−2.76664	−0.220802	−0.110401	−	0.993887i	$-0.535214\pi$
$157$	−2.76664	−0.220802	−0.110401	+	0.993887i	$0.535214\pi$
$158$	0	0
$159$	16.0917	1.27615
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	5.94336	0.465520	0.232760	−	0.972534i	$-0.425224\pi$
$163$	5.94336	0.465520	0.232760	+	0.972534i	$0.425224\pi$
$164$	0	0
$165$	−2.30227	−0.179232
$166$	0	0
$167$	4.52778	0.350371	0.175185	−	0.984535i	$-0.443948\pi$
$167$	4.52778	0.350371	0.175185	+	0.984535i	$0.443948\pi$
$168$	0	0
$169$	−9.58931	−0.737639
$170$	0	0
$171$	−6.86133	−0.524699
$172$	0	0
$173$	−10.7003	−0.813531	−0.406766	−	0.913533i	$-0.633343\pi$
$173$	−10.7003	−0.813531	−0.406766	+	0.913533i	$0.633343\pi$
$174$	0	0
$175$	1.83657	0.138832
$176$	0	0
$177$	5.78999	0.435203
$178$	0	0
$179$	11.3590	0.849008	0.424504	−	0.905426i	$-0.360448\pi$
$179$	11.3590	0.849008	0.424504	+	0.905426i	$0.360448\pi$
$180$	0	0
$181$	20.2162	1.50266	0.751328	−	0.659929i	$-0.229414\pi$
$181$	20.2162	1.50266	0.751328	+	0.659929i	$0.229414\pi$
$182$	0	0
$183$	−31.9842	−2.36434
$184$	0	0
$185$	−4.40509	−0.323868
$186$	0	0
$187$	2.61422	0.191171
$188$	0	0
$189$	−3.25271	−0.236600
$190$	0	0
$191$	1.97054	0.142583	0.0712916	−	0.997456i	$-0.477288\pi$
$191$	1.97054	0.142583	0.0712916	+	0.997456i	$0.477288\pi$
$192$	0	0
$193$	14.7995	1.06529	0.532645	−	0.846339i	$-0.321198\pi$
$193$	14.7995	1.06529	0.532645	+	0.846339i	$0.321198\pi$
$194$	0	0
$195$	−6.93674	−0.496751
$196$	0	0
$197$	15.3496	1.09362	0.546808	−	0.837258i	$-0.315843\pi$
$197$	15.3496	1.09362	0.546808	+	0.837258i	$0.315843\pi$
$198$	0	0
$199$	13.7071	0.971668	0.485834	−	0.874051i	$-0.338516\pi$
$199$	13.7071	0.971668	0.485834	+	0.874051i	$0.338516\pi$
$200$	0	0
$201$	11.1490	0.786392
$202$	0	0
$203$	−10.2406	−0.718748
$204$	0	0
$205$	11.3418	0.792148
$206$	0	0
$207$	−1.45975	−0.101460
$208$	0	0
$209$	−2.88105	−0.199287
$210$	0	0
$211$	22.0376	1.51713	0.758567	−	0.651596i	$-0.225900\pi$
$211$	22.0376	1.51713	0.758567	+	0.651596i	$0.225900\pi$
$212$	0	0
$213$	4.67941	0.320628
$214$	0	0
$215$	19.4294	1.32507
$216$	0	0
$217$	−10.2181	−0.693647
$218$	0	0
$219$	−11.8845	−0.803081
$220$	0	0
$221$	7.87664	0.529840
$222$	0	0
$223$	−3.01419	−0.201845	−0.100922	−	0.994894i	$-0.532179\pi$
$223$	−3.01419	−0.201845	−0.100922	+	0.994894i	$0.532179\pi$
$224$	0	0
$225$	−2.68094	−0.178729
$226$	0	0
$227$	−23.3269	−1.54826	−0.774130	−	0.633026i	$-0.781813\pi$
$227$	−23.3269	−1.54826	−0.774130	+	0.633026i	$0.781813\pi$
$228$	0	0
$229$	27.8111	1.83781	0.918904	−	0.394482i	$-0.129076\pi$
$229$	27.8111	1.83781	0.918904	+	0.394482i	$0.129076\pi$
$230$	0	0
$231$	1.29443	0.0851671
$232$	0	0
$233$	−22.2000	−1.45437	−0.727185	−	0.686442i	$-0.759172\pi$
$233$	−22.2000	−1.45437	−0.727185	+	0.686442i	$0.759172\pi$
$234$	0	0
$235$	0.969185	0.0632227
$236$	0	0
$237$	5.21100	0.338491
$238$	0	0
$239$	13.2136	0.854714	0.427357	−	0.904083i	$-0.359445\pi$
$239$	13.2136	0.854714	0.427357	+	0.904083i	$0.359445\pi$
$240$	0	0
$241$	11.9586	0.770322	0.385161	−	0.922849i	$-0.374146\pi$
$241$	11.9586	0.770322	0.385161	+	0.922849i	$0.374146\pi$
$242$	0	0
$243$	13.9963	0.897865
$244$	0	0
$245$	1.77860	0.113631
$246$	0	0
$247$	−8.68061	−0.552334
$248$	0	0
$249$	−12.3403	−0.782035
$250$	0	0
$251$	15.0805	0.951872	0.475936	−	0.879480i	$-0.342109\pi$
$251$	15.0805	0.951872	0.475936	+	0.879480i	$0.342109\pi$
$252$	0	0
$253$	−0.612946	−0.0385356
$254$	0	0
$255$	−16.0197	−1.00319
$256$	0	0
$257$	−12.8822	−0.803573	−0.401786	−	0.915733i	$-0.631611\pi$
$257$	−12.8822	−0.803573	−0.401786	+	0.915733i	$0.631611\pi$
$258$	0	0
$259$	2.47671	0.153895
$260$	0	0
$261$	14.9487	0.925302
$262$	0	0
$263$	−25.4881	−1.57166	−0.785831	−	0.618442i	$-0.787764\pi$
$263$	−25.4881	−1.57166	−0.785831	+	0.618442i	$0.787764\pi$
$264$	0	0
$265$	−13.5527	−0.832534
$266$	0	0
$267$	−0.0157639	−0.000964737 0
$268$	0	0
$269$	−15.0772	−0.919270	−0.459635	−	0.888108i	$-0.652020\pi$
$269$	−15.0772	−0.919270	−0.459635	+	0.888108i	$0.652020\pi$
$270$	0	0
$271$	10.4014	0.631841	0.315920	−	0.948786i	$-0.397687\pi$
$271$	10.4014	0.631841	0.315920	+	0.948786i	$0.397687\pi$
$272$	0	0
$273$	3.90011	0.236045
$274$	0	0
$275$	−1.12572	−0.0678834
$276$	0	0
$277$	−9.01848	−0.541868	−0.270934	−	0.962598i	$-0.587333\pi$
$277$	−9.01848	−0.541868	−0.270934	+	0.962598i	$0.587333\pi$
$278$	0	0
$279$	14.9158	0.892988
$280$	0	0
$281$	1.91724	0.114373	0.0571864	−	0.998364i	$-0.481787\pi$
$281$	1.91724	0.114373	0.0571864	+	0.998364i	$0.481787\pi$
$282$	0	0
$283$	9.49457	0.564394	0.282197	−	0.959357i	$-0.408937\pi$
$283$	9.49457	0.564394	0.282197	+	0.959357i	$0.408937\pi$
$284$	0	0
$285$	17.6548	1.04578
$286$	0	0
$287$	−6.37682	−0.376412
$288$	0	0
$289$	1.19028	0.0700165
$290$	0	0
$291$	16.0248	0.939388
$292$	0	0
$293$	−16.6936	−0.975249	−0.487624	−	0.873053i	$-0.662136\pi$
$293$	−16.6936	−0.975249	−0.487624	+	0.873053i	$0.662136\pi$
$294$	0	0
$295$	−4.87643	−0.283917
$296$	0	0
$297$	1.99374	0.115688
$298$	0	0
$299$	−1.84681	−0.106804
$300$	0	0
$301$	−10.9240	−0.629647
$302$	0	0
$303$	−24.4767	−1.40615
$304$	0	0
$305$	26.9376	1.54244
$306$	0	0
$307$	25.9285	1.47982	0.739908	−	0.672709i	$-0.234869\pi$
$307$	25.9285	1.47982	0.739908	+	0.672709i	$0.234869\pi$
$308$	0	0
$309$	1.40735	0.0800615
$310$	0	0
$311$	−25.4581	−1.44360	−0.721798	−	0.692104i	$-0.756684\pi$
$311$	−25.4581	−1.44360	−0.721798	+	0.692104i	$0.756684\pi$
$312$	0	0
$313$	17.5490	0.991928	0.495964	−	0.868343i	$-0.334815\pi$
$313$	17.5490	0.991928	0.495964	+	0.868343i	$0.334815\pi$
$314$	0	0
$315$	−2.59632	−0.146286
$316$	0	0
$317$	−4.42250	−0.248392	−0.124196	−	0.992258i	$-0.539635\pi$
$317$	−4.42250	−0.248392	−0.124196	+	0.992258i	$0.539635\pi$
$318$	0	0
$319$	6.27693	0.351440
$320$	0	0
$321$	−14.6221	−0.816126
$322$	0	0
$323$	−20.0470	−1.11544
$324$	0	0
$325$	−3.39179	−0.188143
$326$	0	0
$327$	−8.57952	−0.474449
$328$	0	0
$329$	−0.544914	−0.0300421
$330$	0	0
$331$	31.9340	1.75525	0.877626	−	0.479345i	$-0.159126\pi$
$331$	31.9340	1.75525	0.877626	+	0.479345i	$0.159126\pi$
$332$	0	0
$333$	−3.61538	−0.198122
$334$	0	0
$335$	−9.38990	−0.513025
$336$	0	0
$337$	−7.31791	−0.398632	−0.199316	−	0.979935i	$-0.563872\pi$
$337$	−7.31791	−0.398632	−0.199316	+	0.979935i	$0.563872\pi$
$338$	0	0
$339$	26.5933	1.44435
$340$	0	0
$341$	6.26312	0.339167
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.75608	0.202220
$346$	0	0
$347$	14.0287	0.753101	0.376551	−	0.926396i	$-0.377110\pi$
$347$	14.0287	0.753101	0.376551	+	0.926396i	$0.377110\pi$
$348$	0	0
$349$	−26.8241	−1.43586	−0.717930	−	0.696116i	$-0.754910\pi$
$349$	−26.8241	−1.43586	−0.717930	+	0.696116i	$0.754910\pi$
$350$	0	0
$351$	6.00713	0.320637
$352$	0	0
$353$	−7.37529	−0.392547	−0.196274	−	0.980549i	$-0.562884\pi$
$353$	−7.37529	−0.392547	−0.196274	+	0.980549i	$0.562884\pi$
$354$	0	0
$355$	−3.94108	−0.209171
$356$	0	0
$357$	9.00689	0.476695
$358$	0	0
$359$	−19.5656	−1.03263	−0.516317	−	0.856397i	$-0.672697\pi$
$359$	−19.5656	−1.03263	−0.516317	+	0.856397i	$0.672697\pi$
$360$	0	0
$361$	3.09316	0.162798
$362$	0	0
$363$	22.4365	1.17761
$364$	0	0
$365$	10.0093	0.523912
$366$	0	0
$367$	−8.63696	−0.450846	−0.225423	−	0.974261i	$-0.572376\pi$
$367$	−8.63696	−0.450846	−0.225423	+	0.974261i	$0.572376\pi$
$368$	0	0
$369$	9.30858	0.484585
$370$	0	0
$371$	7.61984	0.395602
$372$	0	0
$373$	−13.1338	−0.680042	−0.340021	−	0.940418i	$-0.610434\pi$
$373$	−13.1338	−0.680042	−0.340021	+	0.940418i	$0.610434\pi$
$374$	0	0
$375$	25.6787	1.32604
$376$	0	0
$377$	18.9124	0.974037
$378$	0	0
$379$	2.07360	0.106514	0.0532569	−	0.998581i	$-0.483040\pi$
$379$	2.07360	0.106514	0.0532569	+	0.998581i	$0.483040\pi$
$380$	0	0
$381$	27.1442	1.39064
$382$	0	0
$383$	7.61051	0.388879	0.194439	−	0.980915i	$-0.437711\pi$
$383$	7.61051	0.388879	0.194439	+	0.980915i	$0.437711\pi$
$384$	0	0
$385$	−1.09019	−0.0555611
$386$	0	0
$387$	15.9463	0.810595
$388$	0	0
$389$	−15.6243	−0.792183	−0.396092	−	0.918211i	$-0.629634\pi$
$389$	−15.6243	−0.792183	−0.396092	+	0.918211i	$0.629634\pi$
$390$	0	0
$391$	−4.26501	−0.215691
$392$	0	0
$393$	31.0326	1.56539
$394$	0	0
$395$	−4.38879	−0.220824
$396$	0	0
$397$	4.53335	0.227522	0.113761	−	0.993508i	$-0.463710\pi$
$397$	4.53335	0.227522	0.113761	+	0.993508i	$0.463710\pi$
$398$	0	0
$399$	−9.92623	−0.496933
$400$	0	0
$401$	25.8569	1.29123	0.645616	−	0.763662i	$-0.276601\pi$
$401$	25.8569	1.29123	0.645616	+	0.763662i	$0.276601\pi$
$402$	0	0
$403$	18.8708	0.940020
$404$	0	0
$405$	−20.0064	−0.994126
$406$	0	0
$407$	−1.51809	−0.0752490
$408$	0	0
$409$	−14.3124	−0.707702	−0.353851	−	0.935302i	$-0.615128\pi$
$409$	−14.3124	−0.707702	−0.353851	+	0.935302i	$0.615128\pi$
$410$	0	0
$411$	40.6109	2.00319
$412$	0	0
$413$	2.74172	0.134911
$414$	0	0
$415$	10.3932	0.510182
$416$	0	0
$417$	12.9786	0.635566
$418$	0	0
$419$	−23.4145	−1.14387	−0.571936	−	0.820299i	$-0.693807\pi$
$419$	−23.4145	−1.14387	−0.571936	+	0.820299i	$0.693807\pi$
$420$	0	0
$421$	15.4946	0.755159	0.377580	−	0.925977i	$-0.376756\pi$
$421$	15.4946	0.755159	0.377580	+	0.925977i	$0.376756\pi$
$422$	0	0
$423$	0.795439	0.0386756
$424$	0	0
$425$	−7.83299	−0.379956
$426$	0	0
$427$	−15.1454	−0.732935
$428$	0	0
$429$	−2.39056	−0.115417
$430$	0	0
$431$	34.6016	1.66670	0.833349	−	0.552747i	$-0.186420\pi$
$431$	34.6016	1.66670	0.833349	+	0.552747i	$0.186420\pi$
$432$	0	0
$433$	−18.8051	−0.903713	−0.451857	−	0.892091i	$-0.649238\pi$
$433$	−18.8051	−0.903713	−0.451857	+	0.892091i	$0.649238\pi$
$434$	0	0
$435$	−38.4644	−1.84423
$436$	0	0
$437$	4.70034	0.224848
$438$	0	0
$439$	−37.7000	−1.79933	−0.899663	−	0.436586i	$-0.856187\pi$
$439$	−37.7000	−1.79933	−0.899663	+	0.436586i	$0.856187\pi$
$440$	0	0
$441$	1.45975	0.0695120
$442$	0	0
$443$	−13.6027	−0.646284	−0.323142	−	0.946350i	$-0.604739\pi$
$443$	−13.6027	−0.646284	−0.323142	+	0.946350i	$0.604739\pi$
$444$	0	0
$445$	0.0132766	0.000629373 0
$446$	0	0
$447$	14.0782	0.665877
$448$	0	0
$449$	3.95586	0.186689	0.0933443	−	0.995634i	$-0.470244\pi$
$449$	3.95586	0.186689	0.0933443	+	0.995634i	$0.470244\pi$
$450$	0	0
$451$	3.90865	0.184051
$452$	0	0
$453$	2.26534	0.106435
$454$	0	0
$455$	−3.28473	−0.153991
$456$	0	0
$457$	19.3668	0.905939	0.452969	−	0.891526i	$-0.350365\pi$
$457$	19.3668	0.905939	0.452969	+	0.891526i	$0.350365\pi$
$458$	0	0
$459$	13.8728	0.647529
$460$	0	0
$461$	−27.0772	−1.26111	−0.630554	−	0.776145i	$-0.717172\pi$
$461$	−27.0772	−1.26111	−0.630554	+	0.776145i	$0.717172\pi$
$462$	0	0
$463$	−20.6172	−0.958164	−0.479082	−	0.877770i	$-0.659030\pi$
$463$	−20.6172	−0.958164	−0.479082	+	0.877770i	$0.659030\pi$
$464$	0	0
$465$	−38.3798	−1.77982
$466$	0	0
$467$	6.54603	0.302914	0.151457	−	0.988464i	$-0.451603\pi$
$467$	6.54603	0.302914	0.151457	+	0.988464i	$0.451603\pi$
$468$	0	0
$469$	5.27937	0.243778
$470$	0	0
$471$	5.84263	0.269214
$472$	0	0
$473$	6.69580	0.307873
$474$	0	0
$475$	8.63250	0.396086
$476$	0	0
$477$	−11.1231	−0.509291
$478$	0	0
$479$	−22.8457	−1.04385	−0.521924	−	0.852992i	$-0.674786\pi$
$479$	−22.8457	−1.04385	−0.521924	+	0.852992i	$0.674786\pi$
$480$	0	0
$481$	−4.57400	−0.208557
$482$	0	0
$483$	−2.11181	−0.0960908
$484$	0	0
$485$	−13.4963	−0.612836
$486$	0	0
$487$	−29.3458	−1.32979	−0.664893	−	0.746938i	$-0.731523\pi$
$487$	−29.3458	−1.32979	−0.664893	+	0.746938i	$0.731523\pi$
$488$	0	0
$489$	−12.5513	−0.567587
$490$	0	0
$491$	40.7264	1.83796	0.918978	−	0.394309i	$-0.129016\pi$
$491$	40.7264	1.83796	0.918978	+	0.394309i	$0.129016\pi$
$492$	0	0
$493$	43.6762	1.96708
$494$	0	0
$495$	1.59140	0.0715283
$496$	0	0
$497$	2.21583	0.0993935
$498$	0	0
$499$	−10.9806	−0.491559	−0.245780	−	0.969326i	$-0.579044\pi$
$499$	−10.9806	−0.491559	−0.245780	+	0.969326i	$0.579044\pi$
$500$	0	0
$501$	−9.56183	−0.427191
$502$	0	0
$503$	9.67314	0.431304	0.215652	−	0.976470i	$-0.430812\pi$
$503$	9.67314	0.431304	0.215652	+	0.976470i	$0.430812\pi$
$504$	0	0
$505$	20.6147	0.917341
$506$	0	0
$507$	20.2508	0.899370
$508$	0	0
$509$	15.7619	0.698634	0.349317	−	0.937005i	$-0.386414\pi$
$509$	15.7619	0.698634	0.349317	+	0.937005i	$0.386414\pi$
$510$	0	0
$511$	−5.62764	−0.248952
$512$	0	0
$513$	−15.2888	−0.675019
$514$	0	0
$515$	−1.18530	−0.0522303
$516$	0	0
$517$	0.334003	0.0146894
$518$	0	0
$519$	22.5971	0.991903
$520$	0	0
$521$	−13.4428	−0.588938	−0.294469	−	0.955661i	$-0.595143\pi$
$521$	−13.4428	−0.588938	−0.294469	+	0.955661i	$0.595143\pi$
$522$	0	0
$523$	−33.5415	−1.46667	−0.733334	−	0.679868i	$-0.762037\pi$
$523$	−33.5415	−1.46667	−0.733334	+	0.679868i	$0.762037\pi$
$524$	0	0
$525$	−3.87849	−0.169271
$526$	0	0
$527$	43.5801	1.89838
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.00223	−0.173682
$532$	0	0
$533$	11.7767	0.510108
$534$	0	0
$535$	12.3150	0.532423
$536$	0	0
$537$	−23.9880	−1.03516
$538$	0	0
$539$	0.612946	0.0264015
$540$	0	0
$541$	−22.7411	−0.977718	−0.488859	−	0.872363i	$-0.662587\pi$
$541$	−22.7411	−0.977718	−0.488859	+	0.872363i	$0.662587\pi$
$542$	0	0
$543$	−42.6927	−1.83212
$544$	0	0
$545$	7.22581	0.309520
$546$	0	0
$547$	35.4551	1.51595	0.757975	−	0.652284i	$-0.226189\pi$
$547$	35.4551	1.51595	0.757975	+	0.652284i	$0.226189\pi$
$548$	0	0
$549$	22.1085	0.943567
$550$	0	0
$551$	−48.1342	−2.05059
$552$	0	0
$553$	2.46755	0.104931
$554$	0	0
$555$	9.30271	0.394878
$556$	0	0
$557$	−37.3269	−1.58159	−0.790796	−	0.612080i	$-0.790333\pi$
$557$	−37.3269	−1.58159	−0.790796	+	0.612080i	$0.790333\pi$
$558$	0	0
$559$	20.1744	0.853288
$560$	0	0
$561$	−5.52074	−0.233086
$562$	0	0
$563$	−7.80022	−0.328740	−0.164370	−	0.986399i	$-0.552559\pi$
$563$	−7.80022	−0.328740	−0.164370	+	0.986399i	$0.552559\pi$
$564$	0	0
$565$	−22.3973	−0.942262
$566$	0	0
$567$	11.2484	0.472388
$568$	0	0
$569$	−13.4450	−0.563643	−0.281822	−	0.959467i	$-0.590939\pi$
$569$	−13.4450	−0.563643	−0.281822	+	0.959467i	$0.590939\pi$
$570$	0	0
$571$	−23.6776	−0.990877	−0.495438	−	0.868643i	$-0.664993\pi$
$571$	−23.6776	−0.990877	−0.495438	+	0.868643i	$0.664993\pi$
$572$	0	0
$573$	−4.16141	−0.173845
$574$	0	0
$575$	1.83657	0.0765903
$576$	0	0
$577$	31.6414	1.31725	0.658625	−	0.752471i	$-0.271138\pi$
$577$	31.6414	1.31725	0.658625	+	0.752471i	$0.271138\pi$
$578$	0	0
$579$	−31.2537	−1.29886
$580$	0	0
$581$	−5.84347	−0.242428
$582$	0	0
$583$	−4.67055	−0.193435
$584$	0	0
$585$	4.79490	0.198245
$586$	0	0
$587$	44.5237	1.83769	0.918844	−	0.394621i	$-0.129124\pi$
$587$	44.5237	1.83769	0.918844	+	0.394621i	$0.129124\pi$
$588$	0	0
$589$	−48.0283	−1.97897
$590$	0	0
$591$	−32.4155	−1.33340
$592$	0	0
$593$	−0.774107	−0.0317888	−0.0158944	−	0.999874i	$-0.505060\pi$
$593$	−0.774107	−0.0317888	−0.0158944	+	0.999874i	$0.505060\pi$
$594$	0	0
$595$	−7.58575	−0.310986
$596$	0	0
$597$	−28.9467	−1.18471
$598$	0	0
$599$	22.1838	0.906404	0.453202	−	0.891408i	$-0.350282\pi$
$599$	22.1838	0.906404	0.453202	+	0.891408i	$0.350282\pi$
$600$	0	0
$601$	−19.5006	−0.795445	−0.397722	−	0.917506i	$-0.630199\pi$
$601$	−19.5006	−0.795445	−0.397722	+	0.917506i	$0.630199\pi$
$602$	0	0
$603$	−7.70657	−0.313836
$604$	0	0
$605$	−18.8964	−0.768248
$606$	0	0
$607$	8.99336	0.365029	0.182515	−	0.983203i	$-0.441576\pi$
$607$	8.99336	0.365029	0.182515	+	0.983203i	$0.441576\pi$
$608$	0	0
$609$	21.6262	0.876338
$610$	0	0
$611$	1.00635	0.0407126
$612$	0	0
$613$	−5.21656	−0.210695	−0.105347	−	0.994435i	$-0.533595\pi$
$613$	−5.21656	−0.210695	−0.105347	+	0.994435i	$0.533595\pi$
$614$	0	0
$615$	−23.9518	−0.965831
$616$	0	0
$617$	−21.1908	−0.853111	−0.426555	−	0.904461i	$-0.640273\pi$
$617$	−21.1908	−0.853111	−0.426555	+	0.904461i	$0.640273\pi$
$618$	0	0
$619$	10.6076	0.426355	0.213177	−	0.977014i	$-0.431619\pi$
$619$	10.6076	0.426355	0.213177	+	0.977014i	$0.431619\pi$
$620$	0	0
$621$	−3.25271	−0.130527
$622$	0	0
$623$	−0.00746465	−0.000299065 0
$624$	0	0
$625$	−12.4441	−0.497766
$626$	0	0
$627$	6.08424	0.242981
$628$	0	0
$629$	−10.5632	−0.421182
$630$	0	0
$631$	0.393777	0.0156760	0.00783802	−	0.999969i	$-0.497505\pi$
$631$	0.393777	0.0156760	0.00783802	+	0.999969i	$0.497505\pi$
$632$	0	0
$633$	−46.5393	−1.84977
$634$	0	0
$635$	−22.8613	−0.907224
$636$	0	0
$637$	1.84681	0.0731731
$638$	0	0
$639$	−3.23456	−0.127957
$640$	0	0
$641$	−25.8922	−1.02268	−0.511341	−	0.859378i	$-0.670851\pi$
$641$	−25.8922	−1.02268	−0.511341	+	0.859378i	$0.670851\pi$
$642$	0	0
$643$	−42.5479	−1.67793	−0.838963	−	0.544188i	$-0.816838\pi$
$643$	−42.5479	−1.67793	−0.838963	+	0.544188i	$0.816838\pi$
$644$	0	0
$645$	−41.0312	−1.61560
$646$	0	0
$647$	48.7238	1.91553	0.957765	−	0.287551i	$-0.0928411\pi$
$647$	48.7238	1.91553	0.957765	+	0.287551i	$0.0928411\pi$
$648$	0	0
$649$	−1.68053	−0.0659664
$650$	0	0
$651$	21.5786	0.845733
$652$	0	0
$653$	19.7238	0.771854	0.385927	−	0.922529i	$-0.373882\pi$
$653$	19.7238	0.771854	0.385927	+	0.922529i	$0.373882\pi$
$654$	0	0
$655$	−26.1362	−1.02122
$656$	0	0
$657$	8.21496	0.320496
$658$	0	0
$659$	6.48604	0.252660	0.126330	−	0.991988i	$-0.459680\pi$
$659$	6.48604	0.252660	0.126330	+	0.991988i	$0.459680\pi$
$660$	0	0
$661$	37.1204	1.44382	0.721909	−	0.691988i	$-0.243265\pi$
$661$	37.1204	1.44382	0.721909	+	0.691988i	$0.243265\pi$
$662$	0	0
$663$	−16.6340	−0.646011
$664$	0	0
$665$	8.36003	0.324188
$666$	0	0
$667$	−10.2406	−0.396517
$668$	0	0
$669$	6.36540	0.246101
$670$	0	0
$671$	9.28329	0.358377
$672$	0	0
$673$	39.9763	1.54097	0.770486	−	0.637457i	$-0.220013\pi$
$673$	39.9763	1.54097	0.770486	+	0.637457i	$0.220013\pi$
$674$	0	0
$675$	−5.97384	−0.229933
$676$	0	0
$677$	12.1898	0.468492	0.234246	−	0.972177i	$-0.424738\pi$
$677$	12.1898	0.468492	0.234246	+	0.972177i	$0.424738\pi$
$678$	0	0
$679$	7.58815	0.291207
$680$	0	0
$681$	49.2620	1.88773
$682$	0	0
$683$	22.0880	0.845174	0.422587	−	0.906322i	$-0.361122\pi$
$683$	22.0880	0.845174	0.422587	+	0.906322i	$0.361122\pi$
$684$	0	0
$685$	−34.2032	−1.30684
$686$	0	0
$687$	−58.7318	−2.24076
$688$	0	0
$689$	−14.0724	−0.536115
$690$	0	0
$691$	29.7055	1.13005	0.565025	−	0.825074i	$-0.308866\pi$
$691$	29.7055	1.13005	0.565025	+	0.825074i	$0.308866\pi$
$692$	0	0
$693$	−0.894750	−0.0339887
$694$	0	0
$695$	−10.9308	−0.414629
$696$	0	0
$697$	27.1972	1.03017
$698$	0	0
$699$	46.8822	1.77325
$700$	0	0
$701$	−37.4972	−1.41625	−0.708125	−	0.706087i	$-0.750459\pi$
$701$	−37.4972	−1.41625	−0.708125	+	0.706087i	$0.750459\pi$
$702$	0	0
$703$	11.6414	0.439063
$704$	0	0
$705$	−2.04674	−0.0770846
$706$	0	0
$707$	−11.5904	−0.435901
$708$	0	0
$709$	13.4163	0.503861	0.251931	−	0.967745i	$-0.418935\pi$
$709$	13.4163	0.503861	0.251931	+	0.967745i	$0.418935\pi$
$710$	0	0
$711$	−3.60201	−0.135086
$712$	0	0
$713$	−10.2181	−0.382669
$714$	0	0
$715$	2.01337	0.0752956
$716$	0	0
$717$	−27.9046	−1.04212
$718$	0	0
$719$	−5.14090	−0.191723	−0.0958616	−	0.995395i	$-0.530561\pi$
$719$	−5.14090	−0.191723	−0.0958616	+	0.995395i	$0.530561\pi$
$720$	0	0
$721$	0.666419	0.0248187
$722$	0	0
$723$	−25.2544	−0.939220
$724$	0	0
$725$	−18.8076	−0.698495
$726$	0	0
$727$	33.3224	1.23586	0.617929	−	0.786234i	$-0.287972\pi$
$727$	33.3224	1.23586	0.617929	+	0.786234i	$0.287972\pi$
$728$	0	0
$729$	4.18752	0.155093
$730$	0	0
$731$	46.5908	1.72322
$732$	0	0
$733$	−37.2546	−1.37603	−0.688015	−	0.725696i	$-0.741518\pi$
$733$	−37.2546	−1.37603	−0.688015	+	0.725696i	$0.741518\pi$
$734$	0	0
$735$	−3.75608	−0.138545
$736$	0	0
$737$	−3.23597	−0.119198
$738$	0	0
$739$	−6.52222	−0.239924	−0.119962	−	0.992779i	$-0.538277\pi$
$739$	−6.52222	−0.239924	−0.119962	+	0.992779i	$0.538277\pi$
$740$	0	0
$741$	18.3318	0.673436
$742$	0	0
$743$	−15.6151	−0.572864	−0.286432	−	0.958101i	$-0.592469\pi$
$743$	−15.6151	−0.572864	−0.286432	+	0.958101i	$0.592469\pi$
$744$	0	0
$745$	−11.8569	−0.434404
$746$	0	0
$747$	8.53001	0.312097
$748$	0	0
$749$	−6.92396	−0.252996
$750$	0	0
$751$	−50.7568	−1.85214	−0.926071	−	0.377350i	$-0.876835\pi$
$751$	−50.7568	−1.85214	−0.926071	+	0.377350i	$0.876835\pi$
$752$	0	0
$753$	−31.8472	−1.16058
$754$	0	0
$755$	−1.90790	−0.0694358
$756$	0	0
$757$	−35.5904	−1.29355	−0.646777	−	0.762679i	$-0.723884\pi$
$757$	−35.5904	−1.29355	−0.646777	+	0.762679i	$0.723884\pi$
$758$	0	0
$759$	1.29443	0.0469847
$760$	0	0
$761$	2.59153	0.0939429	0.0469715	−	0.998896i	$-0.485043\pi$
$761$	2.59153	0.0939429	0.0469715	+	0.998896i	$0.485043\pi$
$762$	0	0
$763$	−4.06263	−0.147077
$764$	0	0
$765$	11.0733	0.400357
$766$	0	0
$767$	−5.06342	−0.182830
$768$	0	0
$769$	23.9098	0.862208	0.431104	−	0.902302i	$-0.358124\pi$
$769$	23.9098	0.862208	0.431104	+	0.902302i	$0.358124\pi$
$770$	0	0
$771$	27.2049	0.979760
$772$	0	0
$773$	−12.0909	−0.434879	−0.217440	−	0.976074i	$-0.569771\pi$
$773$	−12.0909	−0.434879	−0.217440	+	0.976074i	$0.569771\pi$
$774$	0	0
$775$	−18.7662	−0.674101
$776$	0	0
$777$	−5.23035	−0.187638
$778$	0	0
$779$	−29.9732	−1.07390
$780$	0	0
$781$	−1.35818	−0.0485996
$782$	0	0
$783$	33.3097	1.19039
$784$	0	0
$785$	−4.92076	−0.175629
$786$	0	0
$787$	10.7979	0.384905	0.192453	−	0.981306i	$-0.438356\pi$
$787$	10.7979	0.384905	0.192453	+	0.981306i	$0.438356\pi$
$788$	0	0
$789$	53.8260	1.91626
$790$	0	0
$791$	12.5926	0.447743
$792$	0	0
$793$	27.9705	0.993263
$794$	0	0
$795$	28.6207	1.01507
$796$	0	0
$797$	−20.5076	−0.726417	−0.363209	−	0.931708i	$-0.618319\pi$
$797$	−20.5076	−0.726417	−0.363209	+	0.931708i	$0.618319\pi$
$798$	0	0
$799$	2.32406	0.0822194
$800$	0	0
$801$	0.0108965	0.000385010 0
$802$	0	0
$803$	3.44944	0.121728
$804$	0	0
$805$	1.77860	0.0626875
$806$	0	0
$807$	31.8401	1.12083
$808$	0	0
$809$	37.4089	1.31523	0.657614	−	0.753355i	$-0.271566\pi$
$809$	37.4089	1.31523	0.657614	+	0.753355i	$0.271566\pi$
$810$	0	0
$811$	19.2121	0.674627	0.337314	−	0.941392i	$-0.390482\pi$
$811$	19.2121	0.674627	0.337314	+	0.941392i	$0.390482\pi$
$812$	0	0
$813$	−21.9658	−0.770375
$814$	0	0
$815$	10.5709	0.370282
$816$	0	0
$817$	−51.3463	−1.79638
$818$	0	0
$819$	−2.69588	−0.0942017
$820$	0	0
$821$	30.3982	1.06091	0.530453	−	0.847714i	$-0.322022\pi$
$821$	30.3982	1.06091	0.530453	+	0.847714i	$0.322022\pi$
$822$	0	0
$823$	−19.8890	−0.693286	−0.346643	−	0.937997i	$-0.612678\pi$
$823$	−19.8890	−0.693286	−0.346643	+	0.937997i	$0.612678\pi$
$824$	0	0
$825$	2.37731	0.0827673
$826$	0	0
$827$	22.2952	0.775279	0.387640	−	0.921811i	$-0.373291\pi$
$827$	22.2952	0.775279	0.387640	+	0.921811i	$0.373291\pi$
$828$	0	0
$829$	25.1144	0.872260	0.436130	−	0.899884i	$-0.356349\pi$
$829$	25.1144	0.872260	0.436130	+	0.899884i	$0.356349\pi$
$830$	0	0
$831$	19.0453	0.660676
$832$	0	0
$833$	4.26501	0.147774
$834$	0	0
$835$	8.05313	0.278690
$836$	0	0
$837$	33.2364	1.14882
$838$	0	0
$839$	14.8696	0.513355	0.256677	−	0.966497i	$-0.417372\pi$
$839$	14.8696	0.513355	0.256677	+	0.966497i	$0.417372\pi$
$840$	0	0
$841$	75.8696	2.61619
$842$	0	0
$843$	−4.04885	−0.139450
$844$	0	0
$845$	−17.0556	−0.586729
$846$	0	0
$847$	10.6243	0.365055
$848$	0	0
$849$	−20.0508	−0.688140
$850$	0	0
$851$	2.47671	0.0849005
$852$	0	0
$853$	−46.0320	−1.57611	−0.788053	−	0.615607i	$-0.788911\pi$
$853$	−46.0320	−1.57611	−0.788053	+	0.615607i	$0.788911\pi$
$854$	0	0
$855$	−12.2036	−0.417353
$856$	0	0
$857$	−23.0239	−0.786480	−0.393240	−	0.919436i	$-0.628646\pi$
$857$	−23.0239	−0.786480	−0.393240	+	0.919436i	$0.628646\pi$
$858$	0	0
$859$	7.46024	0.254540	0.127270	−	0.991868i	$-0.459378\pi$
$859$	7.46024	0.254540	0.127270	+	0.991868i	$0.459378\pi$
$860$	0	0
$861$	13.4666	0.458942
$862$	0	0
$863$	41.0366	1.39690	0.698450	−	0.715659i	$-0.253873\pi$
$863$	41.0366	1.39690	0.698450	+	0.715659i	$0.253873\pi$
$864$	0	0
$865$	−19.0316	−0.647095
$866$	0	0
$867$	−2.51365	−0.0853680
$868$	0	0
$869$	−1.51247	−0.0513072
$870$	0	0
$871$	−9.74996	−0.330365
$872$	0	0
$873$	−11.0768	−0.374894
$874$	0	0
$875$	12.1595	0.411068
$876$	0	0
$877$	50.4646	1.70407	0.852034	−	0.523487i	$-0.175369\pi$
$877$	50.4646	1.70407	0.852034	+	0.523487i	$0.175369\pi$
$878$	0	0
$879$	35.2537	1.18908
$880$	0	0
$881$	39.7525	1.33930	0.669648	−	0.742679i	$-0.266445\pi$
$881$	39.7525	1.33930	0.669648	+	0.742679i	$0.266445\pi$
$882$	0	0
$883$	2.64149	0.0888933	0.0444467	−	0.999012i	$-0.485848\pi$
$883$	2.64149	0.0888933	0.0444467	+	0.999012i	$0.485848\pi$
$884$	0	0
$885$	10.2981	0.346167
$886$	0	0
$887$	−27.0485	−0.908199	−0.454100	−	0.890951i	$-0.650039\pi$
$887$	−27.0485	−0.908199	−0.454100	+	0.890951i	$0.650039\pi$
$888$	0	0
$889$	12.8535	0.431094
$890$	0	0
$891$	−6.89465	−0.230980
$892$	0	0
$893$	−2.56128	−0.0857099
$894$	0	0
$895$	20.2031	0.675314
$896$	0	0
$897$	3.90011	0.130221
$898$	0	0
$899$	104.639	3.48990
$900$	0	0
$901$	−32.4987	−1.08269
$902$	0	0
$903$	23.0694	0.767700
$904$	0	0
$905$	35.9565	1.19524
$906$	0	0
$907$	21.5751	0.716390	0.358195	−	0.933647i	$-0.383392\pi$
$907$	21.5751	0.716390	0.358195	+	0.933647i	$0.383392\pi$
$908$	0	0
$909$	16.9191	0.561171
$910$	0	0
$911$	−47.6282	−1.57799	−0.788996	−	0.614398i	$-0.789399\pi$
$911$	−47.6282	−1.57799	−0.788996	+	0.614398i	$0.789399\pi$
$912$	0	0
$913$	3.58173	0.118538
$914$	0	0
$915$	−56.8871	−1.88063
$916$	0	0
$917$	14.6948	0.485264
$918$	0	0
$919$	−42.5481	−1.40353	−0.701766	−	0.712408i	$-0.747605\pi$
$919$	−42.5481	−1.40353	−0.701766	+	0.712408i	$0.747605\pi$
$920$	0	0
$921$	−54.7560	−1.80427
$922$	0	0
$923$	−4.09221	−0.134697
$924$	0	0
$925$	4.54866	0.149559
$926$	0	0
$927$	−0.972807	−0.0319512
$928$	0	0
$929$	−48.6564	−1.59636	−0.798182	−	0.602416i	$-0.794205\pi$
$929$	−48.6564	−1.59636	−0.798182	+	0.602416i	$0.794205\pi$
$930$	0	0
$931$	−4.70034	−0.154047
$932$	0	0
$933$	53.7627	1.76011
$934$	0	0
$935$	4.64966	0.152060
$936$	0	0
$937$	−6.35816	−0.207712	−0.103856	−	0.994592i	$-0.533118\pi$
$937$	−6.35816	−0.207712	−0.103856	+	0.994592i	$0.533118\pi$
$938$	0	0
$939$	−37.0602	−1.20941
$940$	0	0
$941$	−22.4618	−0.732234	−0.366117	−	0.930569i	$-0.619313\pi$
$941$	−22.4618	−0.732234	−0.366117	+	0.930569i	$0.619313\pi$
$942$	0	0
$943$	−6.37682	−0.207658
$944$	0	0
$945$	−5.78529	−0.188195
$946$	0	0
$947$	−42.2098	−1.37163	−0.685817	−	0.727774i	$-0.740555\pi$
$947$	−42.2098	−1.37163	−0.685817	+	0.727774i	$0.740555\pi$
$948$	0	0
$949$	10.3932	0.337376
$950$	0	0
$951$	9.33948	0.302853
$952$	0	0
$953$	60.6146	1.96350	0.981750	−	0.190177i	$-0.0609061\pi$
$953$	60.6146	1.96350	0.981750	+	0.190177i	$0.0609061\pi$
$954$	0	0
$955$	3.50481	0.113413
$956$	0	0
$957$	−13.2557	−0.428496
$958$	0	0
$959$	19.2303	0.620980
$960$	0	0
$961$	73.4087	2.36802
$962$	0	0
$963$	10.1073	0.325702
$964$	0	0
$965$	26.3224	0.847348
$966$	0	0
$967$	−52.1364	−1.67659	−0.838297	−	0.545213i	$-0.816449\pi$
$967$	−52.1364	−1.67659	−0.838297	+	0.545213i	$0.816449\pi$
$968$	0	0
$969$	42.3354	1.36001
$970$	0	0
$971$	−50.8055	−1.63042	−0.815212	−	0.579162i	$-0.803380\pi$
$971$	−50.8055	−1.63042	−0.815212	+	0.579162i	$0.803380\pi$
$972$	0	0
$973$	6.14573	0.197023
$974$	0	0
$975$	7.16283	0.229394
$976$	0	0
$977$	8.63804	0.276355	0.138178	−	0.990407i	$-0.455875\pi$
$977$	8.63804	0.276355	0.138178	+	0.990407i	$0.455875\pi$
$978$	0	0
$979$	0.00457543	0.000146231 0
$980$	0	0
$981$	5.93044	0.189344
$982$	0	0
$983$	25.0179	0.797946	0.398973	−	0.916963i	$-0.369367\pi$
$983$	25.0179	0.797946	0.398973	+	0.916963i	$0.369367\pi$
$984$	0	0
$985$	27.3009	0.869879
$986$	0	0
$987$	1.15076	0.0366290
$988$	0	0
$989$	−10.9240	−0.347362
$990$	0	0
$991$	11.4716	0.364406	0.182203	−	0.983261i	$-0.441677\pi$
$991$	11.4716	0.364406	0.182203	+	0.983261i	$0.441677\pi$
$992$	0	0
$993$	−67.4387	−2.14010
$994$	0	0
$995$	24.3794	0.772879
$996$	0	0
$997$	−14.9322	−0.472908	−0.236454	−	0.971643i	$-0.575985\pi$
$997$	−14.9322	−0.472908	−0.236454	+	0.971643i	$0.575985\pi$
$998$	0	0
$999$	−8.05603	−0.254882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	644.2.a.d.1.1	✓	5
3.2	odd	2		5796.2.a.t.1.3		5
4.3	odd	2		2576.2.a.bb.1.5		5
7.6	odd	2		4508.2.a.f.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.d.1.1	✓	5	1.1	even	1	trivial
2576.2.a.bb.1.5		5	4.3	odd	2
4508.2.a.f.1.5		5	7.6	odd	2
5796.2.a.t.1.3		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +O(q^{10})\)	\(q-2.11181 q^{3} +1.77860 q^{5} -1.00000 q^{7} +1.45975 q^{9} +0.612946 q^{11} +1.84681 q^{13} -3.75608 q^{15} +4.26501 q^{17} -4.70034 q^{19} +2.11181 q^{21} -1.00000 q^{23} -1.83657 q^{25} +3.25271 q^{27} +10.2406 q^{29} +10.2181 q^{31} -1.29443 q^{33} -1.77860 q^{35} -2.47671 q^{37} -3.90011 q^{39} +6.37682 q^{41} +10.9240 q^{43} +2.59632 q^{45} +0.544914 q^{47} +1.00000 q^{49} -9.00689 q^{51} -7.61984 q^{53} +1.09019 q^{55} +9.92623 q^{57} -2.74172 q^{59} +15.1454 q^{61} -1.45975 q^{63} +3.28473 q^{65} -5.27937 q^{67} +2.11181 q^{69} -2.21583 q^{71} +5.62764 q^{73} +3.87849 q^{75} -0.612946 q^{77} -2.46755 q^{79} -11.2484 q^{81} +5.84347 q^{83} +7.58575 q^{85} -21.6262 q^{87} +0.00746465 q^{89} -1.84681 q^{91} -21.5786 q^{93} -8.36003 q^{95} -7.58815 q^{97} +0.894750 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9	\( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 + 2 * q^11 + 13 * q^13 + 4 * q^15 + 4 * q^17 + 12 * q^19 - 3 * q^21 - 5 * q^23 + 19 * q^25 + 15 * q^27 + 13 * q^29 - 3 * q^31 + 24 * q^33 - 2 * q^35 - 4 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - 16 * q^45 + 5 * q^47 + 5 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 12 * q^59 + 20 * q^61 - 10 * q^63 - 12 * q^65 - 12 * q^67 - 3 * q^69 + 9 * q^71 - 9 * q^73 + 35 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 - 28 * q^83 + 16 * q^85 - 15 * q^87 + 32 * q^89 - 13 * q^91 - 15 * q^93 - 36 * q^95 + 4 * q^97 + 28 * q^99

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